Tropical Storm Don (2011)

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)

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)

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)

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

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

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

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! Therefore, choices of rotation and the choice of which pairs of faces are to be attached are both 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

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

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

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