Professor Quibb

Infinity: Ordinal Numbers

2012-02-18T06:45:00.002-05:00

This is the sixth post in the Infinity Series, the first of which is found here. For all available posts, see the Infinity Series Portal.

In order to further explore the concept of infinity, one must temporarily move away from the concept of the cardinal number, and consider yet another type of number: the ordinal. The system of ordinal numbers is again entwined in set theory, and, unlike cardinal numbers, they can be expressed as sets. It is also important to know that any cardinal number can alternatively describe a family of sets, i.e. all of the sets that have that cardinality. For example, the cardinal number 3 not only describes a number of members that a set can have, but also represents the family of all sets with three members.

Formally, ordinal numbers are very similar to cardinal numbers in that, in finite situations, they represent properties of classes of sets. The property that an ordinal number represents is known as order type. However, the class of sets whose order type is described by an ordinal number much more specific than that of a cardinal number. To understand the type of set that has an order type can represent, one must first understand the concept of a well-ordered set.

A well-ordered set is a set that can be well-ordered by a relation. A relation, in this case, is just a quantity that connects pairs of elements. A good example is the relation "<" or "less than". To understand well-ordering, observe the example below.

Consider the set S = {2,5,7}, and the relation R = <
If "<" is restricted to the set S, it simply means that the two elements of each ordered pair of the relation must come from S, i.e. chosen from 2, 5, or 7. Of the six possible ordered pairs, (2,5), (2,7), (5,7), (5,2), (7,2), and (7,5), three are members of the relation <, namely (2,5), (2,7), and (5,7). This is because 2<5, 2<7, and 5<7. Obviously, the remaining pairs are not, as 5 is not less than 2, and so on.

In this case, "<" well-orders S, meaning that each (non-empty) subset of S has a "minimal" element. For the relation "<", the "minimal" element of a subset is simply its smallest member. Since it is easy to find the smallest number of any subset of S, it is a well-ordered set.

In contrast, consider the set of all whole numbers N = {1,2,3...}. The above relation "<" does well-order the set, because, for any subset, one can state the smallest element. However, this does not hold with the operation "greater than" or ">". For example, if the subset N' is the set of natural numbers greater than 3, i.e. {4,5,6...}, then there is no "minimal" element under the relation ">", as there is no element in the set that is greater than all the others. However, overall, the set is still "well-ordered", because there is at least one relation that does satisfy the needed conditions.

Now, returning to the class of sets a certain order type, consider all the well-ordered sets that can be put into one-to-one correspondence (i.e. all the well-ordered sets with a given cardinality). However, this correspondence is of a rather special type, matching up the "minimal" elements with each other, followed by the "next-to-minimal" elements, and so on. Every class of sets for which all the members are connected is an equivalence class, and all sets in this class have an identical order type, which is a certain ordinal number.

For instance, the above set S = {2,5,7} under "<", and the set T = {b,c,d} under the relation that will be defined as "alphabet" (which puts alphabet in order), can be related by a correspondence. The minimal element of set S under "<" is clearly 2, and the minimum of T under "alphabet" is b, with b being the letter in the set closest to the beginning of the alphabet. Since the other elements can be paired in a similar way, these two sets are therefore members of an equivalence class and are of the same order type (intuitively, they are this order type because they each have three elements).

However, for each equivalence class, it is convenient to chose one, and only one set to be representative of the entire class, and thus, the ordinal. The first such set is the empty set, Ø, having no elements and being the only member of its equivalence class. We identify this set with the ordinal number "0". Therefore,

0 = Ø = {}

Hence forth, the representative set of each equivalence class is defined as the set containing all of the previous representative sets. For example, the successor of 0 is denoted "1", or the second ordinal number, and is defined as

1 = {Ø}

Note that this set is not the empty set, but rather the set containing only the empty set. It is obvious that this set can be well-ordered, as, trivially, there is only one element to order. It is equally clear that this set is representative of the class of sets with a single element and that it can be put in a one-to-one correspondence with each set in that class. Continuing the pattern of each representative set containing all of those previously, we have

2 = {Ø, {Ø}},
3 = {Ø, {Ø}, {Ø, {Ø}}},
and so on. Note that "2", has 0 and 1 as members, and "3" has 0, 1, and 2 as members.

Henceforth, "ordinal number" will refer to a set of this type, and 0, 1, 2, 3,... will refer to ordinal numbers unless otherwise specified. These ordinal numbers are well-ordered by the membership relation, or the relation set up between two sets with the second containing the first. In other words, each element of an ordinal number set is a member of all subsequent elements in the set. For example,

4 = {Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}}}

In this set, each member is a member of each subsequent member, namely, Ø, being the first element of 4, is a member of {Ø}, {Ø, {Ø}}, and {Ø}, {Ø, {Ø}}}, which, of course, are just the ordinal numbers 1, 2, and 3. The minimal element for each ordinal number is Ø.

It is clear that one can go on to define any natural number as a set of this type, namely as the set of all previous ordinals. As with cardinal numbers, the first infinite ordinal number corresponds to the set of all natural numbers. This ordinal is denoted ω, and has every (ordinal) natural number as a member. It is clear that this set can be put in a one-to-one correspondence with the set of "regular" natural numbers, and it therefore has a cardinality of aleph-zero. However, the world of infinite ordinals is very different than that of infinite cardinals.

One crucial difference is the idea of succession in ordinals. From any ordinal α (greek letters are often used for ordinal variables), there is a successor to α, denoted α', which is defined as the next ordinal after α. Formally, α' is the smallest ordinal which has α as a member. Furthermore, α' can be constructed using the formula

α' = α ∪ {α}

where ∪ is the symbol for a union of sets. The concept of union entails combining all the members of two sets into one, and deleting any duplicates. For instance,

{1,4,6} ∪ {3,6,13} = {1,4,6,3,6,13} => {1,4,6,3,13}

Returning to the succession formula above, note the distinction between α and {α}. The first is an ordinal, but the second is not, being rather a set containing a single ordinal, α, as its only member. To more easily visualize this process, consider the simplest example:

0 = Ø
0' = Ø ∪ {Ø}

Now, the union of the empty set with any other set is simply the other set, as the empty set contributes no members to the result. Therefore,

0' = Ø ∪ {Ø} = {Ø},

the final term of which is, in fact, the ordinal 1! Though ordinal addition has not been defined, we can observe that the succession operation, in a more general case, identifies with the process of adding 1 (0+1=1 and 0'=1, etc.). The role that succession plays in infinite ordinals, as well as in operations on ordinals, is considered in the next post, coming February 26.

Sources: Axiomatic Set Theory by Patrick Suppes

Infinity: Uncountable Sets

2012-02-10T10:46:00.008-05:00

This is the fifth post of the Infinity Series, beginning here. For all available posts, see the Infinity Series Portal.

In previous posts, the idea of an "uncountable set" has arisen, namely a set that has a cardinality greater than that of the natural numbers. We have already revealed that the sets of real numbers and complex numbers are uncountable, specifically having a cardinality that is two to the power of aleph-zero, the cardinality of the natural numbers. However, there are other sets which have this cardinality, as well as sets with one greater than this.

First, sets with the cardinality of the continuum will be listed. The sets of real and complex numbers fit into this category, along with any interval of either of them. For example, the set of real numbers between 1 and 5 (inclusive or exclusive) will have the cardinality of the continuum.

A somewhat more interesting case is the Cantor set, a famous entity in mathematics. It is obtained geometrically by beginning with a solid line segment, (mathematically, this segment is the real numbers on [0,1]) and on each step removing the middle third of every existing line segment. This process is illustrated below.

The first seven steps in producing the Cantor Set. This process is repeated infinitely many times. It appears that the entire original line segment will eventually be removed after an infinite number of the above steps, and this is, in a way, true. During the first step 1/3 of the total bar is removed, followed by 1/9 of the total for each remaining segment on the second step, then 1/27 of the total for each remaining part on the third step, and so on. The sum of these removals is 1/3+2/9+4/27+... since there are 2^n segments on the nth step (the amount removed on second step equals 2*1/9=2/9, and on the third, 4*1/27=4/27). The above sequence sums, in the limit as n increases without bound, to 1. In other words, 100% of the original segment is taken away as the Cantor Set is constructed.

Despite the apparent paradox, it is clear that for any segment remaining at any step of the above process, it will have two endpoints that are untouched even after an infinite number of steps. This is because only the middle of all such segments are removed. The remaining points, which all become endpoints at some step, are the points of the Cantor Set. For example, the points 0 and 1 are never removed, along with 1/3, 2/3, 1/9, 2/9, 7/9, 8/9 and many others. The cardinality of this set can be discovered by expressing all its points in base-3 (ternary) decimal notation. In it, 1/3=.1=.02222..., 2/3=.2=.12222..., 1=.22222..., 7/9=.20222..., etc. Many of these numbers have two forms, i.e. 1/3=.1=.0222..., but it can be shown that the Cantor Set contains only points whose base-3 decimal representation has at least one form (out of the two) that consists of only 2's and 0's (a more full discussion is found here). If one took all of these decimals and changed all of the 2's into 1's, a set of binary sequences would be produced. The Cantor set has all base-3 points consisting of only 2's and 0's, but after the function, which is clearly a bijection, it is now the set of all binary sequences, which has previously been shown to have the cardinality of the continuum. The Cantor Set does as well.

Mathematically, this is remarkable. Even though the Cantor Set is an infinitesimal fraction of the line segment [0,1], it still has the cardinality of the continuum. Since no two points of the final Cantor Set are "adjacent" to each other, the Cantor Set is defined to have the property of being nowhere dense. Simply put, this means that any interval containing two points of the Cantor Set does not necessarily contain a third. In contrast, there are an infinite number of rational numbers between any two one could chose, and the set of rational numbers is therefore dense. The fact that such a "nowhere dense" set can have the cardinality of the continuum is also incredible, because all of the discrete sets (natural numbers, integers, etc.) that we have examined have not had this cardinality. These are clearly nowhere dense, as there are real numbers between any two natural numbers, or any two integers.

Another very interesting case is the cardinality of the set of continuous functions on the real numbers. To find this cardinality, it is useful to use a proof of trichotomous exclusion. Such a proof confirms that a set must have a cardinality greater than or equal to that of a certain set, and then proves that it must simultaneously satisfy the condition of having a cardinality less than or equal to that of the same set. Since one quantity cannot be less than and greater than another, the cardinality of the two sets must be equal.

To go about the above proof, it is useful to reevaluate the definition of a continuous function. The "local" definition is of most use here. It states that for any value x on a continuous function, there exists a sufficiently small ε such that f(x+ε) approximates f(x) to an arbitrary accuracy σ. In addition, as ε goes to 0, σ will as well, as point Q becomes a better and better approximation to point P. This is all summarized in the figure below.

Compare the above to the discontinuous function below, where the open circle on the curve represents a discontinuity, the true function value f(x) being located at point P. Even with ε miniscule, the approximation to f(x) never reaches the desired accuracy of σ.

Having defined what it means to be continuous, it is now possible to find the cardinality of the set of continuous functions. The key concept is that knowing the value of a continuous function on all rational numbers is enough to determine it uniquely, i.e. to find its value at any real number.

To see this, consider the value x from above to be a certain real number (for our purpose, make it irrational). This number can be approximated to an arbitrarily high accuracy with a rational number, e.g. π≈3.14159=314159/10000, etc. and therefore ε can be made as small as desired. In the limit as ε goes to 0, the approximation to f(x) becomes exact, and the function value is uniquely determined. Therefore, the knowledge of the value of a continuous function over the rational numbers is enough to determine it.

The main idea is that the above shows that the any specific continuous function needs no more than a countable set of real numbers to define it. The cardinality of the set of all continuous functions is therefore either equal to or less than that of the set of the countable sets of real numbers, or the power set of real numbers, (a power set is the set of all combinations of elements of a given set, discussed in an earlier post) whose cardinality is 2 to the power of the cardinality of all countable sets: aleph-zero. The set of continuous functions therefore has no more than the cardinality of the continuum.

To confirm that this is the cardinality, we must still prove that the set of continuous functions is uncountable. There is a very simple way of doing this. Take a subset of the continuous functions, the constant functions, for which f(x)=k, k being a real number. Clearly, if one defines a set of all constant functions, including every possibility of k, the set will be equivalent in cardinality to the real numbers: the cardinality of the continuum. However, the constant functions are a subset of all continuous functions, the true cardinality must be equal to or greater than this. Since we have already confirmed that it is no greater, and now it is known that the cardinality of the set is no less than that of the real numbers, it follows that the two must be equal.

In contrast, discontinuous functions cannot be pinned down without their value at each and every real number. For example, the discontinuous (piecewise) function that defines f(x) to be equal to 1 for all x except π, and at π for the value to be 12, no degree of accuracy in the rational numbers can confirm that f(π) will equal twelve. The best approximations will simply yield f(π)=1, as they "expect" the function to be continuous at π. More generally, some discontinuous function could potentially have a jump at any or every real number. Therefore, it takes a set of real numbers to uniquely determine such a function.

The total set of functions (including both continuous and discontinuous) on the real numbers is then the power set of the real numbers, with a cardinality greater than any yet discussed. It is 2 to the power of the cardinality of the continuum, which is also known as beth-two, the symbol of which is illustrated below.

This is the third member of what is called the beth sequence. Beth-zero is equal to aleph-zero, and each subsequent element of the sequence is defined recursively as 2 to the power of the previous one. Beth-one is equal to 2 to the power of aleph-zero, or the cardinality of the continuum, and so on. Equivalently, each element is the cardinality of the power set of the previous element. The distinctions between the aleph series and the beth series are revealed in a later post. The next post is on a new type of number, and is coming February 18.

Sources: http://en.wikipedia.org/wiki/Cantor_set, etc.

Infinity: Operations on Cardinals

2012-02-02T07:52:00.005-05:00

Before reading this post, make sure you have read the first three parts of the Infinity Series, the first of which is found here. For all available posts, see the Infinity Series Portal.

Having found the mathematical relationship between aleph-zero and the cardinality of the continuum, one wonders if it is possible to perform other operations with infinity cardinals, and whether these equations create any numbers not yet discussed. To start, take a simple addition from cardinal arithmetic:

(1)

What exactly does this equation mean? We are asked to "add" two quantities, one of which is an infinite cardinal, and one is a simple number. That it is possible to evaluate the sum (1) follows from the fact that aleph-zero and 1 are both cardinal numbers. Since they are of the same number system, they are "compatible" in a way, and can be combined by the use of sets. We have already proved (1) in the first post of the series, but let us recap. It has been discussed that adding the element 0 to the set of natural numbers does not change its cardinality, since the function y=x-1 from one set to the other is a bijection. In set notation, with vertical lines representing cardinality: |{0}|+|{1,2,3...}|=|{0,1,2,3...}| Note that if one replaces the values in this equation with the actual cardinalities, one obtains the equation (1) above! The equation in simple sets that has the same meaning as a cardinal equation will henceforth be known as the Corresponding Set Equality (CSE).

The generalization of (1) for any natural number n is

(2)

This can also be transformed into a CSE, since any subset of integers also has cardinality aleph-zero. The the set {-n+1,-n+2,...-2,-1,0} clearly has n elements, and can be added to the natural numbers to form the CSE, namely: |{-n+1,-n+2,...-2,-1,0}|+|{1,2,3...}|=|{-n+1,-n+2,...-2,-1,0,1,2,3...}|. By evaluating the cardinalities on both sides, one obtains equation (2). It is easy to continue on to other arithmetic operations, such as multiplication:

(3)

This particular equation also has a CSE. First consider the set of integers. It is clear that it can be split into two components, both of which have a cardinality of aleph-zero, namely the whole numbers: {0,1,2,3...} and the negative integers: {...-3,-2,-1} However, when these two sets are combined, the set of integers results, which we already know has cardinality aleph-zero as well. The CSE here is |{0,1,2,3...}|+|{-1,-2,-3...}|=|{...-3,-2,-1,0,1,2,3...}|. All three of the cardinalities are evaluated as aleph-zero, and (3) results. Now we shall prove the general multiplication result

(4)

for any natural number n. We have already determined that the rational numbers, and any infinite subset of them have cardinality aleph-zero. Consider the set {a/n,1+(a/n),2+(a/n),...}. This is simply the set of natural numbers with the rational number a/n added to each term. For a constant n, one could consider creating a set for each integral value of a from 0, to n-1, inclusive. This produces n sets of cardinality aleph-zero.

An example will make this more clear. Consider the case with n=3. For a=0, the set produced is simply the whole numbers ({0/3,1+(0/3),2+(0/3)...}={0,1,2...}) For a=1, the set is {1/3,1+(1/3),2+(1/3)...} and for a=2, the set is {2/3,1+(2/3),2+(2/3)...}, all of which have cardinality aleph-zero. The sum of these sets is {0,1/3,2/3,1,1+(1/3),1+(2/3),2...}, or the set of all multiples of 1/3, which also has the same cardinality. We have just proved (4) for n=3. The more general CSE for (4) is |{0/n,1+(0/n),2+(0/n)...}|+|{1/n,1+(1/n),2+(1/n)...}|+
|{2/n,1+(2/n),2+(2/n)...}|+...+|{(n-1)/n,1+((n-1)/n),2+((n-1)/n)...}|=
|{0,1/n,2/n...,1,1+(1/n),1+(2/n),...}|, the final set being the set of multiples of 1/n.

This equation is tedious, but it simply is the division of the set of multiples of 1/n into n parts, all of which have cardinality of aleph-zero. Since the set of multiples on the right hand side of the equation as an equivalent cardinality, this proves (4). Moving on, even the multiplication of two infinite quantities is possible.

(5)

The CSE for this equation follows from the countability of the set of ordered pairs. The set of ordered pairs (x,y) with natural numbers x and y can be split into components by setting a value for x, for example 1, and letting y vary among the natural numbers. For each constant value of x, a set of cardinality aleph-zero is generated, and since there are aleph-zero choices for x, the above result (5) follows. In CSE form, |{(1,1),(1,2),(1,3)...}|+|{(2,1),(2,2),(2,3)...}|+...=[the cardinality of the set of ordered pairs]. Each quantity in the equality has value aleph-zero when evaluated, and since there are as many members on the left side, it follows that the cardinality of the natural numbers, when multiplied by itself, yields the same quantity. This result can too be generalized to any positive integer power:

(6)

Since the case n=2 makes use of ordered pairs, it is natural to assume that higher powers will involve the corresponding ordered n-tuplet. This is correct. There are aleph-zero possibilities for each element of an integral ordered n-tuplet, and each choice of element contributes an aleph-zero to the product. The end result is aleph-zero to the nth power, but since it has already been said that the cardinalities of the sets of ordered n-tuplets for finite n are all aleph-zero, the equality (6) is a direct result.

Summarizing the above, no additions, multiplications, or nth powers, when applied to aleph-zero, change its value. However, it has already been shown that two taken to the power of aleph-zero produces a different infinite cardinal, namely the cardinality of the continuum. But what about a general natural number n taken to the same power, or even aleph-zero taken to the power of itself?

(7)

Remarkably, we find that this quantity is equal to the cardinality of the continuum! This can be derived intuitively from the result (6). Since aleph-zero to the power of n is equal to the cardinality of the set of the ordered n-tuplets, one obtains (7) by letting n increase without bound to aleph-zero, at which point one obtains the cardinality of the set of infinite sequences, which has previously been shown to be greater than aleph-zero, and named the cardinality of the continuum. Any other n taken to the aleph-zero power is also equal to the cardinality of the continuum, as such quantities would clearly be greater than 2 to that power and less than the left side of (7). Since these both have the same value, those of the general case do as well. This is all summarized below.

The next post explores uncountable sets, namely stating what other sets besides the real or complex numbers have a cardinality equal to, or even greater than, the cardinality of the continuum, coming February 10.

Sources: http://en.wikipedia.org/wiki/Power_set

Infinity: The Cardinality of the Continuum

2012-01-25T06:26:00.011-05:00

Before reading this post, make sure you have read Infinity: The First Transfinite Cardinal, and Infinity: Countable Sets.

In the previous posts of this series, it was established that the sets of natural numbers, integers, rational numbers, and even algebraic numbers have an equivalent cardinality: aleph-zero. However, not all real numbers fall under the umbrella of algebraic numbers. All of the numbers that are real but non-algebraic are irrational, and are specifically known as transcendental. Numbers such as e and π are transcendental.

To determine the cardinality of the real numbers, this problem can be again simplified to a problem involving ordered n-tuplets. This is done by considering the construction of an arbitrary real number. The general real number has a finite whole number part, followed by an infinite decimal expansion. For example, the real number π has a whole number part of 3, and a decimal expansion of .14159265... It is simpler to just ignore the whole number part, and focus on the real numbers on the interval (0,1). All of these are defined uniquely (almost, as we will see below) by their infinite decimal expansion. Therefore, each of these numbers is defined by an ordered n-tuplet, with n being infinite, and of the form

(a1,a2,a3...)

Since all values in this sequence are place values, each must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. However, no clarity is lost if these real numbers are converted to binary, and each number still as a unique infinite decimal expansion, this time only incorporating 1's and 0's. We have the just simplified the problem to determining whether the set of all infinite sequences consisting of 1's and 0's is countable, i.e. whether it has a cardinality of aleph-zero.

Georg Cantor was the first to devise this method, and through infinite binary sequences found a very elegant way to find the cardinality of the real numbers, through proof by contradiction. It is called the diagonal argument. To understand this argument, consider all possible infinite binary sequences as making up a set, called S. Each element Sn is then an infinite binary sequence. If the cardinality of the real numbers is aleph-zero, then each element in the set can be numbered Sn, with n being a natural number. The first few elements of the set are shown below:

The actual ordering of this set is arbitrary, since if the cardinality of S is aleph-zero, all sequences with be covered eventually. For the next part of the proof, consider a sequence Sx, which is constructed by taking the nth element of each Sn and reversing it, i.e. 0 becomes 1, and 1 becomes 0. This is illustrated below:

The nth element of every nth sequence is bolded, and for each bolded element, the opposite one is placed in the sequence Sx. The resulting sequence is clearly an infinite binary sequence, and therefore is a member of the set S. However, by definition, it is different from any sequence in the set (S1,S2,S3...) because for any sequence Sn, with n a natural number, the nth element of the sequence is different from that of Sx. Therefore, assigning a natural number to each infinite binary sequence does not cover all such sequences, and the function from the natural numbers to S is not a bijection. Therefore, the set S has a cardinality greater then aleph-zero. Sets with cardinalities greater than aleph-zero are also known as uncountable.

When extending this back to our real number problem, there are a few slight glitches in this system, one of which being that an infinite binary decimal expansion such as .001111... with infinite 1's is actually equal to another, namely .010000... Therefore, each real number does not quite have a unique decimal expansion. However, this problem can be resolved.

Only numbers with terminating decimal expansions can be expressed in the two ways shown above, and in binary, the only such numbers are those whose denominators involve only a power of 2. For example, 1/2=.1000...=.0111..., and 5/8=.101000...=.100111... These numbers can be collected into a set of their own, called A, the first few members of which are {1/2, 1/4, 3/4, 1/8, 3/8,...}.

This is a subset of the rational numbers, and therefore has cardinality aleph-zero. The set of infinite binary sequences with infinite 0's or infinite 1's (which starts {.1000...,.0111...,.01000...,.00111...,...}) merely has two elements for each element of set A, and still has a cardinality of aleph-zero. (just as the sets {1,2,3...} and {1,-1,2,-2...} have the same cardinality, even though there are two elements in the latter whose absolute values correspond to the former) Because of this, a bijection can be set up between them, corresponding .1000... to 1/2, .0111... to 1/4, and so on. Adding this to the original set S, one finds that each real number on (0,1) now corresponds to a single unique infinite binary sequence.

Clearly, the cardinality of the entire set of real numbers must be greater than or equal to that of the real numbers on (0,1), and so we have now proved that

The cardinality of the real numbers, known as the cardinality of the continuum and denoted by , is strictly greater then the cardinality of the natural numbers, aleph-zero. In other words

> ℵ₀

More number systems still remain, including the complex numbers. However, it is fairly easy to see that the complex numbers still have the cardinality of the continuum, as any complex number can be defined uniquely as an ordered pair of two real numbers (a,b). Also, through a similar method that was used for integral ordered n-tuplets that is not detailed here, it can be proved that sets of ordered n-tuplets of real numbers or of complex numbers both have the cardinality of the continuum. This result also seems intuitively correct from previous examinations of ordered n-tuplets.

Using Cantor's diagonal argument, it was established that the cardinality of the set of real numbers was greater than that of natural numbers, in other words that there is no bijection between them. However, it has not been found exactly what this cardinality of the continuum actually is, or how to relate these quantities to each other in any way.

In order to accomplish this, one must first understand the concept of a power set. A power set is denoted P(S), where S is any ordinary set. Every set has a corresponding power set, and the above statement says that the power set of S is called P(S).

To construct the power set for any set S, take each unique combination of the elements in S, and put it into its own set. Then compile all of these sets and place them within another set. This is the power set, P(S). For example, the set {1,2,3} includes eight unique combinations of the elements contained within it, namely: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3}. (the first of these is included as a choice of zero elements from the set) All of these are then included in a larger set, yielding {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. In conclusion:

P({1,2,3})={{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

If one compares the cardinality of these two sets, where each subset of the latter is a single element, it is easy to see that they are 3 and 8, respectively. One also notices that 2^3=8. This is no coincidence. The total number of combinations of n elements is (2^n)-1, and when one adds the empty set to this total, 2^n. Expressed in formula form, with the cardinality of a set S written as |S|:

Applying this knowledge to what we know about the set of natural numbers, it is easy to identify the set of all real numbers as the set of combinations of natural numbers, using the technique of binary sequences that was used previously. Combinations of natural numbers can be interpreted at the decimal expansion of real numbers through binary sets. For example, one can draw the combination {1,4,7} from the set of natural numbers {1,2,3...} and take it to mean .1100111000... which is a decimal sequence in which the binary expansions of 1 4, and 7 are joined, and then followed by zeros. There are also infinite subsets of the natural numbers, such as the even numbers {2,4,6...}, which will provide an infinite decimal expansion. This is not a precise mathematical way of doing this, but it serves as an intuitive glimpse into the cardinality of the continuum.

