Irrational Numbers

Of the infinite set of numbers, the most known are the counting numbers, 1,2,3... The negative numbers and zero are combined with the counting numbers to form the wider set of integers. Integers can be divided to form fractions, therefore expanding the system to rational numbers.

All of the above is common knowledge, but the number system can be further expanded by taking the infinite sum of a convergent sequence of rationals, or, more commonly, simply taking the square root of a non-perfect square. Either method results in a number that has an infinite decimal expansion that never repeats. Mathematically, this means that the number cannot be expressed as a fraction. Numbers of this type are called irrational numbers.

Irrational numbers can also be subdivided further into two groups: the algebraic irrationals, and the transcendental irrationals. The algebraic irrationals are simply the ones that solve a polynomial equation with integer coefficients. This simply means any equation like x^2+3x+5=0, x^5-97x^3+4=0, etc. Other numbers can also be algebraic, and all rational numbers are. (for equations like 3x-2=0, the solution is 2/3. All fractions, and therefore all rationals, can be expressed this way) It is also obvious why square roots are algebraic, they all fit equations of the form x^2=n, where the solution will always be the square root of n. Cube roots, and all other roots come about in the same way. Finally, all numbers of the form a+square root of b can be reached by means of the quadratic formula. Some other irrationals are algebraic as well, but it is a lot less obvious.

Transcendental numbers are all irrational, and these are simply the remaining numbers that do not fit the above category. Among these are π (pi), e, and any nonzero powers of either of them. There are still some numbers, like the square root of two to the power of the square root of two, π-e, and others that it is not known whether they are algebraic or transcendental.

Another method for representing the irrational numbers is the continued fraction. A continued fraction looks like this:



In the above picture, the first constant a0 must be an integer, and the remaining numbers must be positive integers. If this conditions are filled, the expression is a continued fraction. Continued fractions can represent all real numbers, rational and irrational (technically, complex numbers are the sum of two continued fractions, one real, one imaginary). Unlike decimal expansions, where rational numbers can go on forever (Ex. 1/3=.333333...) every continued fraction for a rational number is finite, meaning that there are a limited number of constants in the continued fraction. Conveniently, all irrational numbers have infinite continued fractions. The continued fraction for the square root of 2 has the constants 1,2,2,2,2,2... with the two repeating indefinitely. Another interesting property of continued fractions is that the constants of a algebraic irrational number repeat (Ex. square root of three has constants 1,1,2,1,2,1,2,1,2) while transcendental irrationals have a non-repeating sequence of constants (Ex. π has the constants 3, 7, 15, 1, 292, 1, 1, 1, 2, 1...). In a sense, the number is a irrational irrational!

There are infinite irrational numbers, but three notable ones are listed below.

π (pi)
A detailed article on pi in available elsewhere on this blog, see here.

e
Type: Transcendental irrational
Decimal Expansion: 2.71828182845904523536...
Continued Fraction Constants: 2,1,2,1,1,4,1,1,8,1,1,16...(this sequence of constants does have a pattern, but it still does not repeat, making e a transcendental irrational)
Description: The constant e is important in many aspects of mathematics, particularly calculus, and it appears in many functions. e is the limit of the function (1+1/x)^x as x approaches infinity and is the base of the function e^x, which is the only function of x that when differentiated stays the same. e is also the base of the natural logarithm, denoted ln(x) which has many properties. e is related to the notable constants π, i, 1, and 0, through Leonhard Euler's famous eqaution e^πi+1=0. Overall, the constant e shows up in many unlikely places in various regions of mathematics.

φ (phi)
Type: Algebraic irrational
Decimal Expansion: 1.6180339887
Continued fraction constants: 1,1,1,1,1,1,1...
Description: φ is an irrational number that is also commonly known as the golden ratio. This is because it is the ration between two line segments that makes it also the ratio between the longer segment and the sum of the two segments. The image below illustrates this property.



φ is also the only number whose reciprocal is itself minus one (1/φ+1=φ). Interestingly, φ is also related to the Fibonacci Sequence (the sequence that starts 1,1 and each new number is the sum of the previous two). The ratio of two consecutive terms of the Fibonacci sequence approximates φ, and the approximation becomes more accurate with higher Fibonacci numbers. φ has numerous properties, and is the only number to satisfy many formulae throughout mathematics.

There are many other irrationals but the above are the most notable.

Sources: http://en.wikipedia.org/wiki/Continued_fraction (info and image), http://en.wikipedia.org/wiki/E_(mathematical_constant) (info)

Notable Hurricanes: Longevity; Part 3

This is the conclusion of a three part post. For the first part see here. For the second part see here.

#3 Hurricane Inga (1969) September 20-October 15 24.75 days (from formation to final extratropical transition)

The depression that became Inga formed on September 20 from a tropical wave that moved off of Africa a few days before. The system quickly became a tropical storm, but then was a depression again until the 28th. Upon turning northward, Inga became a hurricane and made a very tight loop southeast of Bermuda, before making a bigger loop and becoming a major hurricane, reaching its peak intensity of 115 mph winds and a pressure of 964 millibars. Still moving slowly, Inga started to weaken halfway through its loop and dissipated on October 15. Although Inga was very long lived, it didn't affect land and no damage resulted. Inga had an ACE of 31.64.



Track of Inga.

