Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Recall that in our unit on topology, we learned about the classification of 2-D surfaces. This classification was supported by looking at different 2-D surfaces under various deformations and seeing that some surfaces really are just like other surfaces. For example, both inflated and deflated beach balls are really just spheres. The theorem that we explored in that earlier discussion was that every 2-D surface can be turned into either a sphere or a torus with varying numbers of holes. We will be concerned only with orientable surfaces in this present discussion.
We can characterize the global curvature of these surfaces by examining what sort of shapes would be necessary to cover the surfaces completely. These tiles should have 90-degree angles at all vertices so that whenever four vertices meet, the angular sum is 360 degrees. This requirement ensures that every surface appears flat when viewed up close, one of the key criteria of a manifold. For a sphere, the only tiles that will satisfy this condition are equilateral triangles in which each angle is 90 degrees.
Notice that the angles of each of these triangles sum to 270 degrees, a bit more than the Euclidean 180 degrees that we would expect. This reminds us that the surface of a sphere has positive curvature.
Notice that a single-holed torus can be completely tiled with four quadrilaterals (rectangles). The angle sum, as Saccheri would no doubt recognize, is 360 degrees, which is consistent with flat, Euclidean geometry.
Notice that the two-holed torus is tiled by hexagons whose angles are all equivalent to 90 degrees. A hexagon in flat space has an angle sum of 720 degrees. Looking at our hexagons, we can see that this is not true on the surface of our two-holed torus. The angle sum of one of our two-holed-torustiling hexagons is 540 degrees, decidedly less than sum expected from a Euclidean-based geometry. This means that our surface admits only hyperbolic geometry.
We can extend this thinking to tori having any number of holes. Hexagons can be used to make any other genus of torus. This means that although there are many ways to construct a hyperbolic 2-D surface, there is only one way to construct a flat surface, and only one way to construct a spherical surface. If we want to identify the large-scale geometry of any multi-dimensional surface, without any clues we would do well to guess "hyperbolic."—the odds of being right would be in our favor.
We have seen that the vast majority of surfaces that our ant can inhabit, the 2-D universes so to speak, are hyperbolic. What about 3-D manifolds? What are the possible geometries of the space that we inhabit? This is related to a longstanding question, asked first by Poincaré at the turn of the nineteenth century and resolved only in the first few years of the twenty-first century.
As you might expect, 3-D manifolds can be curved in analogous ways to the 2-D surfaces we have seen—that is, they can be spherical, flat, or hyperbolic. Spherical space behaves like a spherical surface in that if you travel in a straight line far enough away from your starting point, you will always return to where you started, without having to turn around. This implies that the space is bounded, like the surface of a sphere. This obeys the "no-parallels" version of Euclid's fifth postulate. Furthermore, in analogy to the ant's circles, if we create a sphere and compare its surface area to its radius, we will get a smaller number than we would expect in flat space. As we create larger and larger spheres, this ratio shrinks.
Flat space behaves in a nice, Euclidean way. It obeys all five postulates; there is only one parallel line through a given point; lines extend to infinity. This is probably how most of us envision space. Since it is unbounded, we can think of it as larger than spherical space. Spheres of all sizes exhibit the same ratio of surface area to radius.
Hyperbolic space is a strange place, indeed. It can be thought of as much larger than both spherical and flat space. If we were to make spheres of a given radius, we would find that they have much larger surface areas than we would expect. Furthermore, the larger the sphere, the greater the discrepancy we would find.
Note that we said earlier that we actually live in a 4-D manifold called spacetime. So, to be clear about what we are talking about when we ask about the geometry of space, start by imagining all of space and all of time. Now, imagine taking what's called a "time-slice" of this spacetime; in other words, let's take a snapshot by looking at just one particular moment in time.
By taking this time slice, we have basically frozen the entire universe in time. People stop walking mid-step without falling; planets stop in their orbits around stars; galaxies cease their rotation. Geometrically, we now have a pure 3-D manifold with pockets of more or less curvature, depending on the mass present. Now, let's get rid of all the mass.
With no mass, we are left with "pure" space. This is a 3-D manifold that has some sort of geometry. With our now homogeneous 3-D manifold (the inhomogeneous curving effects of mass have been mitigated), the possible geometries are analogous to the geometries of the surfaces of the 2-D objects that we examined before. Now, however, instead of being polygons, the tiles will be polyhedra, and the type or types of polyhedra that will tile or "pack" a particular space are determined by the specific geometry of that space.
So, the question remains: what are the possible geometries? In 1982 William Thurston, one of the most influential modern geometers and topologists, proposed that there are eight possible geometries, Euclidean, spherical, hyperbolic, and five other systems. In the early and mid-2000s Perelman proved Thurston's claims to be correct. This result also proved the Poincaré conjecture, which considered only spherical 3-D manifolds.
Now that Thurston's geometrization conjecture has been proven to be correct— and has earned the title of "theorem"—we have essentially a complete list of possibilities for the fundamental geometry of our space. The task now is to determine which geometry actually governs the real world we live in. This is essentially what Gauss tried to do on his mountaintops so many years ago. The problem, as we have seen, is that to reach a definitive answer, we need to be able to look at extremely large shapes, much larger than anything on Earth or even in our galaxy, perhaps.
So we are, indeed, much like the ant on its surface: we know what is happening with the local curvature, but we are looking too closely to be able to discern much about the large-scale geometry of our system. If we had to guess the specific geometry of our space, we, like the ant, would do well to guess hyperbolic. Indeed, Thurston's Geometrization Theorem confirms that most spaces are spaces that obey the "many parallels" version of Euclid's fifth postulate.