One can also choose the first element of the subset to represent the whole number part, which is always a finite natural number for all positive reals. Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers. This proves that

In other words, the cardinality of the continuum is two to the power of aleph-zero. But how can one even comprehend arithmetic operations with cardinal numbers, and infinite ones at that? What other properties does aleph-zero have? The answers can be defined through sets, and are discussed in the next post.

Sources: http://en.wikipedia.org/wiki/Cardinality_of_the_continuum, http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Infinity: Countable Sets

2012-01-17T02:47:00.002-05:00

Before reading this post, read Infinity: The First Transfinite Cardinal.

At the end of the previous post, it was stated that any infinite set or subset of a number system defined by integral ordered n-tuplets, where n is a natural number, has a cardinality of aleph-zero. This statement is more easily understood with examples.

We have already seen that natural numbers can be put in one-to-one correspondence with any set of n-tuplets that contains only integers. When n=1, the resulting number system is simply the integers themselves: {...-3,-2,-1,0,1,2,3...} Furthermore, the above guarantees that any infinite subset of integers will have equivalent cardinality. The even numbers {2,4,6...}, multiples of 10, {10,20,30...}, and even the powers of 2 {1,2,4,8,16...} all can be accessed from the natural numbers through a simple bijection (in this case the functions y=2x, y=10x, and y=2^x, respectively) and therefore have the same cardinality, aleph-zero.

However, the above theorem states that a number system defined by integral ordered n-tuplets for any finite n are also allowed. First, consider the ordered pairs. How does one define a number system with the general integral pair (a,b)? One way, is to view these numbers as the solutions to polynomial equations with coefficients a and b, namely

ax+b=0

Solving for x, one obtains x=-b/a. The general solutions of these equations are the rational numbers, any numbers that can be formed by the ratio of two integers. Admittedly, using the spiral method to pair each natural number with an ordered pair does not hold up well when converted into rational numbers. For instance, the ordered pairs (-2,1) and (-4,2) both produce the same solution, namely 1/2, and pairs such as (0,1) are not defined at all! Therefore, the function from natural numbers to rational numbers through the spiral method is not a bijection.

These problems can be resolved, however. One way is to simply discard duplicates and undefined ordered pairs. The natural numbers corresponding to unique rational numbers will henceforth be known as unique ordered pair numbers, or UOPN's for short. The first few are

2 -> (1,0) -> 0
3 -> (1,1) -> -1
5 -> (-1,1) -> 1
10 -> (2,-1) -> 1/2
12 -> (2,1) -> -1/2
14 -> (1,2) -> -2

In the above expression, the first number of each row is a natural number, followed by the corresponding ordered pair defined by the spiral rule, and the final number is the rational number that results using the polynomial method on the ordered pair. Since it is clear that there are an infinite number of rational numbers accessed by the above series, one can set up a bijection pairing each natural number n to the nth UOPN. This would change the set {1,2,3,4...} into {2,3,5,10...}. The UOPN's then have the same cardinality as the natural numbers, and therefore the rational numbers do as well.

The result just established is remarkable. Despite there being infinite rational numbers between the natural numbers 0 and 1 alone, the cardinality of both of these sets are identical. But this still isn't the end of it. The theorem also deals with numbers defined by ordered n-tuplets. Continuing the theme of using ordered n-tuplets to define integral polynomial equations, the solutions of the resulting equation give the value of the ordered n-tuplet. For example, the ordered quadruplet (1,0,0,-2) corresponds to

x^3+0x^2+0x-2=x^3-2=0,

the real solution of which is the cube root of two. However, with higher degree polynomials, there may be multiple solutions. Consider the polynomial graph below.

This is the graph of x^3-2x^2-x+1, the ordered quadruplet for which would be (1,-2,-1,1). In this case, there are three solutions to the polynomial equation x^3-2x^2-x+1=0 on the real number line, i.e. the three intersections of the graph with the x-axis. How then does one avoid ambiguity? Which of the three solutions does (1,-2,-1,1) represent? The solution lies in specifying an interval on which the solution is found. For example, the first solution, to the left of the origin, lies on the interval [-1,0], and with this constraint, the unique solution can be specified.

Generally, given a ordered n-tuplet of the form (A1,A2,A3...,An-4,B1,B2,C1,C2), one specifies the corresponding number to be a zero of the polynomial

(A1)x^(n-4)+(A2)x^(n-3)+...(An-3)x+An-4
on the interval whose endpoints are the rational numbers defined by the ordered pairs (B1,B2) and (C1,C2).

It is now clear that any solution of any polynomial equation of integral coefficients has an accompanying natural number. Since these solutions can be set up in a one-to-one correspondence with some subset of the natural numbers, one can be set up with the natural numbers themselves. As before, this implies an identical cardinality, i.e. the set of the solutions to integral polynomial equations has a cardinality of aleph-zero.

Just what are these numbers? We know that they are the general solutions of integral polynomial equations, but what form do they take? This specific set of numbers are called the algebraic numbers, and they include square roots, cube roots, and for that matter any nth root. They also include more complicated sets of nested roots, such as numbers of the form

for integer a, b, and c. Any number with nested roots such as this is algebraic. Specifically, algebraic numbers encompass all rational numbers along with many irrational numbers, but not all real numbers are algebraic. For example, π and e are not algebraic, and cannot be expressed as the solutions of polynomials of any finite degree.

All of the above sets have a cardinality the same as that of the natural numbers, and they are therefore denoted countable sets, named after the ability to count natural numbers. All of the rational numbers, and even some irrational numbers are countable, but one hurdle remains: the real numbers. All of the surprising discoveries above suggest that the idea of real numbers being countable is not an implausible notion. The answer is revealed in the next post.

Sources: http://en.wikipedia.org/wiki/Cardinal_number

Infinity: The First Transfinite Cardinal

2012-01-09T00:00:00.009-05:00

In mathematics, infinity, often denoted ∞, is defined as exceeding all natural numbers, or, conversely, as the limiting value as a variable n increases without bound. ∞ has always been regarded as a sort of mystical quantity, ever out of reach from most mathematical concepts and calculations. However, through set theory, insights into infinity, in fact multiple infinities, can be gained.

A set is a series of elements, such as {1,2,3} or {12,58,-1,4}. The number of elements in a set is known as its cardinality. For example, {1,2,3} has a cardinality of 3, and {12,58,-1,4} has a cardinality of 4. Any number that can represent the cardinality of a set is known as a cardinal number. 3 and 4, as demonstrated above, are examples of cardinal numbers.

In fact, every natural number 1, 2, 3... is a cardinal number, and even 0 is a cardinal number, as it measures the number of elements in the empty set {}, also written Ø.

Now, it is useful to define functions on sets, namely rules for changing one set into another. The most important of these are known as bijections, and they are defined as functions of sets that preserve the cardinality of a set.

The above diagram is a pictorial representation of a bijection, defined as a function that maps each point in set X to exact one point in set Y, in other words a one-to-one correspondence. It is clear that such a function, when applied to a set, will preserve its cardinality. For example, the function y=x-1 maps the set {2,3,4} to the set {1,2,3}, each of the sets having a cardinality of exactly 3. Since this function preserves the cardinality of all sets in the same matter, it is a bijection.

It is also possible to define an infinite set, or a set with an infinite number of elements. The simplest of these is the set of all natural numbers, namely {1,2,3,4...}. This set has infinite cardinality, and the cardinal number representing this set is the first so-called transfinite cardinal. It is denoted aleph-zero, or ℵ₀.

Aleph-zero is not contained within the normal number system that we think of, but rather describes the size of the set containing all natural numbers. One could then wonder whether adding another element, for example 0, to the set would result in a cardinality of ℵ₀+1. This intuitively seems reasonable, but it is not the case. It was earlier shown that the function y=x-1 is a bijection, so that, when applied to the above set X, will preserve its cardinality of aleph-zero. The result is as follows:

X={1,2,3,4...}
Y=X-1={0,1,2,3...}

Therefore, the set containing all whole numbers, which include both the natural numbers and 0, also has a cardinality of aleph-zero.

At first glance, the above result appears paradoxical. It seems that by subtracting all the terms by 1, an element is added to the beginning of the set, but taken off the end. This is certainly true for finite sets of natural numbers. For any finite natural number n, if

X={1,2,3...,n-1,n}, then
Y=X-1={0,1,2...,n-1}

However, as n increases without bound towards ∞, n-1 is ultimately indistinguishable from n, as ∞-1 is still ∞. In addition to this, when one considers any natural number in the infinite set {1,2,3,4...}, it is decreased by 1 when the function is applied, but there is always another number to take its place. When one takes these points into consideration, the result begins to make sense.

One can easily use the functions y=x-2, y=x-3... etc. to incorporate the elements -1, -2, etc. with the same logic as before, generalizing the above statement to include any finite number of negative integers.

This is only the beginning. Next, consider the function defined below.

If x is odd, then y=(x-1)/2
If x is even, then y=-x/2

The above is not a usual function that can be defined in simple operators of x. However, it is still a bijection when applied to the set of natural numbers, as each element of the set {1,2,3,4...} is transferred to a unique number in a second set. This set is {0,-1,1,-2,2-3,3...}. Remarkably, the output of this function covers all integers! Again, it is easy to see that for any integer one could choose, there is always a natural number that produces it with the above function. Therefore, the total set of integers still has the same cardinality, namely aleph-zero, as the natural numbers.

Nor does the fun stop there! Next consider the mapping of a set to a set of ordered pairs, namely assigning a set of two integers to each natural number. This can be done in the following way:

1 -> (0,0), 2 -> (1,0), 3 -> (1,1), 4 -> (0,1), 5 -> (-1,1), 6 -> (-1,0), 7 -> (-1,-1), 8 -> (0,-1), 9 -> (1,-1), 10 -> (2, -1)...

The exact pattern of these ordered pairs can take several different forms, the above being one of these. Initially, the above sequence seems to have no clear pattern, but it does have a clear geometric significance.

The above pattern lists all the integral ordered pairs (white circles) reached as one follows a rectangular counterclockwise spiral beginning at the origin (the black path). Clearly, by following this path for a sufficient distance, we will visit any ordered pair of integers (a,b) that we could choose!

Before exploring the implications of the above statement, we must make one more logical step. Consider an ordered triplet of the form (a,b,c), again with a, b, and c integers. One can use a similar system to the one above to set up a one-to-one correspondence between the natural numbers and these triplets. The first few terms are listed below:

1 -> (0,0,0), 2 -> (0,0,1), 3 -> (0,1,1), 4 -> (1,1,1), 5 -> (1,0,1), 6 -> (1,-1,1), 7 -> (0,-1,1), 8 -> (-1,-1,1), 9 -> (-1,0,1), 10 -> (-1,1,1)...

Just as before, this can be visualized as a rectangular spiral in the three dimensional coordinate system, where each point is given coordinates (x,y,z). In fact, this pattern, and its geometric interpretation, continue for any order n-tuplets, each consisting of elements (a1,a2,a3...an) and representing a spiral in the n-dimensional Cartesian system.

But what does this mean in terms of the number systems? Clearly, the set of all integral n-tuplets, for any finite n, has a cardinality of aleph-zero. From this, we draw the similar result that

Any infinite set or subset of a number system whose members can be represented by integral ordered n-tuplets, with n a natural number, has the same cardinality as the set of natural numbers, namely aleph-zero.

The implications of the above statement are explored in the next post.

Sources: http://en.wikipedia.org/wiki/Cardinal_number

Juno

2012-01-01T00:00:00.000-05:00

Juno is a NASA spacecraft whose mission is to orbit Jupiter and gain further insight to its composition and formation. It is named for the goddess Juno, wife of Jupiter in Roman mythology.

The spacecraft launched on August 5, 2011 to start its six year mission, culminating in a Jupiter arrival in 2016. En route, the probe will flyby Earth (2013) in order to save fuel. Unlike previous missions to the outer Solar System, Juno's energy will come only from solar panels, despite the relative dimness of the Sun at Jupiter's orbit.

The trajectory that Juno will follow on its way from launch in 2011 to arrival at Jupiter in 2016.

In addition, the spacecraft, after reaching Jupiter, will assume an orbit that takes it past the north and south poles of Jupiter with every revolution. This path will take Juno just over 3,000 miles from the cloud tops near the poles, and will allow extensive observations of aurorae and other magnetic field phenomena. However, such an orbit also leaves Juno exposed to high concentrations of radiation, which will slowly degrade the functionality of the spacecraft. Juno is therefore planned to spend just over a year at Jupiter, making 33 orbits, before intentionally crashing into Jupiter late in 2017.

Juno will hopefully provide further information concerning the formation and evolution of Jupiter, specifically about its interior, about which little is known.

2011 Unnamed Tropical Storm

2011-12-22T14:46:00.006-05:00

Storm Active: August 31-September 2

*This cyclone was classified as a tropical storm during the 2011 postseason analysis. It therefore received no name, despite being tabulated in the number of tropical depressions and tropical storms of the 2011 season.

On August 29, a circulation took shape in an area of convection north of Bermuda, some of which had been associated with Tropical Storm Jose a few days previously. The resulting trough organized over the next few days, increasing in shower activity. Late on August 31, a closed low formed on the southeastern edge of the shower activity, and the system became a tropical depression (although it was not then recognized as such). Convection increased markedly on September 1, and gale force winds were recorded, suggesting that the cyclone at this time became a tropical storm.

A banding feature to the southwest of the center formed the same day, and the cyclone strengthened overnight, reaching its peak intensity of 45 mph winds and a pressure of 1002 mb early on September 2. By this time, an approaching front had begun to push the system northeast, away from the U.S. east coast. The proximity of the front caused the unnamed storm to lose definition during the day of September 2, and the system became extratropical that evening. The remnants continued to move north-northeastward, and were fully absorbed on September 2. This event marked the first time since 2006 that a tropical storm was added in postseason analysis.

The unnamed tropical storm weakening on September 2.

Track of the unnamed tropical storm.

2011 Season Summary

2011-12-15T06:29:00.025-05:00

The 2011 Atlantic hurricane season was an above average season, with

20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status*
7 cyclones attaining hurricane status†
and 3 cyclones attaining major hurricane status

*In the NHC postseason analysis, an additional unnamed tropical storm was identified to have formed during the month of September. This means, that although 2011 only reached the letter "S" in tropical cyclone names, and 2010 reached "T", both seasons had the same number of tropical cyclones form.

†Nate was upgraded from a tropical storm to a hurricane during the postseason analysis.

At the beginning of the season, I predicted that there would be

20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status
10 cyclones attaining hurricane status
and 6 cyclones attaining major hurricane status

The tropical depression and tropical storm predictions happened to be exactly correct, although there was a lower number of hurricanes and major hurricanes than I predicted. As with the 2010 season, the 2011 season was tied for third in overall number of tropical storms with 19. This was caused by an ongoing La Nina event that actually intensified towards the latter part of the season. However, many of these storms were short-lived, and this reflects the abundance of favorable conditions for formation, but not for intensification. These conditions included high wind shear over much of the Caribbean for long periods of time, and also large pockets of dry air associated with anticyclones, which worked their way into many developing systems.

Some notable cyclones and facts about the season include:

Hurricane Ophelia, the strongest storm of the season, attained Category 4 status at an unusually high latitude of 32.5° N
2011 was the first season in which none of the first eight tropical storms (Arlene through Harvey) became hurricanes
Hurricane Irene, the first hurricane and major hurricane season, was also the first cyclone of hurricane strength to make landfall in the U.S. since Ike of 2008, and the first cyclone of hurricane strength to make landfall in the state of New Jersey since 1903.
An unnamed tropical storm formed in early September, the first cyclone to be recognized only in the postseason analysis since 2006
Nate was upgraded from a tropical storm to a hurricane in postseason analysis, the first such instance since 2007
Hurricane Philippe was the longest lived storm in the Atlantic basin since 2008, but, despite its longevity, it affected no land.

Overall, the 2011 season was one of numerous, but weak, storms. The U.S. was affected much more than it had been in the previous two years with one hurricane and two tropical storm landfalls, but the damage associated with these systems was not severe.

Tropical Storm Sean (2011)

2011-11-09T18:27:00.004-05:00

Storm Active: November 8-11

On November 4, a frontal boundary moved off of the U.S. east coast. A low pressure system along the front deepened as it moved off the coast of North Carolina later that day. The section of the front to the north of the low had most of the cloud cover associated with it, but the convection moved closer to the circulation of November 6, as the system drifted southeast. After temporarily losing definition, the low strengthened again on November 7. By this time, gale force winds occupied a region around the low, extending hundreds of miles in each direction.

Over the following day, the southern extension of the frontal boundary degenerated, devolving into a banding feature expanding clockwise from the low. Meanwhile, the remainder of the front had moved away to the east, and the leftover moisture became entrenched in the circulation of the cyclone. Early on November 8, convection had circumnavigated the center, and the system was upgraded to Subtropical Storm Sean.

The cyclone continued to increase in organization that afternoon, the eye contracting, and the surface circulation becoming better defined. The movement of the circulation into the lower levels of the atmosphere merited a reclassification of Sean into a tropical storm. Throughout the day, Sean remained nearly stationary, initially revolving around a broader cyclonic center, and later adopting a slow westward motion. Convection developed in earnest during the morning of November 9, and the system intensified into a strong tropical storm, also forming an eye feature.

By this time, Sean had entered the steering currents of the west Atlantic and began to accelerate northward. On November 10, the system curved to the northeast, also reaching its peak intensity of 65 mph winds and a pressure of 983 mb. Late that night, the windfield of Sean enveloped Bermuda, causing tropical storm force winds on the island, along with periods of heavy rain.

As it moved away from Bermuda on November 11 winds shear drastically increase and extratropcal transition began as Sean cam into close proximity with a front. By that evening, the center had elongated, and convective bands associated with the circulation ere stripped away. As a result, the system became extratropical that night. Sean caused minor damage and one fatality in Bermuda.

Tropical Storm Sean near peak intensity on November 10. The Outer Banks of North Carolina are visible on the upper left.

Erratic track of Sean through the Western Atlantic.

Hurricane Rina (2011)

2011-10-24T14:54:00.007-04:00

Storm Active: October 23-28

During the day of October 18, a cold front swept across the Gulf coast. Heading unusually far south, the front continued to moved over the Gulf, crossing the Yucatan Peninsula during the afternoon of October 19. Simultaneously, a tropical wave, moving through the Caribbean, degenerated into a trough, though it still produced showers and thunderstorms. On October 20, these systems merged off of the coast of Nicaragua and a low pressure system formed.

Over the next day, the low drifted south and then west, all the while increasing in organization. Thunderstorm activity became concentrated just off the coast of Nicaragua during the evening of October 21, However southwesterly wind shear displaced thunderstorm to the west of the low pressure center, causing some heavy rainfall in Central America the following day. On October 23, now moving northward off of the coast of Honduras, the low experienced a reformation, with the new center within the southern edge of the convection. That afternoon, sufficient organization had been attained to classify the system as Tropical Depression Eighteen.

An increase in the symmetry of Eighteen later that night merited an upgrade to Tropical Storm Rina. A series of increases in sturutal organization followed early on October 24, with the outflow imporving in all quadrants. Despite this, the eyewall itself did not undergo any major changes until the early afternoon, at which time a contraction caused a well-defined eyewall, as well as a radiating band feature, to form. Rina was therefore upgraded directly from 45 mph to a hurricane.

Shortly afterward, Rina turned to the west and decreased in speed due to the influence of a ridge over the Gulf of Mexico. Meanwhile, Rina's internal structure continued to improve, and an eye appeared on visible, but not infrared images. Therefore, early on October 25, the cyclone underwent another burst of strengthening and became a Category 2 hurricane. Further fluctuations in the strength of the eyewall pushed Rina to its peak intensity that night, achieving winds of 110 mph and a pressure of 966 mb.

The hurricane maintained this intensity through the morning, but a tongue of dry air invaded from the north. This, coupled with increasing wind shear caused rapid deteriorated of the circulation, reducing Rina's intensity to that of a Category 1 hurricane. As the cyclone began to turn north, the area of deep convection associated with the system shrunk. By October 27, interaction with the Yucatan Peninsula to the west and the exposure of the center of circulation weakened Rina to a tropical storm. Later that day, Rina made landfall near Cancun, causing heavy rainfall and a small area of tropical storm force winds.

By the morning of October 28, the system emerged into the Yucatan Channel and weakened into a tropical depression. All convection had been displaced to the north by this time, and Rina degenerated into a remnant low that afternoon. It was absorbed by a front the next day. Due to its rapid degeneration, the cyclone caused only minimal damage in Central America.

Hurricane Rina at peak intensity, with an eye feature apparent on visible imagery.

Track of Rina.

Hurricane Philippe (2011)

2011-09-29T16:21:00.006-04:00

Storm Active: September 24-October 8

A well-defined low pressure area formed west of the African nation of Guinea on September 23. The low deepened rapidly later that day, already having developed large rain bands spread out evenly around the circulation. Convection decreased that night, but became very concentrated near the center, and the system became closed early the next morning. It was then declared Tropical Depression Seventeen. Slow organization followed later in the day as Seventeen moved west-northwest, and the system soon intensified into Tropical Storm Philippe.

A weakness in the ridge to Philippe's north allowed it to turn northwestward and decelerate into the day of September 25, when it reached an intensity of 60 mph winds and a pressure of 997 mb. The tropical storm maintained this strength into September 26, by which time it began to encounter higher wind shear and cooler waters. A gradual weakening ensued, as the circulation became exposed on September 27. That afternoon, Philippe's forward motion slowed to 5 mph and its shallow circulation was steered back to the west-northwest with a restrengthening of the subtropical ridge to its north.

That night, despite the fact that Philippe was a minimal tropical storm with little convection near the center, the outflow remained fairly robust, and thunderstorm activity reappeared just north of the center on September 28. The cyclone even strengthened a little that day. On September 29, the center reformed to the north of its previous position, and once again was completely enveloped in the convection associated with Philippe. The exact center position of Philippe remained difficult to identify overnight, but the circulation became more well-defined on September 30, and the cyclone strengthened further.

As the storm moved farther to the northwest, it encountered very strong shear associated with the outflow of Ophelia, but the tropical storm once again demonstrated resilience to strong upper-level winds, and maintained its intensity, deep convection even increasing near the center of circulation overnight. An analysis of the windfield of Philippe revealed it to be stronger than expected, with winds of 65 mph. Late that night, more intensification occurred, bringing Philippe to the verge of hurricane strength early on October 2.

However, the unfavorable conditions near the cyclone finally began to take their toll that morning, exposing the center and weakening the system. As before, however, Philippe recovered its convection, as Ophelia moved north, and conditions became less hostile. The next day, winds were found to be stronger than previously estimated, and the cyclone was a strong tropical storm yet again. Meanwhile, the center reformed farther to the south than before, and Philippe moved west-southwestward during the day of October 3.

After navigating around the periphery of the ridge steering it, the tropical storm began to turn north on October 4. Recurvature occurred quickly during the following days, and shear gradually decreased. On October 5, the center remained displaced from the deepest convection, but, on October 6, strengthening finally began, bringing Philippe to hurricane strength after over 12 days as a tropical cyclone. An eye feature formed later that evening, particularly visible on infrared imagery, and the hurricane reached its peak intensity of 90 mph winds and a pressure of 976 mb that evening.

Upper-atmospheric conditions rapidly deteriorated on October 7, causing weakening as the cyclone accelerated to the east-northeast. The center quickly became exposed that afternoon, and Philippe's brief period as a hurricane was over. A front encouraged extratropical transition on October 8, and absorbtion completed in that afternoon. Philippe was the longest lived tropical cyclone in the Atlantic since Bertha of 2008, lasting over two weeks. Despite this, it affected no land.

Hurricane Philippe over the open Atlantic.

Track of Philippe.

Hurricane Ophelia (2011)

2011-09-23T16:38:00.009-04:00

Storm Active: September 20-October 3

On September 17, a broad area of showers and thunderstorms formed in association with a large low pressure system in the eastern Atlantic. Three low pressure centers existed in close proximity near the area during the day of September 18, oriented west to east near the 10ºN parallel. Over time, the central low became stronger and dissipated the others, and convection increased. The circulation also increased in definition over the following days, but numerous reformations of the center kept the circulation open. However, late on September 20, thunderstorms became concentrated near a single circulation center, and the low was upgraded directly to Tropical Storm Ophelia.

As the tropical storm moved west over the central Atlantic, shear steadily increased, but despite unfavorable conditions, winds in Ophelia's rain bands increased significantly, and strengthening ensued. On September 21, convection was displaced to the east of the center, but notwithstanding any satellite data to the contrary, winds were reported that suggested a strong tropical storm intensity. By the morning of September 22, Ophelia had reached an intensity of 65 mph winds and a pressure of 994 mb.

Wind shear began to finally take its toll on the system later that day, and the cyclone weakened. By the morning of September 23, Ophelia, had weakened to a minimal tropical storm, and all remaining convection was pushed northeast into a rain band that extended several hundred miles. Meanwhile, Ophelia took a turn to the west-northwest. Later on September 23, shear temporarily relaxed, and shower activity re-ignited near the center of the cyclone, causing it to unexpectedly strengthen that evening.

However, this increase in intensity did not persist, as strong upper-level winds resumed early on September 24. Ophelia weakened once again during that day, and, as a result, turned back to the west. As the system's circulation continued to become shallower, a due west motion was assumed, and by the morning of September 25, Ophelia was a minimal tropical storm. A large area of convection was over 150 miles east of the center during the day, with the center itself bare but for a few showers to the northeast. The Lesser Antilles, to Ophelia's southwest, experienced some gusty winds during the afternoon, as the center of the circulation became more elongated. Subsequently, the system was observed to lack tropical characteristics, and was downgraded to a remnant low.

The exposed center quickly dissipated, but a new swirl quickly became evident in the thunderstorm activity to the east. During the day of September 26, upper-level winds relaxed, and the circulation of the newly formed low became much better defined that evening. As the system continued to organize overnight, it drifted eastward and southward, stalling close to the Leeward Islands. On September 27, moderate rainfall occurred over these islands, as convection increased further during the afternoon, and the low became organized enough to be redesignated as Tropical Depression Ophelia.