#2 Hurricane Ginger (1971) September 5-October 3 27.25 days (from formation to final extratropical transition)

Hurricane Ginger originated from a very large trough of low pressure spanning from the Gulf of Mexico, into the Bahamas, and through parts of the Caribbean. A total of four tropical cyclones, one depression, one tropical storm, and two hurricanes, could be traced back to this low-pressure area. The second of these to form from the trough was Ginger. The tropical depression that became Ginger was recognized as such on September 5, and drifted for much of the next four days. Heading southeast, and then northeast, the depression became Tropical Storm Ginger on September 10. Ginger started to accelerate east-north-east, and rapidly gained intensity before becoming a hurricane. On September 14, now a Category 2 hurricane, Ginger reached its peak intensity of 105 mph winds and a minimum sea level pressure of 959 millibars. Then, a strong high pressure system to the northeast of Ginger slowed its movement, turned it around, and weakened it to a Category 1 hurricane. It stalled for a few days, making a very tight loop on the 19th and 20th of September, before adopting a sluggish, but steady, westward movement. On September 23, Ginger passed its closest to Bermuda. Ginger stalled again a day later and drifted from the 24th to the 29th. During this time, Project Stormfury tried to weaken the storm on September 26 and September 28. Project Stormfury is a program that attempted to dissipate the clouds associated with tropical cyclones by "seeding" them, or dropping chemicals into the eyewall to weaken it. Ginger was the last hurricane to be used for this experiment due to lack of financial support and the fact that it simply didn't seem to work. The idea is sound in theory, but execution isn't feasible for two reasons. Firstly, it costs too much to carry a significant amount of chemicals to high altitudes, and second that the hurricanes quickly redevelop due to the abundance of moisture in ocean air. Soon after Ginger turned to the northwest, and became a Category 2 hurricane again, before weakening to a Category 1. On September 30, Ginger struck North Carolina, still a hurricane, but barely. It weakened inland and was nearly stationary for the first few days of October before becoming extratropical and moving off the coast the east on October 3, joining the Atlantic Ocean once more. Ginger caused some damage, $53 million adjusted for inflation ($10 million at the time), but no deaths resulted. Most of the damage came from the central coast of North Carolina, where a wide area had received over ten inches of rain from Ginger. Ginger had an ACE of 44.19.



Track of Hurricane Ginger, spanning nearly a month.

#1 Hurricane Three (1899) August 3-September 4 28 days (from formation to final extratropical transition)

The tropical storm destined to be named Hurricane San Ciriaco for the place it damaged formed on August 3, 1899. It quickly intensified in the favorable region and was a hurricane by August 5 and a major hurricane soon afterward. It brushed the easternmost islands of the Caribbean shortly before striking Puerto Rico, and its namesake, San Ciriaco, at its peak intensity as a Category 4 of 150 mph winds and an approximate pressure of 930 millibars. After devastating Puerto Rico, the system moved onward to the Dominican Republic, passing north of the island. Over the next four days or so, the hurricane passed through the Bahamas, still a Category 3, before striking North Carolina on August 17. It emerged off of North Carolina as a minimal Category 1 hurricane, and sped off eastward into open waters becoming extratropical on August 22. However, the system was pushed south for a time, and regained enough tropical characteristics to be a tropical storm again on August 26. The tropical storm slowly moved north, and then slowly east. The storm intensified and accelerated east, briefly becoming a Category 1 hurricane on October 3rd before losing its tropical features and becoming extratropical for the last time the next day. Although the damage caused by this storm cannot be calculated in dollars, it can be measured as one of the deadliest tropical cyclones in history, causing 3,433 fatalities, most of them (about 3,200 of them) occurring in Puerto Rico. Hurricane San Ciriaco also had the highest ACE of any Atlantic hurricane, at 73.57.



Track of Hurricane San Ciriaco, a long-lived and deadly hurricane.

Through this and two other posts, the top ten Atlantic hurricanes it terms of longevity have been listed. However, one record remains, the longest lived tropical cyclone in the world. Hurricane San Ciriaco is still second in the world, behind only a remarkable cyclone that was both a hurricane and a typhoon in the Pacific Ocean, due to the fact that it crossed the International Date Line. A summary of the cyclone is shown below.

#1 (world) Hurricane/Typhoon John (1994) August 11-September 10 31 days (from formation to final extratropical transition)

