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
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