Shear still abounded over the region overnight, and strengthening was gradual, bringing Ophelia to tropical storm strength during the morning of September 28. By this time, the cyclone had assumed a definite north-northwestward motion, and it moved away from the Caribbean. Outflow improved throughout the same day, allowing the cyclone to hold its own despite marginally favorable conditions. Further strengthening followed that night, and the system was a powerful tropical storm by September 29. An impressive banding feature formed to the east of the circulation later that afternoon, causing Ophelia to be upgraded to Category 1 hurricane status.

During the same evening, Ophelia's eyewall solidified, and the eye itself became more consistent in its appearance on satellite imagery. Following these structural changes, the cyclone rapidly strengthened. By the morning of September 30, Ophelia was a Category 2 hurricane! However, the strengthening was by no means over, as the hurricane assumed a more rounded appearance, the outflow improving and the eye broadening further. By that afternoon, Ophelia was a Category 3 hurricane. Meanwhile, the storm turned north and started to accelerate, as a large trough exited the U.S. east coast and pushed it farther poleward. Late that night, the system finally stabilized in intensity at 120 mph winds and a pressure of 956 mb.

Early on October 1, the pressure dropped slightly to 952 mb, and the outer bands of Ophelia began to impact Bermuda. During the day, gusty winds and scattered heavy squalls affected the island, but the hurricane passed well to the east, making its closest approach late that afternoon. Just after passing Bermuda, Ophelia unexpectedly underwent rapid intensification, bringing it to its peak intensity as a Category 4 hurricane with 140 mph winds and a pressure of 940 mb. During the morning of October 2, Opheila finally began to weaken as it turned to the north-northeast, reaching a forward speed of 30 mph north of 35ºN latitude. The circulation slowly lost definition as it raced towards Newfoundland ahead of a cold front to its west.

That night, Ophelia lost most of its remaining tropical characteristics, weakening to a tropical storm during the morning of October 3. Any central convection still associated with the system vanished as Ophelia made landfall in the Avalon Peninsula, and the system became extratropical. The cyclone continued northeastward until dissipating on October 5. Minimal damage was the only effect of Ophelia, with no fatalities occurring.

Hurricane Ophelia nearing peak intensity as a Category 4 hurricane.

Track of Ophelia, including time spent as an non-tropical system before reforming.

Hurricane Nate (2011)

2011-09-08T18:08:00.005-04:00

Storm Active: September 7-11

After Tropical Storm Lee became extratropical over the southeast U.S., an extension of its associated frontal boundary dipped into the southern Gulf of Mexico. On September 6, the interaction of this front with a trough of low pressure caused a low pressure system to form in the Bay of Campeche. Over the next day, the low hardly moved, and organized rapidly. On September 7, the low-level circulation of the low became more well-defined and the system was upgraded into Tropical Storm Nate.

Nate formed with a very slow motion to the southeast, and did not move significantly overnight. Despite a low shear environment, there was one inhibiting factor to strengthening: a large area of dry air to the system's northwest, occupying the entire northern Gulf. During the day of September 8, the center drifted further southeast, becoming closer to the convection, which had been displaced to the southwest of the center by moderate wind shear. Nate was sheltered somewhat, and strengthening commenced. By the evening of that day, Nate was completely stationary, and at its peak intensity of 75 mph and a pressure of 994 mb*.

Dry air entered the system during the early morning of September 9, however, and weakening took place. Later that day, Nate began to move slowly northwest, away from the Yucatan Peninsula, as further weakening occurred. By the morning of September 10, rainbands had recovered somewhat on the periphery of the cyclone, but the center remained devoid of convection, giving Nate a hollow appearance. The tropical storm finally adopted a definite motion later that day, moving due west. As it did so, conditions improved slightly and re-strengthening occurred afternoon. Nate reached its secondary peak intensity of 65 mph winds before convection decreased once again early on September 11.

The position of the center became very uncertain as Nate approached the Mexican coast, and the definition of the circulation decreased, lowering Nate's intensity to only 45 mph as it made landfall in Veracruz. The cyclone quickly weakened to a remnant low that night. Not much convection was associated with Nate over its lifetime, but its slow movement still allowed prolonged periods of gusty winds and rain along many parts of the coast of the Bay of Campeche. 7 fatalities were the result of Nate.

*Nate was upgraded to a hurricane during the 2011 postseason analysis, its maximum winds having previously been recorded as only 70 mph.

Tropical Storm Nate at peak intensity on September 8. At this time, Nate was nearly stationary.

Track of Nate.

Hurricane Maria (2011)

2011-09-07T16:45:00.007-04:00

Storm Active: September 6-16

On September 4, a tropical wave off of the African coast began to generate shower activity south of the Cape Verde Islands. This activity became more widespread and more organized with the westward moving wave on September 5, as surface pressures began to drop over the region. A low pressure center developed in association with the wave that day, and the circulation became closed on September 6, announcing the formation of Tropical Depression Fourteen. The depression slowly organized over the next day, as it moved quickly west-northwest. A rapid intensification occurred during the morning of September 7, as an area of deep convection appeared northeast of the center, and the cyclone was upgraded to Tropical Storm Maria, with 50 mph winds.

High wind shear affected the system from its formation, however, and a powerful ridge to Maria's north imparted it with an extremely rapid westward motion of 23 mph during the night and into the morning of September 8. The storm struggled to maintain a circulation during the afternoon, and began to weaken as a result. All but minimal central convection disappeared during the early evening, and Maria became a minimal tropical storm. Overnight, however, a huge area of convection appeared. Despite this new thunderstorm activity, Maria did not intensify significantly on September 9, as the low level center was ill-defined and difficult to locate on satellite imagery.

On September 10, all convection was displaced to the northeast of the center by wind shear. So although being in close proximity to the northeast Caribbean Islands, the storm caused nearly no rain there as Maria moved northwest. The cyclone began to recover organization once again during the morning of September 11, and became a strong tropical storm as it moved west-northwest past the U.S. Virgin Islands that afternoon. A powerful trough coming off of the United States east coast started to affect Maria that evening, and it took a turn to the north on September 12, though slowing to nearly stationary. On September 13, the system began to accelerate, and finally moved away from the upper-level low that had been affecting it for so long with wind shear.

The center of Maria reformed within an area of deep convection late that night, and banding features increased in organization. During the morning of September 14, intensification began, and shear decreased. The decrease was gradual, however, and no further change occurred during the afternoon. By this time, Maria had turned to the north-northeast and already was accelerating extremely rapidly. Conditions in Bermuda deteriorated late that night and into the morning of September 15, as Maria passed to its west at near hurricane strength. Deep convection appeared near the eye that afternoon, and an eye structure began to form. Maria was therefore upgraded to hurricane status.

That night, Maria reached its peak intensity of 80 mph winds and a pressure of 979 mb. During the morning of September 16, the cyclone's forward speed exceeded 50 mph. Its closed circulation vanishing, Maria made landfall in Newfoundland as a minimal hurricane on the cusp of extratropical transition that afternoon, followed by absorbtion by a front a few hours later.

Hurricane Maria moving rapidly northward. The cyclone is already exhibiting some extratropical traits.

Track of Maria.

Tropical Storm Lee (2011)

2011-09-02T07:44:00.009-04:00

Storm Active: September 1-4

During the day of August 30, shower activity increased with a tropical wave moving west-northwestward through the western Caribbean Sea. Meanwhile, a stationary low pressure trough was situated over the Gulf of Mexico. During the next few days, the tropical moisture of these two systems combined, forming a very large area of strong thunderstorms over the eastern Gulf, stretching from the Yucatan Peninsula to Florida. Shear from the west affected the system, but began to slowly abate during the day of September 1. During that afternoon a circulation appeared on the western edge of the clouds, becoming closed by the evening. Tropical Depression Thirteen had formed.

During the night, the depression's center reformed farther south and lost nearly all of its initial motion to the northwest, Due to weak steering currents, the system remained nearly stationary through the morning of September 2. Also, an area of deep convection developed near the center, despite the center itself remaining rather elongated. Due to the system's large size, rain and gusty winds already were moving into southeastern Louisiana. The circulation of the system became more well-defined that afternoon, and Thirteen became Tropical Storm Lee.

Lee had an unusually large windfield even at the time of its formation, and it continued to expand later that day. The cyclone itself also slowly strengthened, moving erratically, but generally northward. Lee still exhibited some characteristics of a non-tropical cyclone even into September 3, with the southwest quadrant still devoid of convection. During that morning, heavy rain continued across Louisiana and Mississippi, and tornadoes were reported within the heaviest bands, located in southern Louisiana.

Lee strengthened further as it slowly moved toward the Gulf coast, reaching its peak winds of 60 mph during the day. However, a second low-level center developed within the system early that afternoon, the original over land, and the second still off the coast. The new formed center took over the circulation while the other dissipated, and Lee therefore stalled off of the coast of Louisiana, with tropical storm conditions still extending far inland all across the central Gulf states. During the night, convection decreased, with the only rain band near the center extending to the southwest. Gradual weakening occurred, but Lee still did not assume any definite motion, and was still hovering over the coast, continuing the already severe flooding of the surrounding areas. Despite the winds having fallen, Lee's central pressure decreased to 987 mb early on September 4. The system finally made landfall in Louisiana later that morning.

The circulation actually became better defined for a brief period over land that afternoon, but the cyclone quickly degenerated, weakening further that evening, and began extratropical transition late that night. The system was fully extratropical by the next day, but the rain was by no means over. During the day of September 5, the remnants of Lee combined with a powerful cold front. With the addition of tropical moisture, a region of heavy rain formed from the tail end of Lee, in Louisiana, through the end of the front, in Canada! Lee's remnants lost their well-defined center on September 8, but rainfall continued for two more days, finally ending on September 10. The several days that Lee spent moving up the coast saw unprecedented flooding in the the mid-Atlantic states, New England, and even in some areas as far north as Canada. Rainfall totals from the combined storm system exceeding 6 inches in widespread areas, with local amounts significantly greater. 21 fatalities resulted from Lee, along with over $250 million in damages.

Lee strengthening over the northern Gulf of Mexico.

Track of Lee.

Hurricane Katia (2011)

2011-08-29T08:23:00.018-04:00

Storm Active: August 29-September 10

A low formed just off of the African coastline on August 27, associated with a tropical wave, and was already showing signs of organization. The broad area of showers and thunderstorms quickly became concentrated over the next day, as the system passed well south of the Cape Verde Islands. Rapid development continued, and the system became Tropical Depression Twelve early on August 29. The depression also formed at 9.4º N, fairly far south for a tropical cyclone. Some shear out of the east affected the system from the beginning, and, as a result, the center remained on the eastern tip of the cloud cover through the day, A ore circular area of convection developed early on August 30, and although shear continued, the system was organized enough to be named Tropical Storm Katia.

Katia adopted a west-northwest motion that morning, and also accelerated somewhat in forward speed. Through the day, shear lessened significantly, and Katia began to rapidly intensify. Deep convection had enveloped the center by that afternoon, and Katia quickly became a strong tropical storm that evening. The intrusion of dry air on the circulation delayed intensification overnight, but Katia resumed a slow strengthening trend by August 31. Meanwhile, the cyclone moved even faster to the west-northwest, under the influence of a ridge to its northeast, its forward speed exceeding 20 mph. Katia continued to slowly organize, and the development of a well-defined eyewall merited the upgrade of the system to a hurricane late that night.

However, some dry air entered the circulation from the south, temporarily weakening the eyewall early on September 1, and causing the cyclone to stabilize in intensity. Katia also returned to a westerly motion that morning. However, shear increased during the day from an upper-level low to Katia's north, due to the close proximity of the low, and Katia again became disorganized weakening back to a tropical storm. A deeper burst of convection appeared with the system early on September 2, but the center remained on the periphery of this cloud cover during the early morning hours. Katia decelerated significantly, and turned once again to the west-northwest. Despite somewhat hostile conditions, Katia regained hurricane status later that morning, as a gradual turn to the northwest began.

Katia continued to struggle against wind shear throughout the day, and even developed an eye for a time that evening! However, the eye remained too close to the edge of convection to survive, and clouded over early on September 3. The cyclone's central pressure dropped, and Katia maintained minimal hurricane intensity. Thunderstorm activity decreased in the eastern half of the circulation that afternoon, and the cyclone became slightly lopsided. As a result, Katia again weakened to a tropical storm. Later in the evening, the fluctuations in intensity continued, as convection once again increased, and another eye formed. Due to this, Katia was once again upgraded to a hurricane during the morning of September 4. Through the night, Katia had still maintained a general northwest motion.

Later that morning, a well-defined eye developed, and Katia underwent rapid strengthening, becoming a Category 2 hurricane. That afternoon, Katia reached an intensity of 105 mph winds and a pressure of 965 mb. However, dry air invaded the system once again, this time from the northeast quadrant, and Katia weakened slightly during the early morning of September 5. Rather than disrupting the circulation in the long term, however, the dry air was incorporated into a large eye that appeared later in the morning. Katia once again strengthened rapidly, as outflow also improved that afternoon. Following these structural changes, the cyclone was subsequently upgraded to a major hurricane. Further intensification ensued late that evening, and Katia quickly reached its peak intensity as a Category 4 hurricane, with 135 mph winds and a pressure of 946 mb.

However, as the storm continued northwest, it began to encounter less favorable conditions, including lower ocean temperatures, and increased wind shear. A general weakening trend began, and Katia soon lost Category 4 status. The hurricane force wind field remained quite broad, though, and even became larger during the morning of September 6. These winds extended up to 55 miles from the center that afternoon. Rip currents and high surf were already beginning to affect Bermuda, and the threat increased on September 7. In the face of dry air and wind shear from the west, the circulation became more lopsided, with any remaining symmetry in the eyewall disappearing by that morning.

Bermuda also received gusty winds and scattered showers, being on the periphery of Katia's powerful east side. By that afternoon, the cyclone's winds had decreased to 85 mph, a Category 1 intensity. A turn to the north followed during the early evening hours, and Katia's forward motion increased. Despite moving into cooler waters, the upper atmospheric conditions near Katia improved that night and it strengthened slightly as it recovered the western half of the eyewall somewhat. The system made its closest approach to Bermuda the following morning, passing well to the west. Once again, Katia's convection actually increased over cool waters during the day of September 8. A turn to the northeast commenced that evening, and Katia's motion rapidly increased. By September 9, the cyclone was speeding away from the New England coast. Extratropical transition began later that day, and Katia finally became extratropical during the morning of September 10, after reaching a forward speed of over 50 mph.

At the time of the last advisory, Katia still packed winds of 80 mph, and became a powerful extratropical cyclone, impacting north portions of the British Isles on September 11 with high winds and rain as it passed to the north. Katia indirectly affected the Lesser Antilles, Bermuda, the east coast of the United States, and the United Kingdom. 2 fatalities, one direct and one indirect, were the result of Katia.

Katia near peak intensity as a Category 4 hurricane.

Tropical Storm Jose (2011)

2011-08-29T07:32:00.006-04:00

Storm Active: August 28-29

On August 19, a tropical wave moved off of Africa, and was immediately monitored for development upon leaving the coast. A broad low pressure system quickly formed in conjunction with the wave, and convection increased, but despite favorable conditions, no closed circulation formed as the system passed near the Cape Verde Island on August 20. The low's circulation became elongated during the day of August 21, and it moved northwest, into cooler waters. THe system did not dissipate, however, and still produced some shower activity over the next few days. The low elongated further on August 25, and degenerated into a trough of low pressure. This trough interacted with another to its west, and a weak low formed from this union on August 27, now located south-southeast of Bermuda. The low deepened and moved generally northwest, but very high shear from the outflow of Hurricane Irene ripped convection from the low faster than it could form. Despite these extremely hostile conditions to tropical development, the circulation, although being nearly devoid of convection, became closed, and gale force winds in the low's southeast quadrant caused it to be named Tropical Storm Jose during the morning of August 28.

Due to its proximity to Bermuda, a tropical storm warning was issued there. Against all odds, Jose's southern side developed some deep convection that afternoon and the system strengthened slightly as it moved north, past Bermuda, and reached its peak intensity of 45 mph winds and a central pressure of 1007 mb. Overnight, Jose accelerated northward and was weakening by the morning of August 29, as it encountered cooler waters. By later that day, the circulation no longer existed, and Jose dissipated, downgraded to a trough of low pressure. In the wake of Jose, to its south, large areas of thunderstorms appeared near Bermuda for the remainder of the day, partly having their origin in tropical moisture from the cyclone, which was displaced to the south by shear. Therefore, Jose indirectly caused some minimal damage to Bermuda.

Track of Jose.

Tropical Depression Ten (2011)

2011-08-25T07:30:00.006-04:00

Storm Active: August 25-26

During the afternoon of August 22, a low pressure system moved into the Atlantic off of the African coast. The circulation of the low was not yet well-defined, but it slowly moved west-southwest over the next few days, gaining convection and organization as it went. As the low passed to the south of the Cape Verde Islands, scattered showers occurred there. By August 24, the system was moving west away from the islands, and was quickly organizing. Early on August 25, a well-defined circulation developed, and the existence of a rain band circumnavigating the center, confirmed the formation of Tropical Depression Ten.

Initially, the depression seemed on the verge of tropical storm strength, but as it tracked west-northwest, the convection decreased, and was minimal by later in the morning on August 25. Despite a return of convection during the day, the circulation became elongated, and badly defined. This trend continued into August 26, keeping the cyclone at tropical depression intensity. Winds near gale force still appeared periodically, but the circulation became so stretched that the depression no longer met the standards of a tropical cyclone by later that night. Tropical Depression Ten officially degenerated into a trough near midnight, losing its definition entirely by August 27. The cyclone affected no land masses, with the exception of a few storms in the Cape Verde Islands.

Tropical Depression Ten over the east Atlantic.

Track of Tropical Depression Ten.

Hurricane Irene (2011)

2011-08-21T07:21:00.012-04:00

Storm Active: August 20-29

On August 15, a tropical wave moved off of Africa, but dry air near the system did not allow significant development. However, on August 17, a small area of convection formed in associated with the wave, followed by a low pressure center on August 19, when a definite circulation appeared in the clouds. Pressure in the area began to fall overnight, but no surface circulation had yet formed. During the evening of August 20, however, rapid intensification occurred, and a very organized surface circulation appeared. Hurricane hunter aircraft found winds high enough for the system to skip the tropical depression stage, and the low became Tropical Storm Irene, packing 50 mph winds even at its formation!

Early on August 21, Irene moved over the Lesser Antilles, and, being a fairly large cyclone, caused gale force winds and tropical storm conditions over a wide area as it entered the Caribbean. The cyclone was still moving westward at a fast clip, with its forward speed exceeding 20 mph. Irene maintained its intensity throughout the day, and made landfall in Puerto Rico that evening, causing heavy rain and very strong winds over the entire island. Despite some interaction with land, Irene rapidly strengthened, its circulation and outflow improving greatly that night. A huge burst of convection engulfed the system early on August 22, and the cyclone became Hurricane Irene, the first hurricane of the 2011 season.

Irene decelerated during the morning, and continued to move west-northwest, leaving Puerto Rico, where over 10 inches of rain fell in localized areas. The eyewall developed further during the evening of August 22, and the Irene rapidly intensified into a Category 2 hurricane with 100 mph winds as it passed just to the north of the Dominican Republic, with tropical storm conditions covering much of the country, and gusts of hurricane force on the northern coasts. Irene slowed down even more during the morning of August 23, with a forward speed of 10 mph to the west-northwest as it approached the Bahamas.

Later in the morning, a ragged eye appeared, and the central pressure of Irene dropped slightly. However, it was not a well-formed structure, and it clouded over in the afternoon, causing a weakening not uncharacteristic of eye replacement, putting the system back to a Category 1 intensity. Land interaction decreased considerably as Irene moved away from Hispaniola during the evening, convection returned in earnest, the central pressure dropped further, and another, more organized eye appeared. The cyclone soon regained its lost strength, and then intensified further, as it encountered the Turks and Caicos Islands during the morning of August 24. The system's winds surpassed 110 mph that same morning, and Irene became the first major hurricane of the season, with 115 mph winds and a pressure of 957 mb.

By later in the day, Irene had entered the Central Bahamas, the eye was tightening, and a new eyewall was forming. As a result of this, the eye began to cloud over, causing fluctuations in intensity. However, the system remained a Category 3, and the central pressure continued to steadily drop. Irene maintained its northwest motion into August 25, as surf began to increase along the U.S. coast. Later in the day, Irene finally left the Bahamas, and made a turn northward, toward the Outer Banks of North Carolina, early on August 26. The pressure of the cyclone dropped to 942 mb, the lowest yet for the cyclone, but another eye replacement weakened the system, decreasing its wind speeds to that of a Category 2 hurricane.

Rain bands from Irene were already beginning to sweep across the U.S. east coast from Florida up through the Carolinas. The internal structure of Irene remained slightly disorganized, and gradual weakening occurred as conditions became less favorable for intensification. Powerful outer bands packing tropical storm force winds penetrated into North Carolina through the day. Late on August 26, the convection became rather lopsided, with the southwest quadrant weak, and Irene subsequently weakened to a Category 1 hurricane early on August 27. Around 8:00 am EDT that morning, Hurricane Irene made landfall with 85 mph winds in eastern North Carolina, just west of the Outer Banks.

The cyclone began to assume a more asymmetrical appearance that afternoon, with most of the rain extending northward, reaching even Pennsylvania and New Jersey by early afternoon. The circulation, however, actually increased in definition as the day wore on. The forward speed of Irene finally began to increase that evening, but the cyclone moved steadily north-northeast, and emerged over water once again later that night, paralleling the Virginia coastline. Dry air entered Irene's southern side, and weakened it slowly through the morning of August 28, as the center of the hurricane made landfall in New Jersey before sunrise. At the time, the cyclone was a minimal Category 1 hurricane with 75 mph winds.

The windfield of the storm, however, was larger than ever, as tropical storm force winds spanned portions of 8 states. During the morning, Irene began to rapidly accelerate to the north-northeast, while quickly transitioning to an extratropical cyclone. By this time, all precipitation associated with Irene was on its north side, with the exception of one rain band, which protruded to the southwest, and affected areas as far south as Maryland even into early afternoon. The cyclone weakened to a tropical storm before making landfall in New York City that same morning. Irene persisted in the same general motion throughout the day, still bringing gusty winds to much of the northeast, before finally becoming extratropical near the U.S. border with Canada late that night. Rainfall continued in Canada as the remnants of Irene continued speeding northeastward, and the low emerged into the north Atlantic on August 30, passing near Greenland by the next day.

Irene had a devastating impact on areas from the Lesser Antilles to Vermont, and caused between 10 and 20 inches of rainfall locally in all regions along its path. The Bahamas, in particular, suffered high wind damages, as Irene was at peak intensity over the islands. Damage is estimated at $10.1 billion, and 54 fatalities resulted from the cyclone. Irene was also the first storm to make landfall in the U.S. at hurricane intensity since Ike of 2008, and the first hurricane to make landfall in New Jersey since 1903.

Irene as a major hurricane entering the Bahamas.

Track of Irene on its path through the Caribbean, the Bahamas, and along the east coast of the United States.

Tropical Storm Harvey (2011)

2011-08-19T08:25:00.011-04:00

Storm Active: August 18-22

On August 10, a vigorous tropical wave emerged off of Africa, and immediately began to produce an area of concentrated thunderstorms southeast of the Cape Verde Islands. On August 11, the system became more organized as a low pressure centered formed. However, the dry air of the eastern Atlantic penetrated the system, and cloud cover rapidly decreased as the low degenerated back into a tropical wave. The wave continued its journey westward, and convection once again erupted over a large area surrounding the system as it entered the moist waters near the Lesser Antilles. A broad upper-level circulation developed on August 16, but the system lacked a surface low, which inhibited further development as it entered the Caribbean Sea. The system moved due west for another few days before developing a surface low late on August 18, and being classified Tropical Depression Eight.

Eight was already producing rain and wind near the Honduras-Nicaragua border at formation, and conditions worsened on the northern coast of Honduras as the cyclone paralleled this coast, just a few miles offshore. Despite its proximity to land, the system developed deep convection and rain bands about the center, and a jog northeast on August 19 made conditions more favorable for strengthening as the day went on. A well-defined eyewall appeared on the south and west sides of the center during the afternoon, and Eight was named Tropical Storm Harvey shortly after.

The cyclone continued on a westward track through the evening of August 19, and strengthened rapidly, reaching an intensity of 65 mph winds and a pressure of 994 mb before temporarily stabilizing in intensity early on August 20. Since the tropical storm force winds extended only about 20 miles south of the center, (they extended 35 miles north at the time) gale force winds missed Honduras for the most part, although heavy rain fell throughout the region. Harvey continued to maintain the above peak intensity through landfall in Belize, which occurred that afternoon.

After landfall, Harvey continued to move west to west-northwest, and maintained a tropical storm intensity for an impressive 12 hours before weakening into a tropical depression over northern Guatemala. Thunderstorm activity continued throughout the region even as Tropical Depression Harvey crossed from inland Guatemala into inland Mexico on August 21. Later that day, the center, which was still well-defined, jogged to the north, and the depression emerged over the Bay of Campeche. With this new found access to water, Harvey strengthened slightly, but did not regain tropical storm intensity before making a turn to the west-southwest and making landfall in the southern Gulf coast of Mexico early on August 22.

Harvey quickly weakened as it moved inland, and dissipated later that morning. Only minimal damage was sustained, but 3 fatalities occurred from flooding, as Harvey dumped many inches of precipitation over Central America. Since Harvey did not reach hurricane strength, the record of consecutive named storms to not became hurricanes to start the season was extended to eight.

Tropical Storm Harvey shortly after formation.

Track of Harvey.