Due to the advanced reanalysis that occurs for all tropical cyclones after (sometimes years after) they form, this particularly powerful hurricane that formed in the Pacific, can have its wave origins trace back to the Atlantic. On July 25, 1994, a tropical wave moved off Africa, and moved westward into unfavorable conditions for development. Therefore, the wave didn't become a low pressure system until it entered the East Pacific Ocean through Central America and became a tropical depression on August 11. It quickly became a tropical storm, but wind sheer was ripping at the system, and Tropical Storm John couldn't strengthen until August 19, when John had nearly passed out of the reach of the sheering system. Upon entering this favorable zone, John became a hurricane, and soon, a major hurricane. On August 20, John passed out of the East Pacific, into the Central Pacific. Unlike the Atlantic, whose boundaries are easily measured by land and the equator, the Pacific is split into three separate basins, the East, the Central, and the West. The East Pacific goes from the west coast of Mexico and the U.S. out to a line. In this area there is an alphabetic name list, similar to the Atlantic. A cyclone that passes from the East to the Central Pacific is still called a hurricane, and retains the same name. However, cyclones that form in this basin follow a different name list that does not go in any alphabetical order, because of the relative lack of cyclones in the region. Therefore, when Hurricane John entered the Central Pacific and switched basins, it was still was Hurricane John, and was still gaining intensity. By the time John was at its closest approach to the Hawaiian Islands it was a Category 5 hurricane, but, fortunately for the islands, John was 345 miles away. Despite this, John's outer rain bands caused a significant amount of damage. However, the system couldn't turn towards Hawaii because the usual high pressure system over the islands was blocking all cyclones from turning north. Later that same day of its closest approach, on August 22, John reached its peak intensity of 175 mph winds and a pressure of 929 millibars. However, the pressure was probably much lower than this, because the pressure had last been taken when John was not at its peak intensity. Now, John took a northwest turn, and steadily weakened to Category 1 hurricane strength. Over the next days, however, John regained some of its intensity and reached Category 4 status. It was at a secondary peak with 135 mph winds when it crossed the International Date Line and entered the West Pacific. The West Pacific had another name list still, with another major difference. The scale of intensity was also different. Instead of going tropical depression, tropical storm, hurricane, it went tropical depression, tropical storm, typhoon, super typhoon. Therefore, when John crossed the date line on August 27, it was still John, but a typhoon, not a hurricane. However, upon crossing the date line, John had nearly stopped moving and by September 1, John was a stationary tropical storm. John went through a slow loop, first east, and then back west, still a tropical storm. By the time this loop ended, a trough was moving in, forcing John quickly to the northeast. On September 8, John had crossed back into the Central Pacific and was still a tropical storm. John attained hurricane strength again and reached 90 mph winds before finally becoming extratropical on September 10. John hadn't caused much damage and the only damage was to Hawaii and to Johnston Atoll in the north-central Pacific. John's ACE was not accurate because it had crossed three basins, but it is estimated to be in the 50's.



Hurricane John at primary peak intensity as a strong Category 5 hurricane.



Track of John.

Some of the above cyclones are the most notable in the world, and many have some of the most erratic tracks of any cyclones ever.

The information on this post can be credited mostly to wikipedia, which, in turn, got its information from the National Hurricane Center that keeps records of Atlantic tropical cyclones dating back to 1850. Any cyclones before that time cannot be on the above list, due to lack of information.

Notable Hurricanes: Longevity; Part 2

This is the second part of a three part post. For the first part, see here.

Most of the hurricanes so far on the longevity countdown have made a loop. This usually is a result of a blocking high pressure, which steers the system, but doesn't always weaken it. However, blocking low pressure systems steer storms and weaken them or are too weak to affect them, and are therefore are less common.

#7 (tied) Hurricane Inez (1966) September 21-October 11 20.75 days (from formation to last extratropical transition)

The tropical wave that was to become Hurricane Inez emerged off Africa on September 18. On September 21, the system became a tropical depression, and on the 24th, it became Tropical Storm Inez. Inez strengthened rapidly into a hurricane and slammed into the most eastern islands in the Caribbean. Strengthening still further, Inez reached its peak intensity of 150 mph winds and a pressure of 929 millibars before making landfall in the Dominican Republic on September 29. The system weakened and emerged over water shortly before striking Cuba. After traveling over Cuba, Inez was barely holding on to minimal hurricane strength and became a tropical storm upon entering the southwestern Bahamas on October 2. Over the Bahamas, Inez became a hurricane again and was forced to the southwest by a subtropical ridge on the 4th. Soon after, Inez crossed the Florida Keys, and began to intensify once again. By this time, Inez skimmed the northern Yucatan Peninsula on October 7, it was a Category 3 and still getting stronger. Then, on October 10, after briefly reaching Category 4 status, the system made landfall in Mexico as a Category 3. Inez weakened rapidly, and dissipated on the 11th of October. Over 1,000 people died form Inez and $222 million in damage resulted ($1.5 billion adjusted for inflation since 1966). Because of this, the name Inez was retired from the circulating name list. Inez had an ACE of 54.58.



Track of Inez.

#7 (tied) Hurricane Carrie (1957) September 2-24 20.75 days (from formation to last extratropical transition)

The tropical depression that eventually became Carrie formed near Africa on September 2. Moving slowly westward, the system gained intensity, first becoming a tropical storm, then a hurricane, and then a major hurricane. Soon after, on September 8, Carrie reached its peak intensity of 155 mph winds and a pressure of 945 millibars. The system turned north weakening to a Category 1, and then turned west strengthening to a Category 4. On September 16, the system passed Bermuda at distance of about 100 miles, not causing any damage. Soon after, the system turned east as a minimal hurricane and became extratropical on the 23rd. The only impact of Carrie was a sunken ship off the Azores on September 21. 86 were aboard the ship and only 6 survived, despite a rescue effort. Carrie had an ACE of 62.59.



Track of Carrie.

#5 Hurricane Nadine (2012) September 11-October 4 21.75 days (spent as a tropical cyclone)
A post on this hurricane is located elsewhere on this blog.

#4 Hurricane Kyle (2002) September 20-October 12 22 days (from formation to final extratropical transition)

The low pressure system that later became Kyle was originally part of a front that moved over the Atlantic Ocean on October 15. A low formed over the dissipating front on September 18 and gained some characteristics of a tropical cyclone. On September 20, the system was midway between tropical and non-tropical features, and was therefore classified as a subtropical depression. The depression, moving slowly northward in the Central Atlantic, was hindered in movement by a blocking trough to its north and another low to its south. Despite all these factors, the depression strengthened into Subtropical Storm Kyle on September 21. Then, getting close to the blocking force in the north, Kyle was forced to make a loop, first going east, then south, and then west. While making this loop, Kyle started to exhibit more tropical features, and Kyle was reclassified as Tropical Storm Kyle on September 22. The system continued generally west over the next few days, intensifying as it went, and soon becoming a hurricane. Soon after, Kyle reached its peak intensity of 85 mph winds and a pressure of 980 millibars on September 26. The next day, however, Kyle started to weaken due to wind sheer. Before long, Kyle was a tropical storm again. Kyle later weakened into a depression, although only briefly, and it regained tropical storm status on October 1. Kyle did not move for the next few days, but soon continued its trek to the west. On October 7, Kyle was stalled again, and was forced southwest, again becoming a depression. It followed a slow arc, curving southwest, then west, and finally northwest due to an approaching front, becoming a tropical storm yet again before striking the coast of South Carolina, and then North Carolina on October 11. The system emerged over water and became extratropical on October 12. The system caused minor damage and one fatality. Kyle had an ACE of 14.44.



Kyle at minimal hurricane status.



The odd-looking track of Kyle.

For the last part of this post, see here.

Notable Hurricanes: Longevity; Hurricane ACE

Among the many Atlantic hurricanes that have formed in the past 150 years or so, most are notable because of their intensity or for their impact. In this post, starting with the 12th, the top twelve hurricanes in terms of longevity will be listed. But first, a little word about ACE, or Accumulated Cyclone Energy, which combines the intensity and longevity of a cyclone into one statistic.

ACE is a formula that adds together the intensities of a cyclone every six-hour period throughout the total history of the cyclone. For each period of six hours, the maximum sustained wind speed is estimated, squared, and divided by ten thousand. This is the ACE figure for that six hour interval. The sum of all the intervals determines the ACE of a storm. However, there are a few extra rules associated with ACE. For one, the ACE is only taken when a cyclone is tropical (tropical meaning that the cyclone has hurricane-like characteristics). When a cyclone is extratropical or subtropical (midway between extratropical and tropical, but there is no given line between these categories) ACE is not taken. Also, when the tropical cyclone has winds less than 40 mph, i.e. a tropical depression, the ACE figure drops to zero. Most long-lived hurricanes have high ACE's, but this isn't always true. Since the statistic applies to longevity and intensity, an intense long-lived storm will have the highest ACE.

#12: Hurricane Ivan (2004) September 2-24 18.75 days (as a tropical cyclone)

Hurricane Ivan was a Cape Verde hurricane that became a tropical depression on September 2, 2004. The system became a tropical depression and then a hurricane quickly. Ivan also set the record for most southerly Atlantic major hurricane when it became a Category 3 on September 5 at 10.6 N latitude. It continued west-north-westward and became a Category 5 hurricane on September 9. Hurricane Ivan, continuing through the Caribbean and fluctuating between Category 4 and 5 once or twice, devastated the Cayman Islands, Cuba, Jamaica, and Grenada. Billions of dollars in damage and over 50 deaths occurred during this rampage, despite the fact that Hurricane Ivan did not make landfall in any of these places, but passed by them. On September 13, the storm entered the Gulf of Mexico and approached the coast of Alabama. The storm made landfall in the United States at Category 3 intensity on September 16 and became a remnant low on the 18th. Causing 54 deaths in the region, the low continued northeastward The low was disregarded while it slowly turned to the south, and emerged back into the Gulf from Florida. On September 22, the system became a tropical depression again, and the National Hurricane Center, recognizing the depression as the remnants of Ivan, called it Tropical Depression Ivan. It strengthened into a tropical storm shortly before making its second landfall in Louisiana and causing minor damage. The system dissipated on the 24th. Although the storm was only the 10th most long lived, it had the second highest ACE of any storm in the Atlantic: 70.38.



Ivan as a Category 5 in the Caribbean.



Track of Ivan. The circles denote the storm as tropical with blue the weakest and red the strongest. The triangles show the remnants of Ivan when it was not tropical.

#11 Hurricane Nine (1893) September 25-October 15 19.25 days (as a tropical cyclone)

Due to the lack of radar and other tracking instruments in 1893, it is not known whether the storm actually originated on the 25th of September. However, it is known that a Cape Verde hurricane formed in late September of this year, and very slowly made its way westward. The system quickly strengthened into a hurricane and then a major hurricane. Since names weren't introduced until 1950, the system was simply called Hurricane Nine. It took a turn north and then once again followed a westerly path as a Category 3 hurricane through the Bahamas. On October 13, the hurricane struck Myrtle Beach, South Carolina, and accelerated northward, dissipating on the 15th. The system caused 28 deaths and had an ACE of 63.5.



Track of Nine.

#10 Hurricane Four (1926) September 2-24 19.5 days (as a tropical cyclone)

A tropical storm formed on September 2 and moved northwest very slowly for the next two weeks. During this time, the system became a Category 4 hurricane, and its movement slowed even more. Then, it turned northeast and made a tight loop. It became extratropical shortly after and dissipated on the 24th. Although the system caused no damage to land, it had the third highest ACE of any Atlantic hurricane, at 67.59.



Track of Four.

#9 Hurricane Alberto (2000) August 3-23 19.75 days (as a tropical cyclone)

The tropical depression that was to become Hurricane Alberto formed very close to the African coast on August 3. The system moved generally westward, and strengthened into a tropical storm and then a hurricane. However, on August 9, the system experienced wind sheer, and weakened to a tropical storm, and turned northeast. After moving out of the influence of the wind sheer, the system strengthened to a hurricane and then reached its peak intensity of 125 mph and 950 millibars on August 12. Then, due to the influence of a subtropical ridge, Alberto began a large loop in the north-Central Atlantic, and once again weakened to a tropical storm on the 14th of August. Then, it began to turn south, and then west again, before becoming a hurricane for the third time on August 18. It became a Category 2 hurricane again, and reached a secondary peak intensity of 105 mph winds Then, Alberto finally turned northward once again, starting slow, and then accelerating, and finally becoming extratropical late on August 23. Although no damage resulted from Alberto, it was notable for its odd track, and it had an ACE of 36.9, which is fairly low for Alberto's longevity, but since it never went beyond Category 3, it wasn't intense for much of its lifetime.