Tropical Storm Gert (2011)

2011-08-15T07:46:00.008-04:00

Storm Active: August 13-16

On August 10, a trough of low pressure developed over the Central Atlantic, associated with a low pressure system to its north. Over the next two days, the low center accelerated northeast, leaving the trough in its wake. The trough formed another weak low pressure center on August 12, as it drifted to the west-southwest at around 10 mph. The convection associated with the low remained very disorganized during the day, but the circulation became more well-defined on August 13, despite the center being partially exposed. More shower activity developed developed later that day, and a deepening of the low sparked the formation of Tropical Depression Seven late that night.

Deeper convection appeared overnight, and Seven intensified into Tropical Storm Gert as the cyclone made a turn to the north on August 14. The outflow of the system improved and the circulation assumed a more rounded appearance early on August 15, and rapid strengthening followed as the cyclone approached Bermuda. Gert reached peak intensity of 65 mph winds and a pressure of 1000 mb later that day. During the afternoon, Gert began a turn to the northeast, passing well to the east of Bermuda, and causing only minimal damage. The cyclone then began to accelerate further, and dry air permeated the system, weakening it overnight and into August 16. Gert quickly lost organization, and was an extratropical cyclone by later that day. Since Gert passed well east of Bermuda, no damage was sustained on the island. Additionally, since Gert did not achieve hurricane status, 2011 became the only year on record in which the first seven named storms did not become hurricanes.

Gert near peak intensity east of Bermuda, on which the cyclone had only minimal impacts.

Track of Gert.

Tropical Storm Franklin (2011)

2011-08-13T06:43:00.006-04:00

Storm Active: August 12-14

On August 10, a large trough of low pressure formed over Florida, with shower activity extending both east into the Atlantic, and west into the Gulf of Mexico. This activity moved generally to the northeast and interacted with a front moving off of the east coast. During the morning of August 12, convection concentrated at a low pressure center on the front, but the elongated nature of the frontal low kept it extratropical through the morning. As it accelerated away from land, the low became well-defined, and became disconnected from the front. As a result, Tropical Depression Six formed that afternoon, just north of Bermuda.

The effects on the island were only to the extent of scattered showers and gusty winds, as Six was tracking quickly away to the northeast. By the morning of August 13, the presence of deep convection within the system allowed it to intensify into Tropical Storm Franklin. Through the morning, thunderstorm activity with this tropical storm continued to increase, and outflow improved in all quadrants, despite increasing shear. Following this increase in organization, Franklin reached its peak intensity of 45 mph winds and a central pressure of 1004 mb during the day. However, Franklin was quickly approaching colder waters, and rapid weakening ensued that evening. By the following morning, Franklin had become extratropical. The remnant low of Franklin tracked east, and was quickly absorbed. The cyclone affected no land.

Tropical Storm Franklin over the open waters of the northwest Atlantic.

Track of Franklin.

Tropical Storm Emily (2011)

2011-08-02T07:03:00.013-04:00

Storm Active: August 1-7

An intense tropical wave left the African coast during the last week of July, producing a large area of scattered showers and thunderstorms as it moved westward. On July 28, a low pressure center formed on the south side of the system. The low continued to increase in organization, and on July 31, the circulation became associated with the convection in a pronounced swirl of clouds. However, the system was not yet closed, and development was delayed as the low approached the Windward Islands. During the evening of August 1, a burst of convection appeared west of the central low's previous position, accompanied by a circulation organized enough to name the system Tropical Storm Emily.

Due to the initial westward shift of the cyclone, its position at formation was west of the Windward Islands, in the Caribbean Sea. Emily's quick westward motion persisted overnight, bringing the system into the warm open waters of the Caribbean. However, conditions were not ideal for strengthening due to wind shear and dry air near the system, and Emily's center underwent a reformation on August 2, temporarily rendering it stationary. After shifting slightly to the north, the cyclone resumed its track and intensified somewhat as it approached Hispaniola.

Emily became disorganized during the morning of August 3, with the center of circulation becoming exposed and all convection being displaced to the southeast due to moderate wind shear. Despite these factors, Emily maintained its 50 mph intensity throughout the day, and scattered shower and thunderstorm activity reappeared near the center. As a result, heavy rain began to sweep across Haiti and the Dominican Republic. Late that night, the center redeveloped farther east, and Emily was once more temporarily stationary. A huge area of convection appeared about the center as Emily resumed a slow west-northwest motion, and powerful storms raged across Hispaniola early on August 4. The system remained offshore during that morning, but tropical storm force winds and rain covered a portion of the Dominican Republic.

During the afternoon, however, the circulation of Emily became entangled with the mountainous regions of Haiti, causing rapid weakening. By late that afternoon, despite never actually making landfall, all traces of organization were lost and Emily dissipated, leaving only a small trough of low pressure in its wake. The remains of the large area of convection that was formerly associated with Emily slid northward over Hispaniola and into the Bahamas overnight, but some thunderstorm activity reappeared near the trough on August 5, situated just to the south of the eastern tip of Cuba. This activity expanded northward during the day as a low pressure reformed just east of Florida. On August 6, the system became more organized, as the low connected to the new convection, and the low once again became a tropical depression, again being named Emily.

The depression became slightly better organized during that evening, but turned more to the northeast overnight, and the circulation became exposed on August 7 as the center was whisked away from the U.S. east coast. Emily continued to lose organization, and during the afternoon became a remnant low, losing tropical cyclone status once again. Over the next day, the low deepened, and briefly concentrated convection near the center, but it did not exhibit sufficient tropical characteristics to reform. The system was monitored for further development through August 11, but no change occurred, and the remnant low was finally absorbed by a front later that day. 5 deaths and at least $5 million in damages are attributed to Emily.

Emily near its peak intensity of 50 mph winds and a central pressure of 1003 mb.

Track of Emily.

Tropical Storm Don (2011)

2011-07-28T06:45:00.007-04:00

Storm Active: July 27-30

Around July 17, a vigorous tropical wave emerged off of Africa, quickly associating convection with it as it tracked over the open Atlantic. On July 22, it began to affect the Lesser Antilles with areas of heavy rain, and briefly developed a low pressure center. However, a small area of unfavorable wind shear passed over the system, and it remained disorganized. Development was hampered further on July 24, as the wave passed directly over the Dominican Republic and Haiti. During the next few days, this was followed by interactions with Jamaica and Cuba, which kept thunderstorm activity to a minimum. However, on July 26 the wave moved over the waters of the Caribbean south of Cuba. Slow organization occurred over the next day, and by July 27, a low pressure center had formed. During that morning, however, the low lacked a closed circulation and the convection was divided into two hemispheres, with the area of low pressure not directly associated with any one part of the system. Finally, during the afternoon of July 27, a circulation became evident at the northern tip of the system's western half, and the low was upgraded to Tropical Storm Don just north of the Yucatan Peninsula.

A ridge of high pressure to the northeast of Don steered it in a generally northwestward course through the Gulf of Mexico. Over the next day, convection increased, particularly on the southern side of the circulation, and modest strengthening occurred despite shear and dry air from the north. On July 29, the ridge became stronger, and turned Don more to the west-northwest, toward southern Texas. During that day, Don reached its peak intensity of 50 mph and a minimum pressure of 997 mb. The presence of Don generated significant tropical moisture along the northwestern Gulf coast, including Louisiana and parts of Texas. However, as the system made landfall, most rain was concentrated near the center and on the south side, and southernmost Texas therefore received the most rain. Tropical storm force winds enveloped a larger area of the coast, but quickly diminished as Don made landfall. Over land, the system weakened rapidly to a remnant low, causing no damage or fatalities.

Don near peak intensity shortly before landfall in Texas.

Track of Don.

Tropical Storm Cindy (2011)

2011-07-21T07:12:00.008-04:00

Storm Active: July 20-23

On July 19, a low pressure center formed along a stationary frontal boundary situated over the central Atlantic, in the vicinity of Bermuda. This front was the same one that spawned Tropical Storm Bret. The low quickly moved to the east, and deepened over the next day. On July 20, its circulation became increasingly disassociated with the frontal boundary, and the appearance of deep central convection was enough to name the system Tropical Storm Cindy that afternoon.

Even as it formed, it began to accelerate northeastward at speeds over 20 mph. Cindy strengthened rapidly as the circulation became better defined, developing a proto-eye feature early on July 21. The cyclone reached its peak winds, 70 mph, during that morning, and maintained its tropical characteristics through the day. The winds began to decrease as Cindy passed over the cold water north of the 40ºN parallel, but the pressure dropped from 1002 to 1000 mb. Additional drops in pressure despite negative changes in wind speed is often a symptom of a cyclone entering extratropical transition, which Cindy did on July 22. However, Cindy's convection was rapidly deteriorating by that afternoon, and the system continued to weaken, becoming a minimal tropical storm by late that night. Cindy was downgraded to a remnant low before actually becoming extratropical early on July 23. The remnant low of Cindy dissipated later the same day, causing no damage.

The low pressure system that formed Cindy mere hours before being classified as a tropical storm.

Track of Cindy.

Tropical Storm Bret (2011)

2011-07-18T07:12:00.011-04:00

Storm Active: July 17-22

Following the departure of a cold front from the U.S. east coast, a west-to-east situated stationary front stalled over the Florida coast and adjacent Atlantic waters. On July 16, a weak low pressure center formed in association with this front, and produced an area of showers and thunderstorms off of the eastern Florida coast. The pressure of the system remained high as it drifted slowly southward over the next day, but a clearly defined closed circulation formed during the afternoon of July 17, and the low was upgraded to Tropical Depression Two. Further deepening quickly followed, and the cyclone achieved tropical storm status a mere three hours later.

The newly formed Tropical Storm Bret experienced almost no motion overnight, drifting southward and then eastward, all the while meandering over the Northern Bahamas. Despite some wind shear out of the west that was bringing dry air into the system, convection persisted, and even developed during the morning of July 18, and the strengthening trend continued. Bret even developed a ragged eye amidst its tight circulation that evening, reaching a strong tropical storm intensity of 70 mph winds and a minimum pressure of 995 mb.

However, dry air penetrated the system late that night, causing weakening as the cyclone continued to move slowly north-northeast. As Bret paralleled the southern portion of the U.S. east coast early on July 19, nearly all convection was lost, and the system packed winds of only 50 mph. Some convection returned that morning, and Bret managed to maintain its intensity despite increasing shear from the northwest. With the exception of the southeastern quadrant, which contained some cloud cover, but even it was struggling, Bret had no associated convection whatsoever, and was essentially a bare circulation. Bret moved northeast through the day, and even into July 20 maintained the same intensity, despite very adverse conditions.

The storm finally began to weaken again that evening as it moved over slightly cooler waters, becoming a minimal tropical storm by July 21. Late that night, the storm was downgraded to a tropical depression while located between the Outer Banks of North Carolina and Bermuda. It also underwent significant acceleration to the northeast, reaching speeds over 20 mph. Yet despite the lack of convection, the system persisted as a tropical cyclone through most of July 22, and finally degenerated into a remnant low that afternoon. Bret's only effects were scattered storms and gusty winds in the northern Bahamas.

Bret at peak intensity just northeast of the Bahamas.

Track of Bret.

Tropical Storm Arlene (2011)

2011-06-29T09:22:00.014-04:00

Storm Active: June 28-30

Arlene originated from one of many tropical waves emerging off of the west coast of Africa during late June. On June 25, this particular wave, at the time moving onto the Yucatan Peninsula, began to interact with a broad trough of low pressure over Central America and the waters to the north. This interaction generated an area of scattered showers and extending westward from the wave itself into the western Caribbean Sea. The next day, on June 26, the wave moved over the central Yucatan, and thunderstorm activity concentrated along it, extending from north to south. On June 27, the wave emerged into the Bay of Campeche, and immediately rain bands began to form about a low pressure center in the southeasternmost area of the Bay, just north of the Mexican coast. The low tracked slowly west-northwestward and strengthened, but upper level winds were not yet favorable, and the circulation was not yet well-defined. However, wind shear diminished further on June 28, and the low was upgraded to Tropical Storm Arlene that evening.

Arlene maintained a slow west-northwest motion overnight, and the rain bands, formerly being sparse, quickly increased in convection early on June 29. The cyclone, which had only minimal tropical storm strength up to this point, intensified as it turned more to the west. By later that day, rain and wind began to sweep across the Mexican coast. Despite its proximity to land, Arlene's winds continued to increase, and the cyclone reached its peak intensity of 65 mph winds and a minimum pressure of 993 mb just before landfall near Cabo Rojo, Mexico. Before losing its water supply, which provided fuel for Arlene's convection, it also developed an eye-like feature.

After landfall early on June 30, Arlene began to quickly weaken over land, becoming a tropical depression that afternoon, and dissipating that evening. The remnants of Arlene caused rain in Mexico for an additional day before moving west into cold Pacific waters. The moisture from Arlene, although causing significant flooding in parts of northeastern Mexico, had a more positive effect on areas of Texas. In that state, numerous thunderstorm activity was generated by the trough associated with Arlene, temporarily relieving drought conditions. 25 fatalities, including 11 direct and 14 indirect, are associated with Arlene.

Arlene near peak intensity shortly after landfall in Mexico. Although a strong tropical storm, Arlene's center (and eye feature) are still not well defined.

Track of Arlene.

Professor Quibb's Picks-2011

2011-05-24T08:50:00.001-04:00

My personal prediction for the 2011 Atlantic Hurricane Season (written May 15, 2011):

20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status
10 cyclones attaining hurricane status
6 cyclones attaining major hurricane status

These predictions are far above the average activity in the Atlantic basin. Several factors contribute to why I have made such a choice. First, the decade of 2000-2009 had shown far above average activity, including the record of most tropical cyclones in a single year (2005, 28 cyclones attaining tropical storm status). This general trend shows no sign of stopping as we enter the 2010's. Second, an ongoing La Nina event that contributed to the 20 tropical storms of the 2010 season is still active, reducing the amount of wind shear present over the Atlantic basin.

Also, I have made the number of hurricanes and major hurricanes quite high in relation to the overall number of cyclones. Last year, a large trough over the Gulf of Mexico prevented storms from tracking all the way through the Caribbean Sea and into the Gulf of Mexico, instead steering them into Central America. This inhibited their strengthening potential, and most landfalling systems were relatively weak. The only major hurricanes of the season meandered out in the open Atlantic. So far this year, there have been a fairly persistent US east coast high pressure systems, and these may serve to steer cyclones on more southward tracks, into the Gulf of Mexico. The Gulf is home to some of the Atlantic Basin's highest ocean temperatures, and in it is the potential for rapid intensification.

Finally, the tendency this year may be toward slower moving and longer lived systems, as the preliminary climatological signs point to weaker upper level steering systems, and the above argument appears to favor longer tracks over water. There is a great deal of uncertainty in slow moving cyclones, and, accordingly, there are many variables to be accounted for in the coming season. The 2011 seasons has the potential to be very active and damaging, but only time will tell whether this is actually the case.

The 2011 Atlantic Hurricane Season will officially begin on June 1, 2011.

Hurricane Names List-2011

2011-05-16T14:45:00.000-04:00

For the Atlantic Basin, the hurricane names list for 2011 is as follows:

Arlene
Bret
Cindy
Don
Emily
Franklin
Gert
Harvey
Irene
Jose
Katia
Lee
Maria
Nate
Ophelia
Philippe
Rina
Sean
Tammy
Vince
Whitney

This list is the same as that of the 2005 Atlantic Hurricane Season, with the exceptions of Don, Katia, Rina, Sean, and Whitney, which replaced Dean, Katrina, Rita, Stan, and Wilma, respectively, as the latter were retired from the circulating names list in the same year.

Manifolds: The Shape of the Universe III

2011-05-08T06:39:00.001-04:00

This is the final post of the Manifolds Series and the third concerning The Shape of the Universe (see the first of the entire series, or the first concerning The Shape of the Universe).

It was previously discussed that the constant Ω represents the ratio of the Universe's actual density to the so-called critical density which makes the Universe Euclidean. As of yet, the most accurate observation of the density of the observable Universe yields an Ω of 1.02, with a possible error of just over .02. This suggests that the Universe is most likely to be elliptic. However, none of the geometries can be eliminated yet, and other clues will most likely be required to definitively determine its shape.

At the moment, it seems that a Universe with an edge can be ruled out definitively, as the presence of boundary would disturb the isotropy (similarity of the view in each direction from any point in the Universe) which seems to be necessary for a Universe of constant density to form. Since this is most likely the case, a Universe with an edge can be ignored.

This leaves two possibilities. Either the Universe is infinite (no identical image points) or it is finite (with at least one image point in the night sky) but without boundary. The location and distance of these images would determine the shape of the Universe.

Take, for instance, that the Ω value is exactly 1.02, as predicted by current measurements. This implies an elliptic Universe. If the Universe is a 3-sphere, the leading theory for an elliptic structure, then a value of 1.02 would imply a radius of 98 billion light-years, meaning that, if one was to look 98 billion light-years into space, they would see the same view in every direction, namely the diametrically opposite pole. In other words, an expanding ball of space in the Universe would first intersect itself when it reaches this radius, known as the injectivity radius for the given Universe. These similar images in all directions would be very easy to spot, as they would be of the same distance, and therefore the same age.

However, for this particular value of Ω, these images are beyond our ability to see. The Universe is only about 13.7 billion years old, and we can therefore only see that far. Despite this, the most distant objects are seen as they were billions of years ago, and they actually have moved farther away since then. Extrapolating backward from the current rate of expansion, one finds that our observable Universe actually measures 46 billion light-years in radius. Although this is a significant portion of the previously discussed 3-sphere, it is by no means enough to identify the shape of the Universe through images.

One encounters interesting phenomena if the speed of light is allowed to reach infinity in an idealized Universe, making it so that multiple images could be detected. If this was the case, the spacing and distance of images, and shape of the "shells" of the images would identify the manifold. Consider the example below.

In this particular case, images of the Earth can be seen in all directions, and all of the same age and appearance. This is because the speed of light is supposed to be infinite, and the light from all images instantly reaches Earth. Each exists at the center of a dodecahedral "cell", each of which, on its own, represents the entirety of the Universe. The other cells outside of the center one are images. The manifold in question is known as the Poincare dodecahedral space. This is an elliptic manifold with properties similar to that of the 3-sphere. Its construction is shown in detail below.

The fundamental polyhedron for the Poincare dodecahedral space is a dodecahedron, hence the name. This polyhedron has twelve faces, so each face can be connected to the one opposite from it. However, to pair any face with its opposite requires a rotation of one of the faces, for they are not in the same orientation, despite being the same size. The faces are pentagons, and the opposite ones are misaligned by a 1/10 turn. Therefore, by first rotating the indicated face A counterclockwise by a 1/10 turn so that the A1 edges line up, one can connect the faces. This procedure is repeated for all of the pairs of opposite faces. Note that a 3/10 or 5/10 turn produces a completely different manifold so choices of rotation, as well as the choice pairs of attachment, are crucial to determining a manifold.

With this construction in mind, further insight can be gained into the Poincare dodecahedral Universe shown above. It is now clear why the cells are dodecahedra, and that the distortion of these cells is by nature of the manifold being elliptic, as it does not "fit" into Euclidean three-dimensional space without distortion. Second possibly only to the 3-sphere, the Poincare dodecahedral space has the most following of any theory for the shape of an elliptic Universe.

For other geometries and manifolds, the image-finding method is even more difficult than that of the 3-sphere case. For a hyperbolic manifold of given curvature, the deviation of the Ω value from 1 produces a higher injectivity radius then an elliptic manifold of the same deviation, and Euclidean manifolds have images that are spaced unevenly and are at many different distances. (see also the discussion of the 3-torus Universe, found here)

Finally, observations of how forces, particularly gravity, affect objects may be helpful in determining the Universe's shape. At (relatively) small scales, when comparing stars, galaxies, and even superclusters, the gravitational pull of these massive objects distorts the local geometry of the Universe. However, when one considers the entire observable Universe, gravity's effect assumes a more uniform state.

To begin an analysis on how gravity's effect is determined by the shape of the Universe, is is useful to consider the mass distribution in its very early stages. The best source for this information is in the Cosmic Microwave Background Radiation. This radiation was emitted approximately 380,000 years after the Big Bang, by the matter present at the time, and, even at that stage, there were slight discrepancies in density and therefore temperature that gravity slowly molded into the structures we see today. The above image is of the temperature variances in this plasma, the precursor of all that we know in the Universe.

But exactly how did this process occur? Different universes and the gravitational differences between them have been analyzed in previous posts, but many of these properties were concerned with the effects of gravity traveling all around the Universe, and if it is of sufficient radius, these effects are not visible. Also, the most solid evidence thus far points to a lack of both images and these effects points to a Universe of very little curvature, if any. The local properties of our Universe closely resemble an infinite flat one, with just a hint of positive curvature.

In conclusion, the Universe is most likely to be elliptic, in the form of a 3-sphere, as this is the simplest of 3-manifolds, and it is not known how early Universe phenomena could have contributed to turning the Universe into a more complicated manifold, such as the Poincare dodecahedral space. The density measurements, with image and gravity evidence taken in mind yield an Ω probably between 1.01 and 1.02. The radius of the Universe is therefore very large, possibly over 100 billion light-years, and since this figure is constantly increasing, it is unlikely that the shape of the Universe can ever be determined through the image method alone.

The study of manifolds and topology is a broad and insightful area of mathematics that the above series of posts has only touched upon. The potential of manifolds in projection, mappings, the abstract and elegant constructions, and many other aspects of manifolds makes it an important area of study, which may even reveal what type of Universe we live in.

Sources: http://www.ams.org/notices/200406/fea-weeks.pdf, http://en.wikipedia.org/wiki/Homology_sphere#Poincar.C3.A9_homology_sphere,

Manifolds: The Shape of the Universe II

2011-04-30T15:22:00.006-04:00

This post is the penultimate segment of the Manifolds Series, and the second part concerning the Shape of the Universe. For the first, see here. For the first post of the entire series, see here.

The 3-torus theory of the Universe is relatively simple and elegant, but it is not the only candidate for the shape. The 3-torus represented finite Euclidean geometry in the debate for the Universe's global topology. This is because the eight corners of the cube eventually coincide when the faces are connected. It is clear that laying out eight corners "fills up" Euclidean 3-space. To see this, consider the 3-dimensional linear coordinate system.

It has three axes, and splits space up into eight sections. At the origin, (the point of coincidence) the corner of each region is the corner of a cube. For more information about 3-dimensional angles (known as solid angles) see the beginning of Polytopes: Part III.

Of course, it is always possible that the Universe is simply infinite, and that it has no notable global topology. However, it is more logical, since the Universe was very probably at a finite size at some point in time, that it remains of measurable size. However, the curvature is not known for sure, and representatives for finite elliptic and hyperbolic geometry exist as well.

If the Universe is elliptic (a perspective which would have the Universe reversing in its expansion at some time in the future) it may be in the form of a 3-sphere, the simplest of elliptic 3-manifolds. Extending off of the common 2-sphere in three dimensions, the 3-sphere is the set of points in Euclidean four-dimensional space that are equidistant from a given fixed point. Its construction can be visualized as follows.

It was discussed perviously that attaching the boundary of one disc to another results in the 2-sphere. Going up a dimension, the same goes for the 3-sphere. Two balls (solid spheres) have their boundaries attached in a one-to-one correspondence (as indicated by the arrows) and the resulting manifold is a 3-sphere, although the process itself cannot be visualized in 3-dimensional Euclidean space.

To imagine traveling through this space, visualize each ball as a set of concentric spheres. Starting at the center of the left ball, one would walk outward until reaching the boundary of the left ball, which, after the 3-sphere is constructed, is the same as the boundary of the right ball. One would then continue to walk in the same direction, reaching the center of the right ball. After that, the process would then reverse, and one would cross the boundary again, this time back into the left ball. It follows from the above construction that the centers of each ball become a pair of poles on the 3-sphere, diametrically opposite from each other.

Using the 2-sphere as an analog to how gravity works in this Universe, one can easily see that gravitational waves travel as arcs of great circles of the sphere. Unlike the torus, only two arcs (the major and minor arcs of a given great circle) connect two points, with an exception if the points are polar opposites, when an infinite number of gravitational rays connect two points. Therefore, the opposite pole is the "hot spot" for this manifold, where the net gravitational force is zero. In addition, due to the presence of the major arc component, the amount of gravity between two points in one direction is less then it "should" be, as the major arc component is subtracted (being in the opposite direction). These results are summarized in the figure below.

In the above figure, the blue object attracts the green object (which has negligible mass) with a force equal to the minor arc gravitational pull minus the major arc gravitational pull in the opposite direction. These gravity vectors emanating from the blue object are the only two that intersect the green object, if both objects are treated as points. Again, this is similar to the sphere, where all pairs of points with the exception of anti-polar pairs have exactly two geodesics connecting them.

Finally, it is possible that Universe is hyperbolic. The leading theory for a hyperbolic Universe is known as the Picard horn. The two-dimensional analog for this manifold is the pseudosphere:

This 2-manifold is infinite in extent, but, remarkably, has finite surface area and finite volume. As an interesting addendum, the surface area of the psuedosphere is equal to that of a sphere of the same radius. The geodesics on this manifold are called tractrices, circles, and rotating tractrices, all of which are illustrated below (click to enlarge).

The view above is actually of the half-psuedosphere, and it is often used to represent a two-dimensional hyperbolic plane. A point on this manifold can be identified by its height off the base, and the angle around the central axis. The geodesic of constant height is the circle, the geodesic of constant angle is the tractrix, and every other geodesic has a change in height proportional to a change in angle, in other words, a linear function of the angle dependent on the height. This general geodesic is a rotating tractrix, and can (as shown above) travel around the entire pseudosphere any number of times.

If two points do not lie at the same height on the pseudosphere, then there are an infinite number of rotating tractrices connecting them. Again taking these to be gravitational waves, the "hot spots" of net zero force are the points 180º separated (on opposite sides) but at the same height. If two points are 180º separated but are not at the same height, then the net gravitational force would be to decrease their separation in height. These and other properties are summarized below.

The properties of the pseudosphere Universe are similar to that of the torus Universe, with the excpetion that there is only one class of non-contractible loops on the surface, (cricles) wheareas a torus has two: one going around the ring, and the other around the hole in the center. Therefore, as shown above, gravitational rays from the blue object to a higher one, namely the red, can only approach it from below, as opposed to the torus, where gravitational rays could approach from all directions.