Hurricane Alberto at its primary peak intensity as a major hurricane.



Track of Alberto.

The top ten is continued on Notable Hurricanes: Longevity; Part 2.

Lightning

Lightning and thunder are the easily observed (and heard) phenomena that are associated with storms, specifically thunderstorms. Although lightning is a very common and well known occurrence, the actual nature of lightning, and the variety of types, are relatively unknown.

The best known type of lightning, called cloud-to-ground lightning, originates when ionized particles in a cloud form an electric field. It is unknown from where the negatively charged particles come from, probably from space in the form of solar wind. These ionized particles form a substance called plasma. (see here) This plasma in the cloud is negatively charged for normal lightning, and forms a large electric field. This electric field is strong enough to attract positively charged particles to cluster on the ground. As the storm moves across the sky, a group of particles follows a similar path along the ground. Soon, enough particles have accumulated and the negatively charged particles flow from the cloud downward. During this process, the flow of particles can split into many different paths, causing the phenomenon we know as fork lightning. We cannot see this part of the lightning's development, because the voltage is relatively low in this stage. In fact, the process that brings streams of negative particles down from the cloud taking a "long" time, relative to the flash, sometimes lasting up to a second. As the particles approach the ground, the attraction becomes powerful enough that the positive particles on the ground start to defy gravity, and soar upward. Then, the particles meet. Suddenly, the positively charged particles can neutralize their counterparts in the cloud easily by moving along the path. A huge movement of positive particles follows, resulting in a flow of energy 500 times larger than previously, heating the surrounding air to a remarkable temperature of 60,000 degrees. This flow, the original stroke of which lasts only a few milliseconds, is the flash we see. Additional flows of particles can occur in the next few hundred milliseconds, which add more flashes to the first.

When this extreme temperature hits the air, the velocity of the air molecules increases hundredfold, causing them to bump into other molecules, creating more heat. All this sudden movement is similar to an explosion, the sound of which creates thunder. Roars of thunder are longer than lightning flashes because the sound comes from each part of the lightning bolt. The sound from the ground area reaches you first, and then the sound from the upper part of the bolt. Additional lightning strokes cause prolonged thunder.

Although cloud-to-ground lightning is the type we know best, there are many different types. Cloud-to-ground lightning causes many visual effects which include the appearance of a ribbon, when lightning strikes move in a direction with each stroke, blending together into a "ribbon". Other effects also occur, such as forked lightning. However, this is all under the category of cloud-to-ground. Dry lightning, or lightning striking the ground without any precipitation, is also under this category.



A bolt of lightning with many forks. The forks are negatively charged paths that never connected with the group of positive particles on the ground. Some positive particles defied gravity and illuminated these forks, but most followed the main path, which is the "easiest" way to reach the cloud.

The most common type of lightning is not cloud-to-ground, but cloud-to-cloud lightning. Most lightning of this type is hidden by clouds, and causes the visual effect called sheet lightning. Hidden by clouds, only a flash can be seen from the lightning, which brightens all the clouds around it. Heat lightning is also a type of cloud-to-cloud lightning that has nothing to do with heat at all, but refers to the occurrence of lightning without accompanying thunder, which is a result of the lightning being too far away to hear.

There is also another type of lightning that is not a visual phenomena, but the nature of the lightning itself is different from normal. In normal lightning negatively-charged cloud bottoms are the source of the electric field. However, in the cloud itself a group of positive particles must exist to even out the charge. Usually this is at the top or the anvil of the cumulonimbus thunderstorm cloud. If this positive group of particles is extremely strong, which is necessary for the event, the negative particles accumulate on the ground and a lightning bolt goes all the way from the top of the cloud to the ground, which can be up to a ten mile span. Also, this type of lightning often travels up to thirty miles to strike an area that is ahead of the thunderstorm. Therefore, this type of lightning can occur when it is sunny, a trait that has earned it the name "bolt from the blue". The voltage on this positive lightning is ten times that of normal lightning and lasts for ten times longer, making this type of a hundred times more powerful overall.



A bolt of positive lightning, from the top of the cloud 50,000 feet high to the ground.

Also, many miles above the cloud tops, upper-atmospheric, or megalightning occurs, often mirroring lightning down below. This is probably due to the mirrored electrical charges that correspond to the lightning below. However, they come in different types. Sprites, which occur less than a second after their companion lightning strokes, are huge, and can occur up to 60 miles from the surface and appear a dim red color. Another sometimes related event is an elve, or a halo that sometimes occurs alone, and sometimes with a sprite. They are the same color as sprites, and are very faint. A third type that occurs in the upper-atmosphere are so-called blue jets that start about 9 miles up, follow a blue arc through the sky stretching to about 30 miles above the surface. The cause of this phenomenon is unknown, and these mysterious blue jets are elusive and difficult to record. Little is known about the nature of these transient luminous events (the name given to them that doesn't strictly relate them to lightning and puts them in a more unique category, because they lack many characteristics of normal lightning) and many people are researching these odd events.



A table comparing the different types of transient luminous events or upper-atmospheric lightning.

Lightning also can occur on other planets, and definite occurrences have been recorded on Jupiter and Saturn. Venus has also had lightning recorded on it by early probes but recently there has been a debate whether the probes actually detected lightning. Hopefully, this mystery will be resolved by future spacecraft.

Before the nature of lightning was known, many different civilizations thought to be struck by lightning was a punishment from the gods. If this was true, the gods obviously had a problem with Roy Sullivan, a man who, between 1942-1977 was struck by lightning seven times. This is a world record according to the Guinness Book of Records, which has stood since.

Lightning is an amazing event, to witness and to study. There is still a lot that we don't know about it that will hopefully be revealed in the future.

Sources: http://upload.wikimedia.org/wikipedia/commons/8/86/Darwin_storms.jpg (image), http://geology.about.com/od/sprites/a/sprites.htm, http://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Upperatmoslight1.jpg/400px-Upperatmoslight1.jpg (image)

Hyperbolic and Other Geometries (Part III)

Note: This is the conclusion of a three part post. For the first part, see Hyperbolic and Other Geometries (Part I) and for the second part, see Hyperbolic and Other Geometries (Part II).

It may seem that hyperbolic and elliptic geometry have no relevant meaning in the physical world, because everything here is purely, and simply, Euclidean. However, the surface of the Universe itself has a basis in the other types of geometry. In fact, there isn't any perfect Euclidean geometry in the physical world at all!

The geometry of the surface of space is not actually the shape of the Universe as a whole. Even if the geometry of the surface of the Universe was hyperbolic, the Universe could still be finite. This is because the surface of space isn't really a physical thing. In essence, it is what all the objects in the Universe "rest on". The gravitationally heavier objects, such as stars, make a dent in this surface and cause nearby objects to roll towards it. For example, the Sun creates a small depression in the center of the solar system, and all the planets roll toward it. However, the planets don't go hurtling into the Sun because each of them has a forward velocity. The closer you get to the Sun, the faster the forward velocity is needed to not roll into the Sun, because the depression is "steeper" in this area. Some objects are theorized to be so "heavy" on this surface, that they actually create a "hole" in space, with a steep well surrounding it. These objects are black holes. The hole in space could actually open a path to a parallel plane of space, an alternate universe!

With all these dents and holes, the Universe clearly isn't homogeneous or isotropic. Therefore, the crucial part is finding what the overall curvature of the Universe's surface is, i.e. what it would be without any of the inhomogeneities. As odd as it seems, the density of the Universe determines its geometry. It is known that there is a specified value in the Universe called the critical density of it. The critical density is the density of the Universe at which the curvature is zero and the surface of the Universe is Euclidean. If the the actual density is smaller than the critical density, the surface of the Universe is hyperbolic. If the density is higher than the critical density, the surface of space is elliptic.



In this image, the omega symbol represents the quotient of the actual density and the critical density of the Universe. For example, if the critical density equals the actual density, the omega would equal 1.

To calculate the density of the Universe, one must take into account not only density from the matter. One must also consider the mass contributed by energy. There is energy in empty space, called dark energy. The famous equation E=mc^2 shows that energy, E is equivalent to a certain amount of mass, m. Therefore, dark energy also adds density to the Universe. Regular matter and dark matter are attracted to each other gravitationally, but, oddly enough, dark energy seems to exert antigravity, or a force that pushes matter apart. The result of this is the creation of more space in the Universe, which in turn creates more dark energy, which accelerates the expansion even further. Despite the fact that the expansion of the Universe lowers the density of matter, the production of dark energy keeps the overall density constant throughout the lifetime of the Universe.

Using the Cosmic Background Radiation, one can easily infer how much matter existed at the beginning of the Universe. Using the speed at which different galaxies are receding from us, one can infer how much dark energy there is. Therefore, the densities of both dark energy and matter have been calculated. They do indeed add up to the critical density. To be exact, matter and dark matter makes up 27% of the actual density and dark energy makes up 73%. At the beginning of the Universe, the density of matter was minutely close to 100% of the critical density, and there was almost no dark energy. Since the amount of matter and dark matter hasn't changed for billions of years (since matter stopped spontaneously annihilating and turning into energy 380,000 years after the Big Bang) its density has steadily decreased. To varies with this change, the density of dark energy has increased. Therefore, the surface of the Universe is indeed flat. What does this tell us? It tells us that the Universe will never stop expanding. A value for the actual density that is the critical density or less dooms us to a fate of an infinitely expanding Universe. Only if the actual density is greater than the critical density will matter dominate. If this was the case, gravitational forces between matter would eventually cause the Universe to contract. Therefore, the Universe will continue to expand, at an ever increasing rate, until the Universe is infinitely large and infinitely old.

Overall, the fallacy of Euclid's parallel postulate has lead to to not only discover other geometries and theorems in mathematics, but has also determined the fate of our Universe.

This post and the others in this series were inspired by the following books: The Road to Reality by Roger Penrose Chapter 2, and Origins by Neil De Grasse Tyson and Donald Goldsmith Chapter 5.

Hyperbolic and Other Geometries (Part II)

Note: This is part II of a post. For the first part, see Hyperbolic and Other Geometries (Part I).

Now that it is true that the Parallel Postulate is false, we can return to the constant C, which is the proportionality constant between the difference between 180 and the angles of a hyperbolic triangle and the area of that triangle. The constant C can differ for different hyperbolic planes. This is because not all hyperbolic planes have the same curvature. To fully explain curvature, one must also look at the Euclidean plane.