The true hyperbolic plane is in some ways different from the psuedosphere, but it serves well as an example, and the three dimensional equivalent is notable for having finite volume, and a Universe of this type would also be finite, despite (again) being infinite in extent.

The above three possibilities are among the most prominent theories for the shape of the Universe. But which of these reflects the current visual evidence? This is the topic of the final post of the Manifolds Series.

Sources: http://en.wikipedia.org/wiki/Shape_of_the_Universe, The Poincare Conjecture by Donal O'Shea, http://www.ams.org/notices/200406/fea-weeks.pdf

Manifolds: The Shape of the Universe

2011-04-22T11:03:00.015-04:00

This post is part of The Manifolds Series.

By use of maps, the surface of the Earth can be definitively defined as a sphere. However, if one only specifies that going in any direction on the Earth will eventually return one to his or her starting point, then many different manifolds qualify. The Earth could have just as easily been a torus, or any other finite manifold without an edge. Only by confirming the curvature within several different "patches" of a manifold can one uniquely determine it.

The same problem exists with the Universe. However, it is a rather more difficult one.

A common misconception concerning the shape of the Universe is that if it is finite, it has an edge or boundary. This is not true. On a 3-sphere, (recall that the surface of the Earth is a 2-sphere) one could go indefinitely through 3-dimensional space in one direction and never reach an edge, although he or she may return to his or her starting point. In fact, the Universe's lack of an edge is mostly agreed upon, as there would be disagreements in density and isotropy that make an edge unlikely.

Also, the curvature of the Universe can be determined to a fairly accurate degree by measuring its density. The density of matter in the Universe determines how fast it continues to expand. This is because the amount of gravity counteracting the expansion of the Universe is dependent on the amount of matter and energy present, and this in turn, determines whether the Universe is expanding, and at what rate. Since the presence of matter also determines how space is bent, the local curvature of the Universe around the aforementioned matter can be calculated as spherical, Euclidean, or hyperbolic.

Throughout all of the Universe that we can see (the observable Universe) the matter seems evenly distributed at a sufficiently high scale. Obviously, on (relatively) small scales, there is a large difference in density between stars and the interstellar medium, and between galaxies and the intergalactic void, but when one considers density on the scale of billions of light-years, the density is remarkably uniform. From this, one can assume that the curvature of the Universe is constant.

Additionally, observations up to the present have suggested that the density of the Universe is very close to the so called "critical density", at which the curvature of the Universe would be exactly 0. The constant Ω, known as the density parameter, representing the ratio of the actual density to the critical density, would then be 1 for a flat Universe, greater than 1 for a spherical Universe, and less than 1 for a hyperbolic one.

As a caveat to assumptions about the finite or infinite nature of the Universe, note that the curvature of the Universe does not determine its global structure topologically. Many people assume that if the Universe has flat geometry, it must be infinite, just as the flat Euclidean plane is. However, it was noted in previous posts that the torus has Euclidean geometry as well.

In fact, when one considers 3-manifolds, there are 10 possibilities for finite Euclidean 3-manifolds alone, the most simple of which is the 3-torus.

The fundamental cube for the 3-torus. Imagine that the upper right end face of the cube is the front. Each face is connected with the opposite one (front to back, left to right, top to bottom) while preserving the orientation of faces. This means that the edge C₁ is attached (facing up) to the corresponding edge also labeled C₁ on the opposite C face.

Many telltale signs would exist if the Universe was a 3-torus. Using the 2-torus (previously known as simply the torus) as an analog, one can explore the remarkable phenomena of a toroidal Universe. To do this, consider life forms occupying the surface of a 2-torus, and then consider the 3-dimensional extension.

The first of these is the closed and finite nature of this Universe. An inhabitant of the 2-torus would see another copy of himself looking around the top of the donut! (see below)

Inhabitants of this Universe would see an image of themselves by looking in the direction indicated by the arrow. The light itself actually leaves the other side of the blue oval, and travels along the orange path, reaching the blue oval again after one revolution. One might argue that the curvature of the torus would obscure the view, but this is not true, as the Universe is the surface of the torus, and light can only travel along this surface. Therefore, light would traverse a closed circle in certain directions. In fact, the number of these directions is infinite!

By looking around the "ring" of the torus, inhabitants of the blue oval might also see themselves. The same goes for an observer looking diagonally, where the light would circle around the bottom as it goes around, any number of times! Infinite images of the same blue oval would exist in their "sky"!

Extending this system to our Universe, the 3-torus shape could easily be identified by images of our galaxy, the Milky Way, in the night sky, right? Unfortunately, it isn't that easy. First, the sheer size of the Universe may be so large that even the closest image point may be many billions of light years away-beyond the scope of the observable Universe. And even if there was an image point within our view, we couldn't easily identify it, as it would be an image of the galaxy from billions of years ago! Although we may not know it, a galaxy looked upon by the Hubble could possibly be our own, simply at a different stage of evolution!

How then, does a donut Universe leave its mark? The signs may be more subtle, but they are there. For example, consider a point in space emitting light uniformly in all directions. Assuming that the intensity of the light dies away over distance, one would expect that the brightness at a constant distance would remain constant. This is not the case.

Returning to our above example, light rays would converge on the opposite side of the torus, and in multiple places in between. By observing from each and every point on the torus, one could construct a contour diagram of the brightness compared to what would otherwise be expected. Judging from their distance from the source, some points receive more light than they "should" in a generic flat, infinite Universe. The two points that deviate most are the diametrically opposite point on the torus, and the point on the bottom of the ring. Other points would be intermediately shaded. These "hot spots" are indicative of a donut Universe, but there are too many light sources for the specific example above to take effect.

Finally, perhaps the most important phenomenon is the discrepancies in gravity that would occur. Treating gravity waves similar to the light example above, one can clearly see that an object, which exerts gravity on its surroundings uniformly, would exert more gravity on some point in the toroidal Universe than others. The only difference here is that direction matters. For an object on the exact opposite side of the torus Universe, the gravity waves converge from all directions in a symmetrical way, adding up to a net zero force. For an object in the vicinity of the original object, the convergence of gravitational waves adds more attraction to the object than one would normally expect given the distance for an infinite Universe. These results are summarized in the figure below with the two-dimensional analogs of both the flat and the toroidal Universe. (click to enlarge)

In both figures, the dotted lines represent gravitational attraction. It is assumed for simplicity that the blue objects are the only bodies exerting gravity, and that the remainder of the objects are of negligible mass. In the flat Universe, only one line connects two points, but on the torus, multiple lines between two points are a result of the finite cyclic nature of the manifold; a line going around the manifold will come back to the vicinity of its original position.

In the next post of the Manifolds Series, other theories are considered.

Sources: http://en.wikipedia.org/wiki/Shape_of_the_Universe, The Road to Reality by Roger Penrose, The Poincare Conjecture by Donal O'Shea, http://www.astronomy.ohio-state.edu/~ryden/ast162_9/notes40.html, http://www.math.brown.edu/~banchoff/STG/ma8/papers/leckstein/Cosmo/torus.html

Manifolds: Curvature and Construction II

2011-04-14T15:00:00.004-04:00

This post is part of the Manifolds series. For the previous post in the series, see Manifolds: Curvature and Construction.

In the previous post, fundamental polygons were introduced. A further exploration of these figures is important in constructing manifolds. However, we find that there are limits to the accuracy to which even two-dimensional manifolds can be represented in three-dimensional space. Take the torus for example. It was found to have a fundamental polygon ABA*B*. All four corners of the four-sided fundamental polygon eventually coincide when the torus is constructed. As a result, the angular measure around the resulting point on the manifold is exactly 360º, and the manifold is Euclidean at that point.

This result can be expanded by considering the symmetry of the torus. In fact, any point can be chosen as the coincidence of the four corners. This means that the geometry around every point on a torus is perfectly flat.

However, an examination of the torus in three-space (above) suggests that this is not the case. What is happening here? The answer turns out to be that the folding of the cylinder to the torus distorts the final result, resulting in a manifold with areas of positive and negative curvature. The torus in its true form can only be expressed in four dimensions and higher, and is called a flat torus. Note that the manifold is not truly "flat", but can be constructed in four dimensions from a flat plane without distorting distances.

Similarly, the sphere (fundamental polygon ABB*A*) connects only two of opposite the corners together. The local geometry of a point is less than 360º, and the sphere therefore exhibits spherical geometry, an obvious fact.

The lapse in the accurate representation of 2-manifolds in 3-space is even more pronounced when one considers the fundamental polygon below.

This fundamental polygon can also be written ABAB*. When constructed, this fundamental polygon takes the form of a manifold known as the Klein Bottle. The first step in construction is the same as the torus, i.e. connecting the sides marked B and forming a cylinder. However, the A sides are facing in opposite directions and the cylinder must be inverted through itself to construct the manifold. This construction is shown below.

The final product is the Klein Bottle.

It is apparent that self-intersection is necessary in order to construct this manifold in three dimensions. As a result, this too gives only a limited view of its actual structure. The addition of another dimension is needed to eliminate the self-intersection, and to show that this manifold too has Euclidean geometry everywhere.

Yet another fundamental polygon ABAB, produces another manifold that cannot be constructed in three dimensions without self-intersection. It is known as the real projective plane. It has many interesting properties, including its geometric construction. It is the most "difficult" to construct out of the fundamental squares because both pairs of opposite sides are orientated in different directions. Geometrically speaking, it is the set of all lines through the origin in Euclidean 3-space connected into a surface, although this gives little insight to its shape.

More complicated manifolds can be constructed from larger fundamental polygons (all with an even number of sides, of course, because one side must line up with another). For example, consider the hexagon ABCC*B*A* (shown below).

This fundamental polygon produces the sphere, just as the ABB*A* does, since only a single diagonal fold is necessary to connect all of the indicated sides. Similarly, ABCB*A*C* produces the torus, ABCB*AC* the Klein Bottle, and ABCBAC, the real projective plane. These are all merely extensions of the fundamental squares, as two sides facing the same direction can be collapsed into one. Take for example the torus, ABCB*A*C*.

Since continuous deformation still preserves the identity of a manifold, the same goes for its fundamental polygon. The above hexagon can therefore be adjusted into a rectangle. The sides A and B are in the same orientation and position relative to each other, they can be combined into a single side while still representing the same idea. The other examples above can be similarly collapsed into fundamental squares.

Fundamental polygon formulas also exist to generate the simplest surface of any given genus, i.e. the n-fold torus. For the two-holed torus, the fundamental polygon is the octagon ABA*B*CDC*D*, which under inspection, is simply two tori fundamental squares "added" together, corresponding to the "gluing" of two tori together to create the double torus. This is equal to ABA*B*+CDC*D*. All eight angles of the polygon coincide where the two tori share a boundary, and the double torus is therefore a hyperbolic manifold. (each angle of a octagon=135º, 8*135º=1080º>360º) With continuous deformation, the point of coincidence can be moved to any point on a double torus, and the manifold therefore has hyperbolic geometry anywhere.

As one adds more tori to the fundamental polygon, the manifold becomes increasingly distorted. It fact, it is "difficult" to draw an n-fold torus with more than three holes in Euclidean 3-space. The general formula is

ABA*B*A'B'A'*B'*A''B''A''*B''*...

which produces an n-fold torus, with each sequence of four letters (ABA*B*, A'B'A'*B'*, A''B''A''*B''*, etc.) represents a single torus that is glued to all others.

As with many aspects of manifolds, the idea of a fundamental polygon may be extended to higher dimensions. For 3-manifolds, fundamental polyhedra take on this role. For example, one might specify a manifold as the resulting figure when opposite faces of a cube are connected. Such connections cannot be visualized in Euclidean 3-space.

Finally, since deformation is permitted, even differently shaped n-faces of any closed polytope of any dimension can be connected to form a manifold. Untold multitudes of manifolds can be produced this way.

The next post of the Manifolds Series deals with the application of manifolds to the shape of the Universe.

Sources: http://en.wikipedia.org/wiki/Fundamental_polygon, The Poincare Conjecture by Donal O'Shea

Manifolds: Curvature and Construction

2011-04-06T08:22:00.008-04:00

This is part of a series on Manifolds. For more information about subjects mentioned, see The Complete Manifolds Series.

On the previous post, it became possible to define unique points on a manifold by the use of coordinates. Consequently, the connecting of these points forms lines.

However, if one insists on staying on the surface of a manifold, the lines will not necessarily be straight. Take the sphere for example. Any lines drawn on the surface will inevitably be curved. But what does curvature mean? And how does one choose the single line that connects two points? The answer is a simple one.

On a surface, the line between two points is the shortest curve that connects them. On a plane, these lines are straight. On a sphere, the lines are arcs, part of great circles whose radius is that of the sphere itself (see below). On a general manifold, the shortest path between two points is known as a geodesic.

Each circle above is defined by a plane passing through a sphere. If the plane passes through the center of the sphere, the resulting circle is a great circle. Otherwise, it is not. As a result, all longitude lines are great circles, but, among latitude lines, only the Equator is.

After identifying the lines on a sphere, one can continue on to produce geometric figures. Take the triangle for example. It is defined as the (minor) area of the sphere contained by three geodesics on the surface. With a little thought, however, triangles on this surface can be identified to have angular measures greater than 180º. Begin at the north pole of the sphere. The 0º and 90º W longitude lines both emanate from this point, and are by definition, perpendicular. These two lines both intersect the Equator at right angles. (longitude and latitude lines are always perpendicular) These are all clearly geodesics, and each pair contains a right angle. Consequently, the total angular measure of the triangles is 270º! The variation from 180º of the angular measure of a triangle on a manifold is known as the angular excess. A basic discussion of angular excess and resulting curvature can be found here.

In the previous post in this series, the construction of manifolds through coordinate systems was discussed. Although some interesting cases are found through this method, it only produces a small number of manifolds. Another method of construction is through the connection of boundaries. A manifold is said to have a boundary if it has an edge.

For example, the circle does not have a boundary, but the disc does, and the sphere does not have a boundary, but the ball does. Also, the plane is infinite in all directions, and therefore doesn't have a boundary.

However, by connecting the boundaries of two discs, as shown above, one obtains a sphere. (remember that bending and stretching is allowed in a mapping, but no breaking) Also, it is clear that connecting two manifolds preserves their dimension, but turns two discs (manifolds with boundary) to a single sphere (a manifold without boundary).

For two-dimensional manifolds, cuts and folds can be made on the surface to reduce the manifold to a polygon with an even number of sides. This polygon is known as the fundamental polygon.

As an example, consider the torus. By making two cuts, it can reduced to a rectangle.

In the first step, the cut goes through the torus and creates two ends. The resulting figure can be straightened out into a cylinder.

A second cut is made parallel to the surface to the cylinder along the plane indicated. The cylinder can then be unrolled into a rectangle. The result of this process is summarized in the following diagram.

The fundamental polygon for the torus. It has four sides, and can therefore be stretched into a square continuously, as it is shown here. The orientation and lettering of the sides indicates which way the boundaries of the square are connected to produce the manifold. The sides with the same letter are connected to each other so in such a way that the arrows face the same direction. Here the top and bottom are connected, and then the left and right. It is easy to see that this is the reverse of the process that we used to decompose the torus.

The same process can be applied to the sphere. The fundamental polygon for the sphere is again a square, but it is connected in a rather different way.

The above cut is made before "flattening" the sphere into the plane figure below.

This fundamental polygon has the same structure as that of the torus, but is connected differently. The points of each arrow must line up in the end result. With a square piece of paper, one only has to fold diagonally to line up the indicated arrows. The figure is then "inflated" to produce the sphere.

The fundamental polygons can be written as follows. Going in a clockwise direction from the upper left corner, the fundamental polygon for the torus is written ABA*B* and, for the sphere, ABB*A* (the * in each case represents an arrow pointing in the counterclockwise direction).

Next, it is possible to extend this system to other two-manifolds, and even to other dimensions. A closer examination of these figures gives insight into curvature as well.

Sources:
http://whites-geometry-wiki.wikispaces.com, http://svr225.stepx.com:3388/sphere, http://en.wikipedia.org/wiki/Fundamental_polygon

Manifolds: Coordinates

2011-03-29T15:01:00.006-04:00

This post is part of a series on Manifolds. Before reading this post, you may refer to the first and second parts of the series for more information.

Another way to define a unique point on a manifold is by the use of coordinate systems. Coordinate systems are collections of numbers (a1, a2... an) that tell one how to identify a point uniquely. There are a number of ways to do this, but within each and every system lies the dimension of the manifold, for the number of coordinates needed to define a point uniquely is the same as the number of dimensions of the manifold.

There are many possible coordinate systems, and some may be more effective on some manifolds than on others. Consider the circle, the simplest of all manifolds.

The circular coordinate system (click to enlarge). The angle "a" uniquely determines a point on the circle, and the circle is therefore a one-dimensional manifold. Note that, similar to the difference between the sphere and the ball, the circle is only the line itself, while the interior of the plane defined by the circle is known as a disc. a is taken to be an angle between 0 and 2π if each point on the circle is to have a unique coordinate.

A similar process works for the line, an infinite manifold where the (signed) distance from 0 defines a unique point. The above is also called the number line.

For two dimensions, one can combine distance and angle coordinates in several combinations.

For the plane, an infinite two-dimensional manifold, two distance coordinates, taken in perpendicular directions, uniquely define a point (see below). The system, usually using x and y for the directions of coordinates, or axes, is called the Cartesian system. Points are found by following the x-axis for a distance b, and then moving perpendicularly (parallel to the y-axis) some distance c. The point is labeled (b,c).

For the aforementioned disc, the angular and linear (distance) systems are combined.

The figure above is essentially equivalent to the circular system mentioned previously, with one exception. An additional coordinate, r, is added to denote the distance from the center O of the disc. In reality, the distance r can be arbitrarily large, and the disc therefore infinite, making the angular-linear or polar coordinate system work for the plane as well. However, the restriction of r to being less than or equal to the radius of the disc emphasizes the different construction of the polar system versus the Cartesian system, and how it builds off the circular coordinate system.

The final simple possibility is the double angular or spherical system, shown above. It expands off the angular system of the circle, constructing another plane perpendicular to the plane containing the circle. The angle of elevation off the plane serves as the second angle, θ, in addition to the original circular angle, φ.

For three dimensions, there are several more cases.

The first is the triple linear system, with three axes, x, y, and z. Distances in the direction of each axis are given to uniquely determine a point. (negative coordinates signify movement in the opposite direction)

The second possibility is a system with two linear coordinates and one angular coordinate. A manifold using this system is the solid cylinder, or rod. The rod system is based off the disc system, with the angular and first linear coordinates defining a disc, and the second linear coordinate defining distance up the rod. In the above views, the rod is not solid, and instead two circular cross sections are shown to emphasize the first linear dimension. If one was to disregard the first linear dimension, a normal cylinder would result, and this is therefore another possibility for the two dimensional angular system. Consequently, the cylinder is a 2-manifold, and the rod is a 3-manifold. Again, both linear dimensions are restricted to finite values for emphasis.

The system with two angular coordinates and one linear coordinate is the ball system, as the ball is the simplest manifold with this system. The ball, as previously mentioned, is simply a solid sphere, with identical angular dimensions. A (restricted) linear dimension is added to denote the distance from the center of the ball, uniquely identifying a point.

The final possibility is the notion of a system with three angular coordinates. Following the pattern of the circle and the sphere, this manifold should also be the set of points equidistant to a fixed point. The set of manifolds with this property are generally called n-spheres, where n is the number of dimensions in the manifold. The circle is then the 1-sphere, the sphere the 2-sphere, and this new manifold, the 3-sphere. However, the full geometry of the three-sphere cannot be expressed with three dimensions, and instead requires four to reveal its curvature.

This generation of manifolds can continue indefinitely through any dimension, but the above process only produces a limited number of manifolds, i.e. the simplest of every dimension. Also, there are multiple possibilities for each system, with the disc and cylinder being one example given above. There are many other similar pairs in higher dimensions when one changes the orientation of the linear and angular dimensions.

Coordinates are necessary in performing mathematical operations on manifolds, and in mapping between them. The differences between angular and linear coordinates, and the number of dimensions needed to express them open the gates to a limitless world of surfaces, which the above method has only begun to explore. For the next post, see here.

Sources: http://www.ibiblio.org/links/devmodules/shared/html/glossary.html, http://www.mathematic.ws/, http://en.wikipedia.org/wiki/Spherical_coordinate_system, Visual Complex Analysis by Tristan Needham,

Manifolds: Mappings and Projection

2011-03-21T15:05:00.005-04:00

To understand some terms used in this post, it is recommended that you first read Manifolds: Geometrically Equivalent vs. Topologically Equivalent.

The world of manifolds is extremely rich and diverse, especially when higher dimensions are considered. A multitude of objects can be explored, most of them fundamentally different from one another, and we are still very far away from identifying all of them, even in relatively low dimensions.

Note: Before exploring manifolds, it is useful to define "dimension" and what it means in terms of a manifold. Recall the definition of a manifold: any surface that appears "flat" at sufficiently small scale. However, "flat" is another word for Euclidean (or plane) geometry, and we can then define the dimension of a manifold as follows:

If a manifold resembles n-dimensional Euclidean space at sufficiently small scale, then it is an n-dimensional manifold.

As an example of this, consider a typical sphere, such as the Earth (the Earth is not exactly spherical, but is often used to represent the abstract mathematical sphere). To someone standing on the Earth, its surface appears to be a flat plane, i.e. two dimensional Euclidean geometry! Therefore, we conclude that the sphere is a two dimensional manifold, or 2-manifold.

Note: The term "sphere" only includes the surface of the sphere (or the Earth) and does not include the inside of the sphere. The region of three dimensional space bounded by the two dimensional sphere is known as the ball, and is a 3-manifold.

With the above clarification of terms, one notices that is easy to define the Earth as a 2-manifold. However, when one attempts to map the surface of the Earth on a flat surface, there are inevitable distortions.

As an example, consider the Mercator projection of the Earth.

When mapping the Earth onto a flat Euclidean plane, one must consider various geometrical properties of the sphere, including area, latitude and longitude lines, and lengths of these lines. A process called projection allows one to view a surface on a flat two dimensional plane. In the above case, a cylinder is used to project the Earth onto a plane (see image below).

For each point on the sphere, a line is drawn from the center of the sphere outward through the aforementioned point, and this line will eventually intersect with the cylinder (the cylinder does not actually have a "top" or a "bottom" but rather goes on forever). After all the points are mapped, the cylinder is unrolled into a plane, resulting in the fact that going off the right edge of the map goes to the left edge and vice versa. However, when one chooses a point near one of the poles, the line must go a large distance before intersecting the cylinder, and the poles themselves cannot be mapped at all! Despite these problems, this mapping is desirable for bearings, as it maps rhumb lines (or lines bearing in a specific direction, e.g. northwest, east southeast, etc.) and therefore also preserves angles.

Another (perhaps even simpler) useful projection for mapping the Earth (and especially mathematics) is the stereographic projection. It again maps the sphere onto a flat Euclidean plane, but in a different way. The figure below denotes this.

To project stereographically, a point of projection is first chosen. For the purposes of this example, the point of projection is always assumed to be the north pole. For each point on the sphere, a line is drawn from the north pole through this point, and is extended until it intersects the plane passing through the equator of the sphere. In mathematics, this is taken to be the complex plane (see here for a basic discussion of i and the complex plane) and the circle at which the sphere intersects the plane is taken to be the unit circle, i.e. the circle centered at 0 with radius 1.

The figure above shows two arbitrary points. Point A, on the plane outside the unit circle, corresponds to a point on the upper half of the sphere, while Point B, on the plane inside the unit circle, corresponds to a point on the lower half of the sphere. Points on the unit circle obviously remain in the same position. (1 and i are shown as examples)

Additional properties include the south pole of the sphere corresponds to the origin of the plane, and the north pole of the sphere does not correspond to ANY point on the normal plane. This is because the point of projection (the north pole) and the point to be projected (also the north pole) coincide, and the line is therefore a tangent line, which is parallel to the plane and never intersects it. The north pole is sometimes called infinity for this reason.

An image of the Earth using another type of stereographic projection where the plane of projection is tangent to the south pole of the sphere. In all other respects, this projection is similar to the one above; it still matches every point on the sphere with one point on the plane, with the exception of the north pole.

The stereographic projection has many properties that make it valuable to mathematics, the most important of which is that it preserves circles and angles (for a proof, see sources, specifically Needham). And since the points are projected continuously with a one-to-one correspondence, the mapping is a homeomorphism. Therefore, we can make the statement that

The Euclidean two dimensional plane is homeomorphic to the two dimensional sphere with one point removed.

The above reflects that the point of projection itself (the north pole in the above examples) cannot be projected onto the plane. In mathematics, it is possible to remedy this, by including infinity itself as a point on the complex plane, and the new entity that results is known as the extended complex plane. Infinity then corresponds to the north pole under stereographic projection. Note that the actual direction, in which we approach infinity does not matter, as all lines heading away from the origin will travel upwards on the sphere and eventually reach the north pole. In a sense, infinity is an endpoint of any straight line in the extended complex plane.

A generalized view on stereographic projection allows one to project from spheres of any dimension to their corresponding planes. For example, elliptic polychora, or four dimensional finite polytopes (see the polytopes series) are actually tilings of the three dimensional sphere which is the surface enclosing the four dimensional ball (this is the higher dimensional analog of the two dimensional sphere being a surface enclosing a three dimensional ball). Therefore, stereographic projection can be used in a similar way as the above to project such polychora onto flat Euclidean three space.

An example of a convex polychoron projected stereographically (specifically, it is the cantellated 24-cell, see here for more information)

Mappings and projection have revealed that the sphere and plane are fundamentally different, despite being members of the same dimension. Further posts in the manifolds series addresses this topic (see next post).

Sources: http://en.wikipedia.org/wiki/Stereographic_projection, Visual Complex Analysis by Tristan Needham, http://en.wikipedia.org/wiki/Mercator_projection

Manifolds: Geometrically Equivalent vs. Topologically Equivalent

2011-03-09T16:39:00.003-05:00

A manifold is the general term for a geometric figure, surface, or space. Manifolds can be of any dimension, and of are great importance in mapping, and in mathematics.