The Euclidean plane is defined to have curvature 0. That means that the surface of Euclidean geometry is flat. Hyperbolic planes, however, have negative curvature. There are different hyperbolic planes because there are different possible values for the curvature of the hyperbolic plane. This is unlike Euclidean geometry, which only has one possible value for the curvature, i.e. 0. But what happens if there is positive curvature? If there is positive curvature another whole type of geometry is introduced, which is even more radical than hyperbolic geometry. This is elliptic geometry. This is the only one of the three types of geometry that has a finite plane, which is one of a sphere. In elliptic geometry, all structures are projected onto a sphere. The theorems resulting include the fact that the angles of a triangle add up to more than 180. A representation of a triangle projected on a sphere is shown below.



This particular triangle is projected onto the Earth. The angles on the triangle add up to 230 degrees. Note that, in the close up image, the relative curvature of the area approximates zero, and the angles do add up to very close to 180 degrees. In elliptic geometry, Euclid's other postulates are violated. In this geometry, all but the fourth of Euclid's postulates are false (the fourth postulate is that all right angles are equal. The first postulate, that exactly one straight line can be drawn between any two points is false because, on a sphere, multiple lines can connect two points. Take longitude lines, for example. All longitude lines are drawn between the North and South poles, and in elliptical geometry, they all all straight lines. The second postulate, that a line can be continued indefinitely without ever intersecting itself is also false, because any latitude line on a globe eventually wraps its way around and intersects with itself. The third postulate (that a circle can be drawn with any center and radius) is false because if a circle is big enough, it will wrap around the sphere and intersect with itself, violating the very definition of a circle because the points on the edge of the circle would not all be the same distance from the center.

Just as the area of a triangle on a hyperbolic plane can be expressed with the equation 180-(x+y+z)=CΔ where C is a constant and Δ is the area of the triangle, the area of an elliptic triangle can also be expressed in terms of its angles, with the equation Δ=R^2(a+b+c-180). Again, Δ is the area of the triangle, while R is the radius of the sphere where the triangle is located. It is apparent that the quantity a+b+c-180 must be positive because the radius must be positive for the area of the triangle to be positive. If you combine these two equations and solve for C in terms of R, you obtain

C=-1/R^2

Going back to hyperbolic geometry, C, as the constant of variation for the area, must have a positive value. In hyperbolic geometry, therefore, for C to be positive, 1/R^2 must be negative (because then -1/R^2 would be positive). But this is impossible because all numbers square to a positive number or 0. Therefore, the solution to this equation for R must be imaginary, or the square root of a negative number. For information on imaginary numbers, see here. This seems impossible. How can a sphere have a radius that isn't positive or even a real number? It turns out an imaginary radius does have a geometrical meaning. The shape produced in this situation is called a pseudo-sphere. A representation of a pseudo-sphere is shown below.



In essence, this structure is kind of that of an inside-out sphere. The pseudo-sphere is wonderful, but what does it mean? It came about because the radius of a sphere that is a hyperbolic plane is a pseudo-sphere. It makes sense to say that hyperbolic geometry is actually projected onto this shape, just as an elliptic triangle is projected on a normal sphere. This may not seem like the case, but a pseudo-sphere is yet another Euclidean representation of hyperbolic geometry.

In conclusion, these are the only three types of geometry that fit the criteria. The criteria is that the geometries are homogenous and isotropic. Homogenous means that each point on the surface of the geometry is the same, and isotropic means that the perspective of the area around each point is the same as from any other point. There are other geometries that are combinations of the above three, and these usually include some regions with one geometry, and some regions with other types.

For the conclusion of this post, see Hyperbolic and Other Geometries (Part III)

Hyperbolic and Other Geometries (Part I)

Most people have some knowledge about regular plane, or Euclidean, geometry. The basis of many geometric theorems relies on five basic rules, or axioms, about Euclidean geometry that cannot be proved or disproved, but are simply guidelines that geometry must follow. People take these statements to be obviously true. The five are as follows (as stated by Euclid in Elements)

1. Exactly one straight line can be drawn between any two points
2. A finite line segment can be extended continuously without intersecting itself
3. A circle can be made from any center and radius
4. All right angles are equal to one another
5. Parallel Postulate: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if continued indefinitely, will eventually meet

In this post, the fifth of the five statements above is the one that will mainly be dealt with. This is the parallel postulate. In simple terms, as illustrated in the above image, if two lines are slightly inclined towards each other, in other words the angles between them and an intersecting line add up to less than 180 degrees, they will intersect eventually, whether it is a hundred units away, a million units away, or a googleplex units away, they will intersect. If alpha and beta in the picture above add up to more than 180 degrees, than the lines will incline away from each other and will meet only if you travel in the other direction. However, if the angles add up to exactly 180 degrees, the lines will be parallel. They will never meet. An equivalent property to this is the fact that all the angles of a triangle add up to 180 degrees. This is because if the lines are parallel, they never intersect. Yet another equivalent statement as the Parallel Postulate is the fact that for a line and a fixed point in a plane (that is a two-dimensional surface), only one line drawn through the point can be parallel to the given line (note that in three dimensions, two lines that never intersect aren't necessarily parallel. This is because they can be on two separate planes. Two lines with this relationship are called skew lines) All of the axioms above seem like common sense, right? However, most new branches of mathematics are opened up when one challenges the most simple axioms and properties of math.

In this case, we are challenging Euclid's axioms. The fifth of the statements, the Parallel Postulate, seemed the least obvious, and for centuries, various mathematicians attempted to prove this. Many of these mathematicians used a method known as proof by contradiction. Proof by contradiction begins by assuming what you're trying to prove is false and then trying to find a contradiction to a known axiom directly resulting from this fact. If a contradiction is reached, then the statement cannot be false and is therefore true. This method has be repeatedly used by mathematicians, and one notable example includes the proof that the square root of two is irrational (see here). The point is that many mathematicians assumed that the parallel postulate is false. If this is assumed to be false, then more than one line can be drawn through a point parallel to a given line. The many theorems that were derived from this assumption were very odd, indeed. However, no contradiction could be found. We now know that the Parallel postulate is neither true or false; it is true only in some cases. While searching for a contradiction, mathematicians found something much more interesting: a whole new type of geometry! In this geometry, a theorem that the angles of a triangle add up to less than 180 degrees can be derived. This new geometry is called hyperbolic geometry. It is very difficult to represent this geometry with images in Euclidean geometry, but one visual representation is shown below.



This is a representation of hyperbolic geometry in a Euclidean circle. In hyperbolic space, each angel or devil are precisely the same size. However, this representation has squeezed the entire plane of hyperbolic geometry into one circle, called the bounding circle. To the inhabitants of this Universe, you can go as far as you want, but you will never reach the bounding circle because the plane goes on forever. Shapes, lines, and other Euclidean structures can all be represented in hyperbolic geometry, although they may look different than in Euclidean space. Here is a triangle as it looks like on the hyperbolic plane.



The values shown are of each angle along with the total sum off to the right. Note the the sum of the angles, 107.1 degrees, is much smaller than what it "should" be, 180. In fact, it was discovered that the value of 180 minus the sum of the three angles is proportional to the area of the triangle, a relation somewhat unexpected by mathematicians. The following equation represents this relation.

180-(x+y+z)=CΔ

x, y, and z represent the three angles of a triangle and the symbol Δ represents the area of the triangle. In this equation, the constant C is not a set value for all hyperbolic planes. In fact, unlike Euclidean geometry, there are different types of hyperbolic planes, which will be mentioned later. Note that if the line segments on the above triangle were continued into lines, they would continue curving and also "reach" the bounding circle (obviously the finite curve that we see intersecting the bounding circle is indeed infinite from a hyperbolic standpoint).

Returning to the origin of this geometry, is the parallel postulate indeed false on this plane? Is is possible to draw more than one parallel line through a given point? Yes. The picture below is a different visual representation of hyperbolic geometry, but the structures are similar.



This representation may look different, but it is viewing the same plane. The lines in the lower right are parallel, but they curve in opposite directions, rather than remaining a fixed distance fro m each other. From this image, it is clear that infinitely many lines could be drawn through the same point as the given line, but would still never intersect it. Euclid's parallel postulate is indeed false in this geometry.

For the next part of this post, see Hyperbolic and Other Geometries (Part II).

2009 Season Summary

The 2009 Atlantic Hurricane Season was below average with eleven depressions forming, nine of which became tropical storms, three of which became hurricanes, and two out of the three became major hurricanes. In total, about $174 million in damages resulted (wealth value for 2009) and fifteen direct fatalities. No hurricanes hit the U.S. this year, and only one made landfall. Although this was a relatively quiet season, there were a few notable storms.

• Tropical Depression One forming in May made a tropical cyclone form in the Atlantic Ocean in May for three consecutive years
• No tropical storms formed until August 12, which was the latest start since 1992
• Hurricane Fred achieved major hurricane status the farthest south and east of any Atlantic cyclone ever recorded
• Tropical Storm Grace formed the farthest north and east of any tropical cyclone in the Atlantic Basin
• Hurricane Ida made landfall in the U.S. as a tropical storm in November, becoming one of only six cyclones to do so

Overall, due to an El Nino event, the 2009 Atlantic Hurricane Season was relatively quiet and not much damage resulted.

Intertropical Convergence Zone

The Intertropical Convergence Zone is an area near the Equator that circumnavigates the globe. Most of it lies between 10ºN and 10ºS. The trade winds, or the winds near the Equator, converge on this zone. Since the winds are rotated in different directions by the Coriolis Effect, no low pressure systems, and therefore no tropical cyclones, can form too near the Equator.



The Intertropical Convergence Zone over the East Pacific. Although no low pressure systems can form in the ITCZ, it is constantly marked by thunderstorm activity.

However, the boundary of the ITCZ, or the ITCZ Axis, is always shifting, and a tropical system can sometimes, but very rarely, from below 5ºN or above 5ºS. A notable example is Typhoon Vamei, which was a Western Pacific cyclone that reached tropical storm strength at a mere 1.5ºN in December of 2001, which is the closest formation to the Equator on record. Also notable is 2004's Cyclone Agni, which formed farther from the Equator than Vamei and moved towards it, eventually reaching the most southerly point of 0.7ºN before turning back northward. Claims on this system are disputed because it was not officially tracked until a few days after the record was set but a fair amount of evidence supports that Agni does indeed hold the record for closest cyclone to the Equator.



Cyclone Agni at record latitude, a mere 45 miles from the Equator.

Although the winds over the ITCZ hinder tropical cyclones from forming within the region, they do provide an important factor for tropical cyclone development: tropical waves. A vast majority of tropical cyclones form from tropical waves, which are areas of convection that typically move west over the ITCZ. After the wave moves northward, it then can develop a low pressure system, and eventually become a tropical system. Therefore, the ITCZ is an important factor in tropical cyclone development, even if tropical cyclones can't form in the area itself.



A map of the formation and progression of tropical waves before becoming tropical cyclones.