The study of the manipulation of surfaces is known as topology. It is very important to understand that geometry and topology, although both dealing with geometric figures, are very different. Geometry is concerned with sizes and shapes, i.e. measuring area, radius, perimeter, volume, and so forth. However, topology, which is more relevant to this post, does not worry about specific shapes. The following definitions are very important distinctions.

Geometrically equivalent: Two objects are geometrically equivalent if they have exactly the same size, shape and dimension. Figures with this property are called congruent. Another identical statement is that if two figures can be placed on top of each other to exactly line them up, then they are congruent.

Homotopic: Two objects are homotopic if they can be continuously deformed into one another. This means that the stretching, bending or twisting of an object does not alter it topologically. Two objects that can be continuous deformed into each other in this way are called homotopic to one another. Therefore, there exists what is called an invariant in the original surface. An invariant is defined in this sense as a feature of a manifold that remains the same when it is changed is some way. The specific example of this for two homotopic manifolds is called a homotopy group.

There are actually several different homotopy groups, each of which defines features of a manifold. The simplest of these is called the fundamental group, which determines the number of holes in a surface. The mug and the torus below both have one hole in their surface; they are homotopic and equivalent topologically.

Another way to express the same idea is to consider a point on a manifold, and, starting from that point, trace any path on the given manifold, with one condition: the endpoint and the starting point of the path must coincide. When this happens, the path is called a loop. If all possible loops can be contracted into a point without leaving the given surface, than the surface has no holes and is what is called simply connected.

A demonstration that the sphere is simply connected. The loop shown, along with any other loop beginning from any other point on the sphere, can be contracted without leaving the surface.

Other surfaces, such as the torus, do not share this property.

An image of a regular torus, with three example loops drawn on its surface. It can easily be seen that some loops, such as c can be contracted to a point without leaving the surface. Others, such as a and b, cannot. Additional analysis of the set of loops on a surface yields the number of holes, known as the genus of a surface.

The triple torus is a manifold with genus 3.

The remaining homotopy groups are, simply put, higher dimensional generalizations of this. For example, the 2nd homotopy group deals with the cutting of spheres out of a surface, unlike the fundamental or first group, which deals with the cutting of circles. Therefore, there are infinitely many homotopy groups possible, each of which corresponding to a specific dimension. The "loops" for the 2nd homotopy groups will be surfaces, rather than lines, that can be contracted to a point. In addition, there is a equivalent for any dimension. For any specific manifold, the homotopy groups will assign an invariant (or a set of invariants if multiple groups are used) that identifies it as homotopic to any other manifold with an equivalent invariant.

Homeomorphic: However, the actual definition of topologically equivalent is even broader than this. Two manifolds are topologically equivalent if they are homeomorphic. Note that if two manifolds are homeomorphic, they are homotopic, but the reverse is not necessarily true.

Two manifolds are homeomorphic if there exists some function that maps each point on one manifold to a corresponding point on the other.

An example of a simple mapping (click the image to enlarge) of the number line x (red) to the number line y=x^2 (blue). Each point on x is mapped to a corresponding point on y (only the points x=1 and x=2, becoming y=1 and y=4, respectively, are shown). The above lines are actually manifolds related by the mapping y=x^2!

However, for two manifolds related by a mapping to be homeomorphic, the mapping must satisfy certain conditions, listed below. For the mapping function f that relates a set of points X to a corresponding set of points Y:

The function f must be continuous.
The function f must be one-to-one.
The function f must be onto.
The inverse of f must be continuous.

Some definitions are in order:

Continuous: Literally that each change in input causes only a small change in output. In other words, there cannot be any "jumps" or discontinuities in the function.
One-to-one: If the function f maps a point a in X to a specific point b in Y, then a is the only value in X that f maps to b.
Onto: Each value in Y corresponds to a point in X.
Inverse: The inverse of a function f is the mapping that undoes f. In other words, f maps X to Y, and the inverse of f maps Y to X.

Armed with the ideas of homotopic, homeomorphic, and mapping, one can begin to explore the world of manifolds. (see the next post)

Sources: http://en.wikipedia.org/wiki/Homotopy_groups and other various wikipedia titles, Visual Complex Analysis by Tristan Needham, The Poincare Conjecture by Donal O'Shea, http://www.regentsprep.org/Regents/math/algtrig/ATP5/OntoFunctions.htm, http://media.web.britannica.com/eb-media/58/96258-004-7747AF96.jpg

Gamma Rays

2011-03-05T07:40:00.000-05:00

Gamma rays are the final type of radiation found on the electromagnetic spectrum. Gamma rays have the highest frequency and smallest wavelength of any radiation, and therefore also have the highest energy.

Gamma rays are powerful enough to penetrate most substances, and are powerful ionizers. The gamma ray spectrum involves wavelengths lower than one trillionth of a meter (less than .000000000001 meters).

This radiation is often produced as a byproduct of atomic decay, in which unstable atoms emit energy through gamma rays and other particles before settling to a stable state. Descriptions of some of these reactions can be found here.

Among the parts of the electromagnetic spectrum, gamma rays are comparatively rare. Few objects are powerful enough to give off high amounts of gamma ray energy. However, this radiation is very important on atomic scales, when atomic decay spontaneously converts mass into energetic photons, i.e. gamma rays. This is possible due to the well known mass-energy relation

E=mc^2

where E corresponds to energy, m to mass, and c a constant (the speed of light: 186282 miles per second). The meaning of this equation is that a certain amount of mass is equal to a certain amount of energy. During the early stages of the Universe, the temperature was sufficiently high that particles collided with their antiparticle counterparts. When they collide, they instantly annihilate each other, releasing energy. As energy, the reverse happened, with particles and antiparticles being spontaneously created. Now these reactions only take place in extreme conditions, such as on the edges of a black hole.

Despite this, astronomical gamma ray sources can still be found, and the closest is the Moon.

An image of the Moon taken by a gamma ray telescope. The gamma rays emitted from the Moon originate when other solar ionizing radiation, such as ultraviolet, hits the Moon and causes the excitation of heavy atoms, and this in turn, causes the emission of gamma rays. In contrast, the Sun is nearly invisible in the gamma ray spectrum, because it consists of very light atoms (mostly Hydrogen and Helium) that cannot be excited by ionizing radiation.

Other gamma ray sources include solar flares, the death of massive stars (through processes such as supernovae) and their remnants (neutron stars and black holes), as well as active galaxies. Galaxies are "active" if large amounts of infalling matter are feeding their central black hole, giving off enormous amounts of radiation. Most information on gamma ray bursts is theoretical, as little is actually known about their sources, and they usually are very distant.

Gamma ray bursts and their sources are some of the most fascinating areas of astronomy.

Sources: http://en.wikipedia.org/wiki/Gamma_ray, http://science.hq.nasa.gov/kids/imagers/ems/gamma.html, http://en.wikipedia.org/wiki/Timeline_of_the_Big_Bang, http://en.wikipedia.org/wiki/Mass–energy_equivalence

X-rays

2011-02-25T07:40:00.001-05:00

X-rays are yet another type of electromagnetic radiation, which is radiation in the form of photons. X-rays are very high energy, and therefore have a very short wavelength and a very high frequency.

X-rays are the first portion of the electromagnetic spectrum to be ionizing, (high frequency ultraviolet rays also have this property) meaning that they have high enough energy to dislodge electrons from the outer shells of atoms, thus converting them into ions.

This part of the electromagnetic spectrum varies in wavelength from .01 nanometers to 10 nanometers, covering three orders of magnitude. X-rays are primarily used for medical purposes, although they can be carcinogens in larger doses. It is estimated that a common dental X-ray does not meaningfully increase cancer risk, but more invasive CAT scans may significantly increase the risk.

The first known instance of a medical X-ray, taken in 1895 by William Roentgen of his wife's hand. The X-rays penetrate skin but not bone, and the lack of X-rays passing through creates the image above. Note the ring present on the ring finger.

Sheets of lead are often used to contain this radiation, as it is the most cost efficient metal for doing so. Varied thicknesses are used, although 10 mm is sufficient for most X-rays.

In astronomy, X-ray telescopes are used to detect sources of radiation throughout the galaxy. The closest source of X-rays is our Sun, (these are often reflected off the moon) but the Sun predominantly emits light at higher wavelengths, and large sources of X-rays are relatively rare. The accretion disc of a black hole emits large amounts of X-rays as it is heated to extreme temperatures. These sources are particularly prominent when the black hole has a large source of material, e.g. a binary star companion.

An artist's conception of a binary star system in which one of the stars has become a black hole. The gravitational pull from the black hole pulls in material from its companion star, and this material rotates the black hole at extreme speeds, (often over a million miles per hour) causing it to emit X-rays. The most well-known example of a binary system containing a black hole is Cygnus X-1, named for being a strong X-ray source in the constellation Cygnus.

X-ray image of Cygnus X-1 (false-color). The blue supergiant companion star is not prominent in this part of the spectrum, and is therefore not visible.

Other astronomical sources of X-rays include very massive stars, and the supernovae they result in, as well as black holes at galaxy centers. In the former case, the massive stars are often very unstable, and shed material in terrific explosions periodically, culminating in a supernova, in which, huge amounts of X-rays are emitted in an extreme explosion. The latter case works on the same principle as stellar black holes: with accreted matter increasing in temperature and energy, and subsequently releasing X-rays before being sucked in to the black hole.

Sources: http://en.wikipedia.org/wiki/X-ray, http://science.hq.nasa.gov/kids/imagers/ems/xrays.html, http://en.wikipedia.org/wiki/Astrophysical_X-ray_source

Ultraviolet Rays

2011-02-17T07:40:00.005-05:00

Ultraviolet Light is another type of electromagnetic radiation. It is known as ultraviolet because the frequency of ultraviolet rays are just larger than that of violet visible light (ultra=beyond).

This area of the spectrum has wavelengths as large as 400 nanometers (right on the cusp of visible light) and as small as 10 nanometers. This area is further subdivided into regions, the most well-known of them being UVA, UVB, and UVC, spanning 400-315, 315-280, and 280-100 nanometers, respectively. Unlike many of the former parts of the spectrum, ultraviolet light is primarily blocked from reaching the surface of the Earth. Of the ultraviolet light that does pass through the ozone layer, (perhaps 2% of the radiation that reaches the ozone layer from the Sun) most of it is UVA rays.

UVA rays are beneficial to health in small quantities, as they cause the production of vitamin D. In larger quantities, they cause tanning of the skin, and in excess, sunburn. UVA rays also are emitted from black lights, lights that are just on the edge of ultraviolet, making them partially visible. However, a majority of the light emitted is within the UVA region of the electromagnetic spectrum. and this radiation can cause chemical reactions, allowing a few substances to radiate a glow under UVA light.

An example of this is the security band on a typical U.S. $20 bill. This band is hard to duplicate, discouraging counterfeit.

UVB light (315-280 nm) is the next type (in increasing frequency) of ultraviolet light. It affects the skin in a more negative way, and is the rarer of the two types (UVA and UVB) that penetrate the atmosphere. Radiation in this part of the spectrum is more likely to cause cancer than UVA.

UVC light (280-100 nm) is blocked by the ozone layer, but is sometimes artificially produced on the earth's surface. This type of radiation serves as a disinfectant, as exposure of a microorganism to UVC rays damages its genetic material, resulting in mutations that cause infertility, and shortly after, death. In large exposure, these rays have harmful effects on humans as well. Despite their health hazard, they have practical applications in the disinfection of water and other materials.

The remainder of the ultraviolet spectrum (100-10 nm) is mainly used for astronomical purposes, although the Universe looks fundamentally different in ultraviolet than in visible. The most prominent feature, or, in this case, lack thereof, is the dimness of most stars in ultraviolet. Only stars with surfaces at higher temperatures (very young stars and stars in the final stages of their evolution) appear brightly in this part of the spectrum. The interstellar medium, or the sparse material occupying the space between stars, can be best seen and studied with ultraviolet telescopes. These telescopes must be mounted in space, however, due to the low amount of the radiation that reaches the surface of the Earth.

An image (false-color) of the galaxy Messier 81 seen in ultraviolet light. To obtain colorful images of other galaxies, these images are often combined with the images in visible light to detect more features.

Sources: http://en.wikipedia.org/wiki/Ultraviolet, http://science.hq.nasa.gov/kids/imagers/ems/uv.html, http://www.germsquad.com/home/faqs/6-what-are-uv-c-rays-and-how-can-they-benefit-us.html

Visible Light

2011-02-09T07:40:00.002-05:00

Visible light is a type of electromagnetic radiation, and is the only type observable through the naked eye.

Rather than being divided into different bands, visible light is divided into colors. The slight differences in the wavelengths of visible waves determine which color that the light is. Visible light is the "smallest" range of wavelengths of the seven main sections (logarithmically speaking) but it makes up everything we see.

Above: The division of visible light into a spectrum of colors. At the edges of the scale are near ultraviolet (the top) and near infrared (the bottom). The discovery of light as a composition of colors may have come in Isaac Newton's time (c. 1665), when the prism revealed sunlight as made up of light on many different wavelengths.

The prism is able to split visible light into its components because there are slight differences in the speeds of light through glass. Higher wavelengths are faster and have greater refraction angles, causing them to bend more and therefore appear at the bottom of the rainbow shown above. Red and other long wavelengths travel slower and refract at a smaller angle, causing them to be at the "top" of the rainbow above. Real rainbows work in a similar way, when water droplets disperse sunlight into the visible spectrum. This also proves that sunlight has many different wavelengths of light, by no means limited to visible light.

Visible, or optical astronomy, is by far the oldest type, and has been going on since antiquity. Only recently have observations in other parts of the spectrum expanded our knowledge of the heavens.

Sources: http://www.windows2universe.org/sun/spectrum/multispectral_sun_overview.html, http://www.juliantrubin.com/bigten/lightexperiments.html, http://en.wikipedia.org/wiki/Optical_astronomy

Infrared Rays

2011-02-01T07:40:00.001-05:00

Infrared radiation is a type of radiation included in the electromagnetic spectrum. Infrared rays still have a longer wavelength than visible light, but one shorter than radio waves and microwaves.

Infrared is best known as being heat radiation. The shortest infrared rays are very close to being visible to humans, hence the name (Latin infra meaning below, "below red"). However, humans can sense this radiation as heat, and infrared makes up a majority of the radiation that the Earth receives at ground level. Animals absorb heat and emit it themselves, and these phenomena can be captured by thermographic images:

A thermographic image showing two people (false-color).

Infrared radiation is subdivided further into three zones, near-infrared (.7 to 5 microns, or .0000007 to .000005 meters), mid-infrared (5 microns to about 35 microns), and far-infrared (35 microns to about 300 microns). Each zone has separate uses, notably near-infrared for telecommunication, mid-infrared for heat-seeking missiles, and far-infrared for lasers.

The zones are labeled in this way because near-infrared is on the cusp of visible light, while far-infrared has a much longer wavelength. Some living things can naturally "see" into the infrared spectrum, such as some snakes that use a special heat-detecting sense to find live prey.

Infrared rays have many other uses, including night vision and the heating of objects. Infrared cameras are also a valuable resource in meteorology, as the surface of the earth emits more infrared radiation than that of clouds, creating an easy way to analyze cloud height and temperature.

An infrared image of the Atlantic Basin. Clouds of increasing height are brighter white. The image also shows Tropical Storm Richard in the Caribbean. Image taken 10/23/10 12:15 UTC.

Infrared astronomy was pioneered by the man who discovered infrared rays: William Herschel. He is also well known for discovering the planet Uranus, along with two of its moons. His discovery was made while studying the spectrum of visible light produced by a prism. He noted that there was a significant temperature increase outside the visible light spectrum, beyond the red. He performed more tests and concluded that it was indeed a new type of radiation, being absorbed and emitted just like visible light, and also being released in massive quantities from the Sun.

In modern times, near-infrared rays can usually be picked up with a normal optical telescope, as these telescopes often have larger visual ranges than the naked eye, seeing into both near-infrared and near-ultraviolet. As one progresses farther into the infrared spectrum, a vast majority of rays do not reach the surface of the earth. However, telescopes at high altitudes in dry environments can pick these rays up with good efficiency. Telescopes operating in infrared can detect objects behind interstellar dust clouds and within nebulas better than visible light, as they can see the heat emitted from the region.

A photo of the Orion constellation in visible (left) and infrared (right). Although the infrared provides little indication to the exact location of the stars, it detects gas clouds throughout the constellation and other features totally invisible in the optical spectrum.

Overall, infrared rays have important uses, such as telecommunication and lasers, as well as transmitting heat throughout the Universe.

Sources: National Hurricane Center, http://www.ipac.caltech.edu/Outreach/Edu/Regions/irregions.html, http://en.wikipedia.org/wiki/Infrared. http://www.nasa.gov/mission_pages/SOFIA/infrared.html

Microwaves

2011-01-24T07:39:00.002-05:00

Microwaves are a type of wave in the electromagnetic spectrum (see here). Their wavelengths are longer than any type of wave except for radio waves (see here).

The wavelength of a microwave can range from 1 millimeter (.001 meter) to 1 meter. This range is actually within the broader term radio wave, but on some scales, they are separate. Accordingly, the frequency range of microwaves is 300 MHz to 300 GHz (300,000,000 to 300,000,000,000 Hz).

Microwaves are specifically used in areas such as communication, power, and radar. Before fiber-optic cables were adopted into the phone system, microwaves were used for the same purpose. Radar, or the ability to map objects by bouncing waves off them, lies predominantly within the microwave region, but perhaps the most famous of microwave uses is the microwave oven. Microwave ovens bombard non-radioactive radiation into food, installing energy into, and therefore heating, it.

The main component of the microwave oven is the magnetron. This is a cross-section of such a device which generates an electric field to produce microwaves. Magnetrons are generally for heating certain substances with microwaves, such as water, sugar, and fats in food, and sometimes, humans. Microwaves, however, are not specifically heat radiation.

Microwaves, just like all waves used for communication are divided into bands to organize their use. WIth microwaves, these bands are denoted by letter, (e.g. the L Band, ranging from 1 to 2 GHz) and each band has specific uses.

Yet another use of microwaves is in astronomy. There are many sources of microwaves in the heavens, but the most notable is the Cosmic Microwave Background Radiation. This radiation was emitted 379,000 years of the Big Bang, when the Universe became transparent due to the cessation of photon-plasma reactions. Plasma is ionized gas, and is therefore made of ionized atoms. When the temperature of the Universe had fallen enough to support complete atoms with electrons, the atoms became neutral, and the reactions stopped. The image of the Universe released at that moment was carried by gamma rays, but over time red shifting transferred these into microwaves, as they are today. More and less intense patches on the radiation indicate the density of different parts of the Universe. The differences were minute at the time, but gravity gradually increased the differences, and all of the structure in the large scale Universe was defined.

An image adjusted from the original Cosmic Microwave Background to show temperature differences in the early Universe.

Microwaves have a variety of uses, from communication to heating to discovering the origins of the Universe.

Sources: http://en.wikipedia.org/wiki/Microwave, http://www.gallawa.com/microtech/mwave.html, http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation, etc.

Radio Waves

2011-01-16T07:39:00.002-05:00

Radio Waves are part of the electromagnetic spectrum (see here) and consist of photons in their wave form. Out of all of the electromagnetic waves, radio waves have the lowest frequency and highest wavelength.

The range of radio waves is approximately 3 Hz, meaning three wavelengths per second, to 300 GHz, or 300,000,000,000 Hz, although the boundary isn't clearly defined, and waves with frequencies even smaller than these exist. This range is equivalent to wavelengths from 1 millimeter (.001 meter) to 100,000 kilometers (100,000,000 meters).

Radio waves are both man-made, and occur from natural sources, and have many uses, due to the fact that most of the radio spectrum pass through Earth's atmosphere intact. The percentage of waves that pass through the atmosphere are shown on the picture below, along with the rest of the spectrum.

Since radio waves can pass through the atmosphere, they are invaluable for communication purposes and information propagation takes place mainly in the radio spectrum. This portion of the spectrum is further divided into powers of 10. For example, the area from 3-30 Hz is known as Extremely Low Frequency (ELF), the area from 30-300 Hz is known as Super Low Frequency (SLF), the area from 300-3,000 Hz (3 kHz) is known as Ultra Low Frequency (ULF) and so on. The chart of frequencies above ULF are shown below.

The first radio waves to be used for communication are AM radio waves, which extend from 148.5 kHz (148,500 Hz) to 30 MHz (30,000,000 Hz). These waves are in the Low, Medium, and High Frequency ranges. However, waves of these frequencies are produced naturally by the ionosphere and magnetic storms, causing interference. As a result, FM radio, falling entirely under the Very High Frequency range, is now used for long distance broad casts. This area of the spectrum is the busiest of all, as television, FM radio and mobile communications all resides within 30-300 MHz.

The very low frequencies, (VLF and lower) are used for military (submarine) communications as they are useful in subaquatic environments.

The higher frequencies also have uses. Super High Frequency (3-30 GHz, or 3,000,000,000-30,000,000,000 Hz) is used for radar, and Extremely High Frequency (30-300 GHz) is used for radio astronomy.

The Very Large Array, located in New Mexico, USA, is used for radio astronomy. Large, parabolic satellite dishes concentrate the signals on a specific sources in the center of each individual telescope. Since many radio waves picked up from space have very long wavelengths, a very large apparatus is needed to capture a reasonable image. The Very Large Area has 27 radio telescope receivers, each 82 feet in diameter, and the images produced by each are compiled into a single one in the center.

This is an optical (visible light) image of the M87 elliptical galaxy. The active center and jet of radiation from it are visible, but in visible light, most of the galaxy is a haze of stars.

However, by use of radio astronomy, a much more specific picture can be taken. Unlike visible light, radio waves are not produced to a great extent by any part of the galaxy except the active center, and it can therefore be identified better in radio astronomy. The above image is taken by the Very Large Array (VLA). However, the bottom image, which is magnified tens of thousands of times more than the above one, is taken by the Very Long Baseline Array (VLBA). Since M87 is over 50 million light years away, a resolution of this magnitude can only be created by use of a planet sized radio telescope, which is essentially what VLBA is. Satellite dishes all over the world are synced together by a central computer in Europe, and therefore at least a small number of radio waves coming in over a Earth-sized area can be put together to get a very precise view, even if all the incoming waves aren't picked up.

Notable sources of radio waves from space include the Sun (only due to its proximity to Earth, because stars do not emit a majority of their energy in radio waves), galactic centers, such as the Milky Way itself, and active ones like M87, neutron stars, and very distant powerfully emitted objects such as quasars. The redshift effect of the quasars' movement away from us causes the waves to lengthen in wavelength as they approach Earth. As a result, many waves that start higher in frequency are radio waves when they reach Earth. This effect is most notable at very long distances, billions of light years away. Among objects at this distance, only quasars are bright enough to be visible, and are therefore excellent subjects of radio astronomy.

Sources: http://www.google.com/imgres?imgurl=http://www.electronics-radio.com/, http://en.wikipedia.org/wiki/Radio_astronomy, etc.

Electromagnetic Spectrum

2011-01-08T13:21:00.008-05:00

The Electromagnetic Spectrum, technically speaking, is the range of frequencies that a photon wave can assume. The photon is essentially the particle of light, and this particle has many interesting properties.

First, a photon has no mass, which means it can travel at a speed unlimited by its mass. The speed at which the light particle can travel in a vacuum defines the fastest known speed in the Universe, namely the speed of light, or 186,292 miles per second. Second, the photon can have properties of both a particle and a wave. But when dealing with the electromagnetic spectrum, the wave form of a photon is most important.

The electromagnetic spectrum deals with all possible photon waves, each of which are composed of the same particle, but have different energies. Along with energy, the three properties of a specific electromagnetic wave are amplitude, wavelength and frequency.

Electromagnetic waves are in a sinusoid form, meaning that they are shaped like the red line above. The distance, A, from the sinusoid axis to the peak of the curve is known as the amplitude of the wave, and the distance between peaks is known as the wavelength, while the distance between crossings of the sinusoid axis are half of a wavelength. The remaining element, frequency, is how many wavelengths the wave goes through over a certain period of time. The horizontal axis (x) on the graph above is the passage of time, and over this time, the wave modulates, or moves from peak to peak, in a cyclic manner.

The different electromagnetic waves are separated by varying amplitude and wavelength. The above chart shows five different wave types. The horizontal axis represents time, as before. On the top of the chart is the wave with the highest wavelength, but lowest frequency, and as one goes down the chart, the wavelength decreases and the frequency increases. The two properties are inversely proportional because an electromagnetic wave is traveling at a constant speed through a vacuum, the speed of light, and therefore the wavelength determines a length of time between wave peaks, namely frequency. Wavelength is measured in meters, while frequency is measured in a unit called a Hertz (Hz), with 1 Hz meaning one wavelength per second.

The entire electromagnetic spectrum (click to enlarge), including names given to specific areas of the spectrum, some uses of particular wavelengths, and the corresponding wavelength and frequency axes. The plot is a log scale of base 10, meaning that instead of counting up 1, 2, 3... the graph counts 10^1, 10^2, 10^3... The frequency increases from the bottom to the top, starting at 10^6, and ending at 10^19, and the wavelength counts in meters (m), using symbols like cm to represent increments of meters.

The waves are as follows, from lowest to highest frequency are: radio waves, microwaves, infrared rays, visible light, ultraviolet rays, X-rays, and gamma rays. Each of these have their own individual posts. Click on one to enter and begin exploring the electromagnetic spectrum.

Radioactive Decay

2011-01-01T00:00:00.003-05:00

Radioactive Decay is the decomposition of an unstable atom into multiple parts. Often, this involves the emission of radiation, hence the name radioactive decay. Some elements of the periodic table have no stable atoms, while some have stable and unstable isotopes, meaning that the number of protons remains the same, but stability can vary with the number of neutrons.

There are 118 known elements on the periodic table, and, among these, numbers 43, 61, and all above 82 possess only isotopes that are radioactive. The remaining 80 have at least one neutron number that results in a stable nucleus, but a vast majority of the possibilities are radioactive, and overcome the force, known as the strong nuclear force or the strong interaction, that binds the particles in a nucleus together.

For example, the element oxygen has 8 protons, and the number of neutrons can range from 4 to 16, because it is known that there are physical limits on how many neutrons can exist for a certain number of protons, called the nuclear drip line. It is simply impossible for 3 neutrons and 8 protons to exist in a nucleus, and if these particles are shoved together, a proton will simply drip out. Similarly, it is impossible for 8 protons and 17 neutrons to be together in a nucleus, because a neutron will "drip" out. Therefore, these nuclei do not qualify as parts of atoms, as they are never in a whole state. Unlike these, isotopes of oxygen between 4 and 16 neutrons can exist. However, all but three of these are unstable, and decay to other atoms within seconds. Atoms with 8 protons and 8, 9, and 10 neutrons, corresponding to Oxygen-16, Oxygen-17, and Oxygen-18 are stable, and occur in nature.

Also, there are different ways that atoms can decay into others. The three most common were the first types of radiation isolated, and there were named alpha, beta, and gamma decay.

Alpha decay involves an unstable nucleus of an atom emitting an entire Helium-4 nucleus, that is, two protons and two neutrons, at once. This nucleus is also known as an alpha particle. The resulting nucleus of the parent atom has two less protons and two less neutrons than it did before alpha decay. Some argue that the Helium nucleus takes two electrons with it when an atom experiences alpha decay, because otherwise there would be an unbalanced charge, and some believe that two electrons are simply released into the environment. An example alpha decay reaction is pictured below.

An example reaction involves Uranium-238, which, by alpha decay loses two neutrons and two protons. Because of this the resulting atom's atomic weight will drop from 238 to 234, and the atomic number will drop by two, becoming Thorium. Therefore the reaction is denoted

238,92U=234,92Th+4,2He

The notation above shows the two balanced sides of the reaction, with the Uranium-238 (first number before the atomic symbol is the entire weight, the second is the number of protons) atom on the left, and the Thorium-234 atom and the Helium nucleus (alpha particle) on the right. In this reaction the two elections are assumed to leave the atom with the Helium nucleus.

Beta decay involves a slightly more complicated chain of events, and there are two types: Beta negative and beta positive, denoted B- and B+ respectively. The B- reaction uses the weak nuclear force (one of the four fundamental forces of the Universe) to convert a neutron into a proton. However, there are some byproducts of this reaction, namely an electron and an electron antineutrino. Since a neutron is only slightly heavier than the proton, the loss of a tiny electron (and an electron antineutrino) is enough to convert it into a proton. Also, the change is mass is so minute that the atomic weight remains the same, but the atomic number goes up. This type of decay actually "increases" the complexity of the nucleus, instead of lowering it.

This image is an example beta decay reaction. The main picture shows only the B- particle (electron) being emitted, while the inset shows the entire reaction. The neutron splits into three parts, of which the proton stays in the nucleus, the B- particle may leave the atom or stay and compensate for the gaining of a positive charge via the proton, and the tiny electron antineutrino is emitted. To fully understand this process, one must break it down into even smaller stages, by use of a Feynman Diagram.

A Feynman Diagram graphically represents quantum phenomena to make them easier to understand. In this diagram, time progresses with respect to the vertical axis. The neutron, n, is shown as its three composite parts, u,d, and d, representing one up quark and two down quarks. The weak nuclear force has an effect on the final down quark only, instantly changing it to an up quark. As a result, the three new quarks are up, down, up (u,d,u) and these are the composite particles for a proton, hence the proton end product. However, to make this change happen, a W- boson, the carrier of the weak nuclear force, must be emitted from the down quark, to change it to an up quark. Since a down quark is heavier than an up one, the loss in mass again makes sense. The W- boson is very short lived, however, and nearly instantaneously splits into an electron and an electron antineutrino, denoted by the e- and ve+ respectively.

An example reaction of B- decay is the process of changing the Caesium-137 atom into the Barium-137 atom

137,55Cs = 137,56Ba + e- + ve+

Neutron-rich nuclei are more likely to undergo B- decay.

The other type of Beta decay is B+ decay. It is basically the opposite (in terms of particles) of the previous process, because a proton is converted into a neutron. Since the neutron is heavier than the proton, the reaction needs outside energy to add mass to the reaction (since energy can at any time by changed into mass and vice versa). This must be provided by the environment, and therefore this reaction cannot occur by itself in a vacuum. The byproducts of the reaction are the opposite of B- decay in charge; instead of a electron and an electron antineutrino being emitted, the positron (anti-electron) and electron neutrino are emitted. An example B+ decay reaction involving the transformation of Carbon-11 into Boron-11 is written below

11,6C = 11,5B + e+ + ve-

This reaction also involves the weak nuclear force, but with use of a W+ boson, rather than a W- one. The W+ boson is emitted when a up quark changes into a down quark. This particle then splits into the two mentioned above.

B+ decay is more likely to occur in proton-rich nuclei.

There are other rarer types of Beta decay, such as electron capture, where an electron is "captured" from the orbitals of the atom, and is pulled to the nucleus, where it combines with a proton to form a neutron and an electron antineutrino. An example reaction is

59,28Ni + e- = 59,27Co + ve+

Although the electron on the left side of this equation is presented as if it is separate from the atom, the electron actually originated from what was part of the atom, namely orbiting electrons.

There are also forms of Beta decay where the process happens twice simultaneously. These are called Double Beta Decay, and a similar double exists for electron capture.

Other simple types of radioactive decay include proton emission, if a nucleus is very rich in protons, and neutron emission, if a nucleus is very rich in protons. Note that this is different from proton and neutron "dripping" discussed earlier, because the nucleus does exist as one unit before the the proton or neutron is emitted.

Another famous type is the emission of a larger particle than an alpha particle, namely a heavier atomic nucleus. However, this type of decay, called cluster decay, only occurs among atoms that usually decay through the emission of an alpha particle. Some unstable atoms decay in different ways, with one occurring a certain percentage of the time, and another in the remaining percentage. An example is the atom Radium-223.

Usually, Radium-223 decays through alpha decay:

223,88Ra = 219,86Rn + 4,2He

but for one out of every one billion reactions, something else occurs, and the atom emits a Carbon-14 nucleus!

223,88Ra = 209,82Pb + 14,6C

The second reaction was the first of its kind known to occur and was discovered in 1984 at Oxford University. The heaviest known nucleus to be emitted in this fashion is Silicon-34, happening only once out of trillions and trillions of alpha decays from Plutonium-240, Americium-241, and Curium-242.

Finally, some atoms simply split into two atoms, through a process called spontaneous fission. This processes occurs when a neutron impacts the nucleus and splits it in two. Uranium, Plutonium, and Californium are three elements that have a chance for spontaneous fission, although they are more likely to decay through other processes. Californium-252 has a relatively high fission rate, with 3.09% of reactions result in fission. However, Uranium-235 decays through fission only seven reactions out of 100 billion! There are higher elements which predominately decay by fission, some of which are isotopes of Mendelevium and Rutherfordium. Some of these reactions emit neutrons in turn, and these can lead to chain reactions. Uranium-235 is one of these, and since each reaction gives out energy, it is one of the isotopes used in the detonation of atomic bombs.

Reactions of this type occur to all of the possible unstable isotopes. The chart that maps all the isotopes is known as the table of nuclides, linked to here.

This is another version of the table of nuclides, which shows the predominant mode of decay for every known nuclide. The center of the band of nuclides are the most stable (those in black are stable), and they tend to get less stable away from the center. Isotopes that undergo B+ decay or Proton Emission occur on the right (proton-rich) side of the stable isotopes, while ones that undergo B- decay or Neutron Emission occur the left (neutron-rich) side of the stable isotopes. Also, alpha decay, cluster decay and fission tend to occur with heavier atoms, toward the upper right of the chart.

Some isotopes do not decay directly into a stable nucleus, and go through multiple steps of decay before reaching stability.

An example is Uranium-238. Its decay chain is imaged above (click to enlarge), with each octagon containing the atomic number and mass of an isotope, while the arrows denote the decay chain, with letters representing the type of decay. The half-life, or average time to decay on each step, is also included under each octagon. Sometimes, the decay chain branches, when there are probabilities for other types of decay, but all of the branches converge on the stable isotope Lead-206.

Radioactive decay produces energy, and is therefore valuable as a potential power source, with drawbacks including lower feasibility and harmful radiation. Radiation is also used for other purposes, such as medical procedures, specifically for eliminating cancerous cells.

Sources: http://education.jlab.org/glossary/betadecay.gif, http://www.nature.com/nature/journal/v307/n5948/abs/307245a0.html, http://en.wikipedia.org/wiki/Table_of_nuclides_(complete)

2010 Season Summary

2010-12-21T15:11:00.009-05:00

The 2010 hurricane season was well above average, with

21 cyclones attaining tropical depression status
19 achieving tropical storm status
12 hurricanes
and 5 major hurricanes

This is higher than my predictions of

18 cyclones attaining tropical depression status
17 cyclones attaining tropical storm status
7 cyclones attaining hurricane status
4 cyclones attaining major hurricane status

particularly in the hurricanes category.

This activity (19 named storms) was tied for the third most ever recorded in the Atlantic basin. The most powerful cyclone of the season was Igor, which attained a peak intensity of 155 mph winds and a minimum pressure of 925 mb. It also was the largest tropical cyclone ever to form in the Atlantic basin in terms of tropical storm wind diameter. Due to its colossal nature, Igor was also the third wettest tropical cyclone every recorded in Canada, dumping 9.37 inches of rain in one location in Newfoundland.

Also:

8 storms formed in September (tied for a record high)
Four cyclones (Alex, Karl, Matthew, and Richard) made landfall in Belize, although two of them at tropical depression status (record high)
Two Category 4 hurricanes (Igor and Julia) existed simultaneously for a brief period of time, an occurrence that has not happened since 1926

Overall, the 2010 Atlantic hurricane season was a very active one, and impacts were mostly in the Caribbean and Central America. The United States, surprisingly, was barely affected, with no hurricane landfalls.

Hurricane Tomas (2010)

2010-10-30T16:52:00.015-04:00

Storm Active: October 29-November 7
On October 25, a tropical wave formed in the extreme southeastern Atlantic Ocean, near 5ºN. The wave produced only scattered shower and thunderstorm activity as it moved west over the next few days. It was a vigorous tropical wave, however, and it developed a low pressure center on October 27. The low adopted a general northwest motion, and deepened significantly over the next two days. By the afternoon of October 29, the system had a very organized circulation and outflow, and the confirmation of a closed low at its center merited the upgrading of the system into Tropical Storm Tomas.

Tropical Storm Tomas was already in a state of rapid intensification, and the winds increased rapidly as the cyclone approached the Caribbean Islands, moving westnorthwest between 10 and 15 mph. During the morning of October 30, Tomas passed directly over Barbados, with peak winds of 70 mph, causing fairly significant damage. Tomas developed a very wide eye feature (about 40 miles across) just before noon on October 30, and it was then organized enough to be upgraded to a Category 1 hurricane.

The system promptly made a direct landfall in St. Vincent during that afternoon, and heavy rain and tropical storm winds affected islands up to 100 miles north and south along the Windward and Leeward Islands. The wide eye clouded over with a flare of convection, and Tomas continued to strengthen, becoming a Category 2 by later that night. However, southwesterly shear and dry air began to impact the west side of the system early on October 31, weakening it to a Category 1 storm by the afternoon. The center became ragged in appearance, and lost definition as a result of harsh atmospheric conditions.

Tomas weakened further into a tropical storm during the night, and only stabilized on November 1, when the winds dropped to 45 mph. Tomas was pushed on a general westsouthwest course during the day. Tomas's intensity fluctuated with large variations in convection over the day of November 2. The cyclone's forward speed also decreased as it reached the edge of a ridge to its north and steering currents weakened. For a brief period on November 3, Tomas degenerated into a wide area of scattered convection covering the entire southwest Caribbean, and was therefore downgraded to a depression, but the conditions for development drastically improved later in the day and Tomas turned towards the north, and the storm underwent a fast strengthening process, regaining tropical storm status. Tomas reached an intensity of 50 mph winds, and maintained it for the next day as it slowly moved northward. Wide rain bands began to sweep across Jamaica, Haiti and Cuba by the afternoon of November 4. As Tomas approached land, it rapidly strengthened into a hurricane, reaching its secondary peak intensity of 85 mph winds and a pressure of 984 mb as it passed just west of Haiti on November 5.

Cuba and Haiti both experienced tropical storm conditions, as well as hurricane force in some areas of Haiti, as the day went on, and as Tomas began to accelerate northeast, it interacted with the land around it briefly, and weakened back to a minimal Category 1 hurricane later in the evening. Tomas passed over the Turks and Caicos islands overnight, but emerged over open Atlantic waters on November 6, weakening back to tropical storm. Unexpectedly, Tomas once again regained hurricane strength late on November 6, but a cold front quickly overtook the system, and Tomas rapidly transitioned into an extratropical low on November 7. 41 fatalities and $572 million in damage directly resulted from Tomas in the Caribbean Islands, but Tomas is also indirectly linked to an epidemic of cholera in Haiti.

Hurricane Tomas intensifying as it enters the Caribbean. A fair amount of wind shear is evident on the south side of the system.

Track of Tomas.

Hurricane Shary (2010)

2010-10-29T14:57:00.007-04:00

Storm Active: October 28-30
A trough of low pressure formed in the Caribbean on October 26. The trough was associated with an area of convection, but strong upper-level winds prevented development. However, on October 28, the shear relaxed enough for a central low pressure to form. However, the convection remained disassociated with this low until late on October 28, when the system developed an eyewall. At this point, the low was upgraded to Tropical Storm Shary.

Tropical Storm Shary sped to the northwest through the night, and shear began to increase on the system once again, giving it a lopsided appearance. This shear was produced by a strong front moving east off of the U.S. and this front also began to turn Shary to the north and then northeast. Despite adverse conditions, Shary strengthened as it began the turn, intensifying to 60 mph winds and a pressure of 1000 mb during the afternoon of October 29. The cyclone then made its closest approach to Bermuda, causing only showers and gusty winds however, as it passed well to the east. Shary's circulation was largely exposed through the coming day, but it continued to strengthen, becoming a minimal hurricane early on October 30.

By this time, Shary was speeding off to the northeast, and it briefly reached its peak intensity of 75 mph winds and a pressure of 989 mb before quickly becoming extratropical and being absorbed by a front later that afternoon. No damage resulted from Shary.

Hurricane Shary at peak intensity.

Track of Shary.

Hurricane Richard (2010)

2010-10-21T16:49:00.012-04:00

Storm Active: October 20-26
On October 16, an area of showers and thunderstorms developed in the extreme southwestern Caribbean. The area drifted to the northwest, and interaction with the Nicaragua-Honduras area inhibited development for a time on October 18 and 19. However, after emerging over open water, the circulation improved, and the system moved slowly northeast. Late on October 20, the system became organized enough to be Tropical Depression Nineteen. By this time, steering currents had weakened, and the cyclone had reverted to a slow southeast movement. Dry air existed near the circulation over the next day, but intensification occurred nonetheless, and Nineteen became Tropical Storm Richard on October 21.

The system did not intensify for almost a day, but convection increased as dry air moved away from the system and a ridge built over the Gulf of Mexico, steering Richard back to the west by October 22. The system finally began to strengthen that day, rapidly intensifying into a strong tropical storm the next morning. Due to its proximity to Honduras, tropical storm conditions began for coastal areas by late on October 22. Richard began accelerating to the westnorthwest late on October 23, and a burst of convection the following morning caused Richard to intensify rapidly, becoming a category 1 hurricane.

The cyclone continued to intensify and made landfall in Belize at its peak strength of 90 mph winds and a minimum pressure of 981 mb during the evening of October 24. It quickly weakened over the next day, becoming a tropical depression by October 25. The system reemerged over the Gulf of Mexico early on October 26, but conditions were hostile for restrengthening and Richard swiftly weakened to a remnant low. The effects of Richard were $24.7 million in damage and 2 fatalities.

Richard as a Category 1 hurricane before landfall.

Track of Richard.

Hurricane Paula (2010)

2010-10-12T16:45:00.009-04:00

Storm Active: October 11-15
On October 7, a broad area of low pressure developed in the southwestern Caribbean. Disorganized showers and thunderstorms remained associated with the system as it drifted generally to the northwest over the coming days. A low pressure center formed on October 9, and deepened thereafter, becoming a tropical depression during the morning of October 11, although not being formally recognized as a tropical system until later that afternoon. By that time, the cyclone was already a strong tropical storm, and was named Paula.

The system was in the midst of rapid intensification, and was a hurricane by the morning of October 12. It turned more to the northnorthwest over the next day, but continued to strengthen, exploding into a Category 2 (albeit a small one) by the afternoon of that same day, as it approached the Yucatan Peninsula. It stalled just offshore to the east later on October 12, still maintaining its peak intensity of 100 mph winds and a pressure of 981 mb. Since Paula was a small storm, only minimal rain and wind affected the Yucatan itself, and a jet stream just to the north of the system started to push Paula to the east and weaken it by the afternoon of October 13. The system accelerated eastward slightly, and made landfall in western Cuba on October 14, as it weakened to a tropical storm. Paula continued to degenerate, becoming a remnant low by October 15. It dissipated the next day. Paula was a very small storm, and damage was therefore limited, with only one fatality recorded.

Paula at peak intensity. The system remains very small, with a correspondingly small eye feature.

Track of Paula.

Hurricane Otto (2010)

2010-10-07T14:49:00.013-04:00

Storm Active: October 6-10
On September 28, a tropical wave over the Central Atlantic began to produce an area of showers and thunderstorms. The next day, another tropical wave to its east also began to be monitored for development. The two systems moved west, but the second caught up with the first and the two waves combined on September 30. The combined disturbance produced a wide area of showers and thunderstorms as it moved westnorthwest, but wind shear increased, and the system remained disorganized. A low pressure center began to form in association with the system, and the low deepened as it passed over the Leeward Islands on October 3-5. By October 6, a surface circulation had formed. However, unlike a tropical cyclone, the center of the system had an upper-level low situated above it, rather than an upper level high, and this fact, combined with the limited convection that was only prevalent on the southeast side of the center, resulted in the classification of the system as Subtropical Depression Seventeen early on October 6. Seventeen's convection wrapped around the center the next day, and the winds reached gale force that evening, meriting the naming of the system as Subtropical Storm Otto.

By late on October 6, Otto's winds had rapidly increased, and the cyclone had reached an intensity of 65 mph winds and a pressure of 990 mb. However, the convection remained very sparse throughout the night, and intensity was difficult to judge, although the ragged appearance of the circulation suggested a slight weakening during the morning of October 7. By later in the morning, the upper-level low that had been shearing the circulation weakened, the core had warmed, and a distinct central eyewall had appeared as the system turned northeast. Otto was now a tropical cyclone, and it was officially classified as such at 11:00 am EDT on October 7. Otto's cloud cover continued to increase as it accelerated eastnortheast, and the system strengthened further, becoming a hurricane by October 8. The system reached its peak intensity of 85 mph and a minimum central pressure of 972 mb, before beginning to weaken. The system picked up speed as it moved out to sea, and it became a tropical storm late on October 9. Otto lost most of its central convection and was displaced to the north over the next 12 hours. As a result, the system was extratropical by midmorning on October 10. The cyclone subsequently impacted the Azores with some rain and wind as it weakened, dissipating on October 12.

Otto caused $20 million in damage but no deaths were reported, most damage being caused by flooding in the Caribbean Islands when the cyclone loitered to the north. As much as 17 inches of rain was reported in parts of Puerto Rico over a six day period from October 3-8 (this and similar rainfall reports courtesy of the Hydrometeorological Prediction Center).

Hurricane Otto at peak intensity speeding off into the open Atlantic.

Track of Otto.

Tropical Storm Nicole (2010)

2010-09-28T16:41:00.009-04:00

Storm Active: September 28-29
During the final week of September, Tropical Storm Matthew dissipated over Mexico. However, the huge amount of moisture left over from this system was accompanying a broad area of low pressure over Central America and the western Caribbean. Disorganized showers and thunderstorms began to appear in this area during the day of September 26, and pressures dropped over the area over the coming days. On September 28, a closed center formed, and the system was upgraded to Tropical Depression Sixteen. The depression's center was broad, with the only area of convection to the southeast of the circulation, but a band formed to the northeast of the center later that day, as well as the center itself becoming slightly more defined, and these factors resulted in the pressure dropping slightly as the system moved northnortheast, but not a promotion to tropical storm status. The depression made landfall in Cuba, but as it was a very asymmetrical and broad cyclone, it was very difficult to classify one way or the other. However, the presence of tropical storm force winds near the center was enough to push the cyclone to Tropical Storm Nicole during the morning of September 29, while still over Cuba.

The system emerged over water but the huge circulation lost the little tropical characteristics that it had, and was declared dissipated later that day, shortly after reaching its peak intensity of 40 mph winds and a pressure of 996 mb. However, the rainfall was by no means over. The remnant moisture of Nicole combined with an extratropical low off of North Carolina and a stationary front over the northeast to bring torrential rainfall to the region from Maine to Florida, with local amounts exceeding fifteen inches. This storm activity finally ceased by October 1. This cyclone caused 13 fatalities and $151.9 million in damage, but this does not include additional damage wreaked by the combined system that impacted the northeast U.S.

Nicole as an odd-looking tropical storm near Cuba.

Track of Nicole.

Tropical Storm Matthew (2010)

2010-09-24T15:27:00.007-04:00

Storm Active: September 23-26
On September 19, a trough of low pressure formed in the south central Atlantic. It slowly organized organization and a low pressure center appeared on September 21. The system gained convection, and the convection showed signs of a organized circulation by September 23, meriting the promotion of the low to Tropical Depression Fifteen. The tropical depression's center was fairly open until later that day, when a band circumnavigated the center for the first time, and the system was named Tropical Storm Matthew.

Matthew moved quickly westward through the southern Caribbean, and strengthened slowly over the next day. However, Matthew only reached an intensity of 50 mph winds and a pressure of 998 mb, before making landfall in northern Nicaragua during the afternoon of September 24. It maintained an intensity of 50 mph over land, and briefly emerged over the Gulf of Honduras on September 25, before beginning to weaken over Belize later that day.

The system quickly weakened to a tropical depression, and was a remnant low by early on September 26. However, the large area of moisture associated with the large circulation of Matthew caused continual heavy rain and flooding through the next few days. Matthew was the direct cause of several mudslides in Central America, causing 109 fatalities.

Matthew at its peak intensity before making landfall in Mexico.

Track of Matthew.

Hurricane Lisa (2010)

2010-09-21T16:04:00.009-04:00

Storm Active: September 20-26
The tropical wave that became Lisa emerged off of Africa on September 16. The cloud cover associated with the wave remained disorganized for a day, but a broad low pressure center formed on September 17. The low took a turn northwest before slowing its motion and drifting to the north over the next few days. This uncertain motion was caused by a weakness in the usual Azores high that pushes cyclones to the west, a weakness that was enlarged by Hurricane Julia a few days earlier. Meanwhile, the system became more organized, and was declared a tropical depression late on September 20. The depression soon strengthened into Tropical Storm Lisa.

The cyclone meandered northeast, then east, then south, and then east again through the next day, with no change in intensity. Some convection was lost during the day of September 21, and the system weakened to a tropical depression, still maintaining a slow east motion. By September 23, convection had organized enough for Lisa to become a tropical storm again, and the system was nearly stationary during that day. However, Lisa adopted a northward motion during the day of September 24, and continued strengthening.

By late on September 24, Lisa had quickly intensified to a hurricane, reaching its peak intensity of 80 mph winds and a pressure of 987 mb just before midnight. An eye feature even made a brief appearance. However, Lisa began quickly weakening the next day, as it moved into cooler waters. The system was a remnant low by the afternoon of September 26. Lisa moved north and quickly dissipated. The cyclone affected no land masses.

Lisa as a hurricane. It is obviously a very small storm.

Track of Lisa.

Hurricane Karl (2010)

2010-09-14T18:46:00.009-04:00

Storm Active: September 14-18
Late on September 9, a low pressure system formed to the east of the northeastern coast of South America. The low stalled near the Windward Islands, and crossed into the Caribbean on September 10. The low produced a lot of disorganized shower and thunderstorm activity which brought stormy conditions to much of the Caribbean over the next few days as it moved west, but little of this convection showed evidence of any circulation. However, on September 14, a closed circulation became evident from an aircraft reconnaissance mission sent to investigate the system, and a small pocket of tropical storm force winds allowed the low to skip depression status and become Tropical Storm Karl.

Karl lost most of its convection during the night, but still strengthened slightly, before a burst of convection near the center caused the cyclone to intensify to 65 mph winds early on September 15 just before landfall in the southeast Yucatan Peninsula. Since Karl made landfall near the Mexican border, portions of both Mexico and Belize were struck by tropical storm force conditions. Karl weakened over land to a minimal tropical storm, but reemerged over the Bay of Campache by early on September 16. Karl immediately began a strengthening trend over water, and the system quickly intensified into a hurricane later on September 16. An eye feature developed over the next day, and Karl rapidly intensified into a major hurricane, reaching its major hurricane peak intensity if 120 mph winds and a pressure of 956 mb on early on September 17, before weakening slightly just before landfall in Veracruz, Mexico with maximum winds of 115 mph.

Karl quickly began to weaken over land, losing its major hurricane status quickly later on September 17. Karl was ripped apart by the mountainous regions of central Mexico as it moved southwest inland, but these same mountainous regions caused a large amount of flooding over the affected area. 22 fatalities and $3.9 billion in damage are direct effects of the cyclone.

Hurricane Karl before striking the Mexican coast.

Track of Karl.

Hurricane Julia (2010)

2010-09-12T16:58:00.012-04:00

Storm Active: September 12-20
Early in the second week of September, a strong tropical wave was already very evident over Africa, and it was monitored for development even before leaving the coast. By the time it did emerge over the Atlantic Ocean on September 11, it was very organized and was already showing tropical characteristics. As a result, the system was declared Tropical Depression Twelve on September 12. As the depression turned westnorthwest, it intensified into Tropical Storm Julia late on September 12.

Early on September 13, the southernmost Cape Verde islands experienced tropical storm conditions, as Julia passed just to the southwest. Julia took a more northward track than the cyclones before it, but it still intensified over the next day, as the predecessor of an eyewall formed near Julia's center, signifying a very healthy cyclone. The storm continued this strengthening trend, and became a Category 1 hurricane early on September 14.

It continued to strengthen through the morning as a structure that was almost an eye appeared, but Julia stabilized later in the afternoon after strengthening rapidly to its peak intensity as a Category 4 hurricane with 135 mph winds and a pressure of 950 mb early on September 15.

The cyclone turned farther to the north and began a general northwest motion. It began to encounter less favorable conditions as it approached cooler water and shear associated with the outflow of Igor. However, during its time as a Category 4, Julia set the record for strongest Atlantic cyclone east of 35ºW, surpassing the record set by Hurricane Fred just a year earlier. Julia continued to weaken over the next day, but wind shear died down slightly later on September 16, as Julia took a more westward turn. Julia maintained its Category 1 intensity over the next day, despite entering the outflow of the much larger and powerful Igor.

Julia continued weakening, and the center became separated from the convection on September 17. As a result, Julia soon became a tropical storm. Julia continued weakening into September 19, as it turned north and then northeast, accelerating over the open ocean. The system became extratropical on September 20, and started to be absorbed over the next day as a frontal boundary associated with Igor engulfed it. Julia caused only minimal damage while passing the Cape Verde islands, and was only notable for being a major hurricane very far east.

Julia near peak intensity.

Track of Julia.

Hurricane Igor (2010)

2010-09-08T18:17:00.019-04:00

Storm Active: September 8-21
On September 6, a strong tropical wave emerged off of Africa. A low became associated with the system on September 7, and the system continued to organize, despite significant shear. On September 8, the system was organized enough to skip the tropical depression stage and intensify directly into Tropical Storm Igor.

The convection and circulation of Igor were at a very high level of organization, even with two low pressure systems in the vicinity. These three lows shared a low pressure trough extending from just off Africa to a few hundred miles north and west, but Igor began to strengthen later on September 8, and asserted its dominance. Igor meandered slowly to the west, before taking a turn north and then northwest on September 9. As it moved into greater shear, it weakened, becoming a tropical depression later that day, as the center became exposed from the east side. Despite this, the circulation deepened, the convection became more organized, and the system regained tropical storm status the next day. The shear had abated once again and Igor began to strengthen. During the morning of September 11, Igor developed an interesting eye feature on the north side of the system, and the storm approached hurricane strength. However, convection decreased during the afternoon before a burst of convection during the evening caused Igor to become a hurricane.

An eye became evident within this convection early on September 12, and Igor underwent explosive strengthening, ballooning from a minimal hurricane to a amazing Category 4 by that afternoon as a result of a drop of 50 mb in 12 hours. Igor's forward motion slowed and the system moved due west throughout that same day. Igor continued to strengthen, reaching an intensity of 150 mph late on September 12, and this remarkable intensity was maintained throughout the day of September 13, as a beautiful symmetric eye dominated the system. By that evening, surf began to increase in the Northern Leeward Islands as Igor approached from the west. Igor made a slight westnorthwest turn that night, and an eye replacement cycle destabilized the system, weakening it. However, it remained a Category 4 through the day of September 14, before organizing further and strengthening once again during the evening. Igor reached a peak intensity of 155 mph and a pressure of 925 mb later that night before the eye clouded over and a weakening trend commenced.

However, Igor organized once again, and the fluctuations in intensity continued as Igor became a powerful Category 4 once more with 145 mph winds. The convection became slightly asymmetrical, with the bulk of the cloud cover on the north side on September 16, but the moisture evened out during the evening as a a pronounced rain band formed south of the center. Also, during the day of September 16, Igor's tropical storm wind field broadened to 506 miles in diameter, making it the third-largest Atlantic hurricane on record. Igor began weakening, however, and finally lost it's Category 4 status during the late afternoon of September 16, after maintaining it for four days. The system continued to weaken over the next day. By the afternoon of September 18, Igor was a Category 2, but the windfield was still broadening, and squally weather was already sweeping over Bermuda. By later that night, Igor's tropical storm force windfield engulfed Bermuda, and the winds increased throughout the day, despite the fact that Igor weakened to a Category 1.

The center of Igor passed just to the west of Bermuda late on September 19, and the island saw sustained winds near hurricane-force as a result. Igor accelerated northeastward, and maintained a minimal hurricane status while the extratropical transition began on September 20. However, this transition wasn't completed by the time Igor passed Newfoundland, and the center passed just to the east of the island, causing sustained winds near hurricane force and dumping over 9 inches of rain in some areas, causing flooding. Igor finally became extratropical on September 21. Igor caused 3 fatalities and about $100 million in damage. The cyclone was also notable for being the largest Atlantic hurricane ever recorded, with a tropical storm force wind diameter of 920 miles.

Igor near peak intensity.

Track of Igor.

Tropical Storm Hermine (2010)

2010-09-06T08:40:00.011-04:00

Storm Active: September 5-7
On September 3, a low pressure system formed in the eastern Pacific and quickly became Tropical Depression Eleven-E. However, the system made landfall in Mexico the next day without becoming a tropical storm. The remnant low of the system moved into the Bay of Campeche on September 5 and began to organize again. The low was associated with a very large area of showers and thunderstorms covering a significant portion of the western Gulf of Mexico. Late on September 5, the system was organized enough to be upgraded to Tropical Depression Ten. Ten quickly strengthened, and became Tropical Storm Hermine early on September 6.

Hermine quickly organized, and began rapidly strengthened as it moved generally to the north. Hermine reached its peak intensity of 65 mph winds and a minimum pressure of 991 before making landfall in extreme north Mexico late on September 6. Hermine crossed the U.S.-Mexico border inland a few hours later, as it steadily weakened. Hermine maintained minimal tropical storm status for a fairly long time inland, but finally weakened to a tropical depression by the evening of September 7. By later that night, it was no longer monitored by the national hurricane center, but it still maintained tropical depression status, as it tracked northward through the central U.S. The depression merged with a frontal boundary on September 9. The remnant moisture combined with the frontal system caused heavy rain from the midwest to the northeast over the next couple of days, before moving off the coast on September 12.

Hermine inland over Texas, still maintaining tropical storm strength and a healthy outflow.

Track of Hermine.

Tropical Storm Gaston (2010)

2010-09-02T08:41:00.010-04:00

Storm Active: September 1-2
A tropical wave emerged off of Africa on August 29, and immediately began to organize, developing a low pressure center rapidly. By the beginning of September, a defined center had formed, and the system became Tropical Depression Nine early on September 1. The depression quickly crossed the border to tropical storm strength and was named Tropical Storm Gaston.

However, some Saharan dry air was still embedded in the system, preventing deep convection in the center. Meanwhile, the cyclone was tracking only very slowly westward, due to the presence of a trough to its north. The dry air present in the system weakened Gaston to a tropical depression and then a remnant low by the afternoon of September 2. However, early on September 3, convection associated with the remnant increased, and organization continued to increase over the next few days. Despite this, the low lost its good circulation, and, although convection persisted, the chance of development was significantly decreased by September 7 as it passed through the Caribbean. On September 8, the low dissipated. Gaston affected no landmasses and therefore had no impact.

Note: It is believed that the remnants of Gaston may have briefly attained tropical depression status again on September 4, but post-season analysis will confirm this after the conclusion of 2010.

Gaston on September 4, possibly a tropical depression.

The track of Gaston, with appropriate changes made from post-storm analysis.

Tropical Storm Fiona (2010)

2010-08-31T07:27:00.010-04:00

Storm Active: August 30-September 3
On August 26, a strong tropical wave emerged off the coast of Africa, adopting the same general track of Danielle and Earl before it. By later that same day, a low pressure center became embedded in the wave. Although the outflow and circulation of the system was very organized from the beginning, convection remained minimal and a tropical depression didn't form during the next few days. On August 28, a burst of convection appeared at the center, but it ebbed away over the next day. By August 30, the system was producing tropical storm force winds, and a center was found, causing the low to be promoted to Tropical Storm Fiona, with no intermediate tropical depression stage.

Its initial intensity was 40 mph winds and a pressure of 1007 mb. Fiona sped off to the west and westnorthwest over the next day, and tropical storm watches were issued for some of the Northern Leeward Islands in preparation for possible tropical storm conditions in areas that were still recovering from Hurricane Earl. Fiona's motion was at least 20 mph until September 1, when the presence of Earl to its east slowed its motion. The outflow of Earl and Fiona kept a certain distance between the two, and Fiona actually became more organized as it slowed down, unexpectedly strengthening into a strong tropical storm by the morning of September 1, with winds of 60 mph. However, later on September 1, Fiona peaked at 60 mph and a pressure of 997 mb, before losing most of its cloud cover and weakening. Meanwhile, it turned to the northwest and sped up again through the morning of September 2. Fiona recovered some convection during the day, but intense shear exposed Fiona's circulation, and as it turned north, it continued to weaken. As Fiona struggled north-northeast, its pressure rose further, and it weakened to a tropical depression and then a remnant low late on September 3, before even reaching Bermuda.

The only effects of Fiona were some showers and gusty winds in the Northern Leeward Islands and Bermuda.

Fiona at peak intensity on September 1, despite the exposure of its circulation.

Track of Fiona, notable for coinciding almost exactly with that of Tropical Storm Colin earlier that year.

Hurricane Earl (2010)

2010-08-26T07:22:00.019-04:00

Storm Active: August 25-September 4
On August 23, a strong tropical wave emerged off of Africa and immediately began to show signs of organization. The wave developed a low pressure center on August 24, and during that day, brought rain and gusty winds to the Cape Verde Islands. However, the system did not possess a closed circulation until August 25, and was then declared Tropical Depression Seven. Upon formation, Seven was already on the verge of tropical storm intensity and in another six hours, during the afternoon on August 25, the system became Tropical Storm Earl.

Overnight, the outflow of the system grew very organized, suggesting strengthening, but the location of the center itself was a moving target, reforming every few hours in a slightly different location relative to the convection. The center became more defined with a burst of convection during the evening of August 26, but the system did not undergo significant intensification overnight. Earl persisted westward during the day of August 27, and tropical storm watches were issued for portions of the northern Leeward islands as a result. Meanwhile, Earl began to strengthen, reaching strong tropical storm intensity by August 28. Despite an early turn north on the models, Earl continued west much longer than expected, and continued strengthening. Earl attained hurricane strength on August 29, and hurricane watches and warnings were issued for parts of the Northern Leeward Islands.

Finally, Earl turned westnorthwest later that day, but the outer bands of Earl began to sweep across the northeasternmost islands of the Caribbean bringing heavy rains and wind, with conditions getting progressively worse into the evening hours. By 8:00 pm EDT that night, tropical storm force surface winds covered the northern Leeward Islands, and hurricane force sustained winds also brushed these areas causing intense storm surge and flooding. Meanwhile, Earl continued to gain strength and rapidly became a Category 2 very late on August 29 and was on the verge of major hurricane strength by the morning of August 30. An eye appeared in Earl during the day as it strengthened rapidly, becoming a major hurricane quickly and then a Category 4 as it passed north of the U.S. Virgin Islands and Puerto Rico and caused tropical storm force winds and rain throughout the regions. Earl's pressure continued to drop, and by August 31, Earl had reached an amazing intensity of 135 mph winds and a 931 mb pressure. It still maintained a general westnorthwest motion, and the eye clouded over somewhat as Earl went through the Eye Replacement Cycle. The pressure rose as the cycle progressed, and Earl turned northwest, passing east of the Bahamas. But Earl maintained a Category 4 intensity until September 1, when it encountered some more significant shear and weakened to a Category 3 hurricane. Earl slowly turned to the north-northwest and recovered an eye, becoming more organized during the afternoon of September 1, and it restrengthened into a Category 4 hurricane. It surpassed its previous peak in intensity and reached its primary peak of 145 mph winds and a pressure of 928 mb early on September 2!

The storm continued to approach the Outer Banks of North Carolina during the day. By that evening, rainbands and tropical storm force winds swept over Cape Hattaras and the surrounding areas, as Earl turned north-northeast. However, hurricane force winds stayed offshore. As Earl approached, the eye clouded over again and Earl began steadily weakening, to a Category 3, and then a Category 2 by the time it passed by Cape Hattaras early on September 3. The weakening continued, and Earl was a tropical storm by the time it brushed passed Cape Cod overnight, bringing tropical storm force winds and rain to that area as well. By September 4, conditions were deteriorating in Nova Scotia. During the day, Earl made landfall in Nova Scotia and then Prince Edward Island as a powerful tropical storm, before entering the Gulf of St. Lawrence and finally becoming extratropical late on September 4 just off the coast of northeast Quebec. In total, Earl caused $150 million in damages and 3 fatalities over the areas it affected.

Earl nearing its peak intensity east of the Bahamas on September 1.

Track of Earl.

Hurricane Danielle (2010)

2010-08-22T07:17:00.017-04:00

Storm Active: August 21-30
On August 19, a broad area of low pressure formed just of the coast of Africa and quickly developed deep convection due to its proximity to the Intertropical Convergence Zone. Over the next day, the system developed two centers along the trough, one on the eastern side, toward Africa, and one on the western side. The one on the western side was 1008 mb, as opposed to the eastern's 1011 mb, and the former soon gained dominance as the other dissipated. During the day of August 20, the system remained disorganized. However, during the afternoon on August 21, a closed circulation formed, and a very apparent spin appeared on satellite images, and the disturbance was classified Tropical Depression Six that day with 30 mph winds and a pressure of 1008 mb. Six strengthened as it moved westnorthwest, and a burst of convection near the center merited an upgrade to Tropical Storm Danielle during the afternoon of August 22.

Favorable conditions with warm water and minimal wind shear allowed Danielle to strengthen significantly through the night and into August 23. By the afternoon of that day, Danielle reached hurricane strength and was still rapidly intensifying. Also, contrary to previous models, Danielle still continued on a generally westnorthwestward track overnight and into the next day. During the early morning of August 24, Danielle reached Category 2 hurricane strength. However, a dry air mass embedded itself in the system during the afternoon, briefly exposing the center! This caused Danielle to weaken to a tropical storm by the evening, but already it had recovered and started to regain strength. By early on August 25, Danielle's movement slowly was shifting to the northwest, although it was still westnorthwest for much of the morning.

The system was also a hurricane again by this time and gaining intensity. Danielle maintained an intensity of 85 mph winds and a pressure of 982 mb through the day, and turned northwest during the evening. Also, an eye feature began to develop during the night, albeit an asymmetrical one, as Danielle once again became a Category 2 hurricane. The eye had been clouded over due to the Eye Replacement Cycle, but Danielle still gained intensity into August 26. Danielle redeveloped a well-formed eye during the day, and its movement to the northwest slowed as a trough interfered with its motion. Danielle continued to strengthen, becoming the first major hurricane of the 2010 season at 2 am EDT on August 27, and became a Category 4 just three hours later with an amazing intensity of 135 mph winds and a minimum central pressure of 946 mb. Later that day, Danielle achieved its peak intensity of 135 mph winds and a central pressure of 942 mb.

After that, Danielle began to be exposed to some wind shear and cooler waters, resulting in some weakening. Danielle lost its major hurricane status early on August 28, and continued its downward trend during that day as it turned to the north. During the day of August 28, Danielle, despite being over 1000 miles from the east coast, influenced the surf along the coastline, and created 3-6 foot waves and rip currents, killing one person in Florida. However, a trough moving off the east coast picked up Danielle and began to steer it to the east. That evening, Danielle's circulation broadened and became asymmetrical, marking the beginning of its extratropical transition. This transition continued into August 29, as the cyclone accelerated northeast and weakened to a Category 1 hurricane. By August 30, it was clear that Danielle was nearly extratropical and barely holding on to minimal hurricane strength, but it somehow stayed tropical through the day and turned more eastward, weakening to a tropical storm. However, by 11:00 pm EDT on August 30, Danielle had become fully extratropical and the last advisory was issued.

Danielle's remnants became embedded in a frontal boundary the next day, and it dissipated soon after as it sped off to the east. No damage and 1 indirect death occurred from Danielle.

Danielle near peak intensity over the open waters of the Atlantic.

Track of Danielle.

Tropical Depression Five (2010)

2010-08-11T18:37:00.006-04:00

Storm Active: August 10-11
On August 8, a stationary frontal boundary off the east coast of the United States developed a low pressure system at its southern end. The force of high pressure systems to the west pushed the low south, where it encountered the typical west to east motions of the lower latitudes, and tracked over Florida on August 9, emerging over the Gulf of Mexico on August 10 as a 1010 mb low. The pressure continued to drop, and the circulation became organized enough to be declared Tropical Depression Five at its peak intensity of 35 mph winds and a pressure of 1007 mb. However, the convection associated with the system never attained definition with respect to the center, and the low weakened as it moved northwestward. The low dissipated before even reaching the Gulf coast on August 11. The broad area of low pressure associated with the dissipated Tropical Depression Five combined with a stationary front inland over the Gulf states in August 13. The system once again tracked southeastward, and approached the Gulf, intensifying as it went. As the low deepened, it became detached from the front, and by early on August 16, the system was a powerful 1010 mb low entering the Gulf with a fairly impressive clump of convection. However, no closed circulation formed, and the low made landfall once again in Louisiana without achieving tropical characteristics. The low moved north and dissipated.

Five in the Gulf of Mexico.

Track of Five.

Tropical Storm Colin (2010)

2010-08-02T16:09:00.009-04:00

Storm Active: August 2-8
On July 29, a low pressure system formed in the southeast Atlantic and slowly drifted westward. The low became associated with a large area of showers and thunderstorms, but the system remained disorganized. A tropical wave accompanied the low as it drifted westward, but the wave disengaged from the circulation, and convection decreased. However, another tropical wave, moving off of Africa combined with the low during the day of July 31, and the systems had totally merged by August 1. However, this time convection persisted within the system and it organized. A flare up of convection that defined the system's center marked the formation of a closed circulation and the system was declared Tropical Depression Four on August 2. Tropical Depression Four's initial movement was swift, to west at 17 mph, due to the steering force of a subtropical ridge to its north. Overnight, Four became more organized, and it was upgraded to Tropical Storm Colin, with 40 mph winds and a pressure of 1006 mb, during the morning of August 3.

Colin accelerated to the west at an even greater speed, reaching a velocity of nearly 25 mph during the day on August 3. Then, the low pressure associated with the system became open, and it degenerated into a remnant low. However, cloud cover persisted within the system, and a new surface low pressure center became evident late on August 4. The system continued to organize, and was redesignated Tropical Storm Colin on August 5. Colin moved northwest and slowed in movement as it encountered a high pressure system, and began to turn east, as with most cyclones in the region. It briefly attained an intensity of 60 mph winds and a pressure of 1005 mb before the circulation became widely separated from the cloud cover again and Colin weakened to a weak tropical storm (45 mph winds) by August 6. Colin became nearly stationary on August 7, and weakened further, barely a tropical storm by the time it resumed movement to the northnortheast later that day. As Colin approached Bermuda, it weakened into a tropical depression, and brought needed rain to the island. Soon after passing west of Bermuda on August 8, Colin lost its circulation and dissipated.

Tropical Storm Colin shortly after reforming on August 5. The circulation, although more organized than before, is clearly exposed.

Track of Colin.

Tropical Storm Bonnie (2010)

2010-07-22T11:21:00.006-04:00

Storm Active: July 22-24
On July 14, a tropical wave emerged off of Africa and moved westward. By July 17, the wave became associated with a broad upper-level low. However, very little storm activity accompanied the system at that time, as it moved westnorthwestward. On July 18, cloud cover increased in the system, but the circulation remained in the upper levels, prohibiting development. Over the next few days, the surface pressure began to drop and heavy rain from the system caused widespread flooding in Puerto Rico and Hispaniola. On July 22, a surface circulation appeared, and the system was upgraded to Tropical Depression Three, with 35 mph sustained winds and a pressure of 1008 mb just north of eastern Cuba. However, shower activity was primarily displaced to the north and east of the center due to wind shear.

Despite wind shear, a small intensification of the system during the evening of July 22 allowed the system to develop into Tropical Storm Bonnie, with 40 mph sustained winds. Due to an upper level ridge to the system's north, Bonnie accelerated to the westnorthwest during the morning of July 23, reaching a forward speed of 19 mph by 8:00 am EDT that morning. Soon after, the system slammed into Florida with 40 mph winds. Bonnie lost most of its convection before entering the Gulf, and was downgraded to a tropical depression. Despite a redevelopment of convection overnight, the surface pressures continued to rise and the system's center was stripped away by shear, leaving a exposed circulation. Bonnie continued struggling northwestward through the Gulf of Mexico during the morning of July 24, but it ultimately degenerated to a remnant low later that day. The low made landfall in Louisiana on July 25 as it dissipated. Damage was minimal, and one death was recorded in association with this system.

Bonnie after landfall in Florida.

Track of Bonnie.

Tropical Depression Two (2010)

2010-07-08T07:45:00.006-04:00

Storm Active: July 7-8
The tropical wave that eventually became Two formed over northeast South America within the ITCZ (Intertropical Convergence Zone). It slowly moved northwestward and as it moved into the western Caribbean, scattered thunderstorm activity began to be associated with it. However, this activity remained disorganized, and the system tracked over the Yucatan Peninsula on July 6. Although the system lost a lot of cloud cover over land, a low pressure center actually developed during this time, allowing the system to be more organized as it emerged into the Gulf of Mexico on July 7. However, the low pressure became elongated over the next few hours, and did not assume the perfect circular shape reminiscent of a healthy circulation. Nevertheless, convection continued to organize, and late that night the system was declared Tropical Depression Two with 35 mph winds and a minimum pressure of 1005 mb.

As had been the trend for a few days, the system lost much of its convection overnight, as it moved towards the Texas-Mexico border at 12 mph, but the circulation remained intact, and even strengthened a little. However, the system did not have enough time to reach tropical storm strength and made landfall in northern Mexico at 11:15 a.m. EST on July 8. The system quickly dissipated over land that evening. Overall, the main effect of Two was flooding, as it hit in an area which had already suffered from Hurricane Alex a few days eariler.

Two at landfall.

Track of Two.

Hurricane Alex (2010)

2010-06-25T19:58:00.016-04:00

Storm Active: June 25-July 1
On June 12, a strong tropical wave moved off of Africa and persisted westward. For the next week, it was held within the Intertropical Convergence Zone, and showed no signs of development. On June 20, the tropical wave moved into the Caribbean, and moved slightly northward on its westerly track. The wave tapped into the large amount of moisture in the Caribbean and developed a broad area of convection with little organization. The wave continued through the Caribbean, encountering increasingly favorable conditions as it went, and slowly organized. On June 24, a low became associated with the tropical wave, but most convection was still to the east of the circulation. However, the convection soon concentrated at the center of the system, and soon developed an apparent circulation. On June 25, an aircraft reconnaissance mission confirmed the existence of a closed circulation and the system was declared Tropical Depression One off the coast of Honduras.

Tropical Depression One continued to gain organization, and was declared Tropical Storm Alex the following morning as it moved westnorthwest at 10 mph towards the Yucatan Peninsula. During the day of June 26, Alex assumed a more westerly track, and gained intensity as its thunderstorm activity concentrated toward the center. As a result, Alex reached an intensity of 65 mph winds and a central pressure of 996 mb, before making landfall in central Belize at approximately 9:00 pm EST June 26. Alex remained over land for the next day, weakening as it went, and by the morning of June 27, Alex was downgraded to a tropical depression. Alex maintained its impressive circulation, but cloud cover depleted, as the system did not receive new water to fuel itself as it moved westnorthwestward at 12 mph. Alex slowed down and turned more to the northwest as it emerged into the Gulf during the evening of June 27, allowing for more time over the favorable Gulf, and more strengthening. As Alex reemerged over water, deep convection immediately started to appear around the center, and the system was once again upgraded to a tropical storm overnight, as it moved slowly northwest. Mild shear affected the system, but it strengthened nevertheless during the day of June 28. However, the shear lessened on June 29, and the system strengthened to a hurricane overnight.

Alex began to turned westward again, as its outer bands swept across the coast of northern Mexico and southern Texas. Alex continued west, and strengthened further, reaching its peak intensity of a category 2 hurricane with 105 mph winds and a minimum pressure of 947 mb just before landfall in northern Mexico at 10:00 pm EST June 30. Alex then began weakening, becoming a category 1 hurricane by very early the next morning, and a tropical storm a few hours later. All warnings and watches were quickly discontinued as the convection associated with Alex continued to weaken. By the late evening of July 1, Alex's circulation and vorticity were gone, and the system had dissipated. However, Alex caused $1.21 billion in damages, and 32 deaths were associated with the system. As a result, Alex was already much more costly and much more deadly than the entire 2009 Atlantic hurricane season! Also, Alex was a fairly rare event climatologically, being the first June hurricane since 1995, and the most powerful in central pressure since 1957.

Image of Alex near peak intensity just before landfall in Mexico.

Track of Alex.

Professor Quibb's Picks-2010

2010-05-21T16:26:00.002-04:00

My personal picks for the 2010 Atlantic hurricane season:

18 cyclones attaining tropical depression status
17 cyclones attaining tropical storm status
7 cyclones attaining hurricane status
4 cyclones attaining major hurricane status

Hurricane Names List-2010

2010-05-19T16:00:00.007-04:00

For the Atlantic Basin in 2010, the names list is as follows

Alex (used)
Bonnie (used)
Colin (used)
Danielle (used)
Earl (used)
Fiona (used)
Gaston (used)
Hermine (used)
Igor (used)
Julia (used)
Karl (used)
Lisa (used)
Matthew (used)
Nicole (used)
Otto (used)
Paula (used)
Richard (used)
Shary (used)
Tomas (used)
Virginie
Walter

This list is the same as the one used in 2004, except for Colin, Fiona, Igor, and Julia, which replaced the four retired hurricanes of 2004: Charley, Frances, Ivan and Jeanne.