While the three-sphere is the fundamental model for spherical geometry, its not the only such space. Just as we built different flat spaces by cutting a chunk out of Euclidean space and gluing it together, we can build spherical spaces by gluing up a suitable chunk of a three-sphere. Each of these glued shapes will have a hall-of-mirrors effect, as with the torus, but in these spherical shapes, there are only finitely many rooms to travel through.
Is Our Universe Spherical?
Even the most narcissistic among us dont typically see ourselves as the backdrop to the entire night sky. But as with the flat torus, just because we dont see a phenomenon, that doesnt mean it cant exist. The circumference of the spherical universe could be bigger than the size of the observable universe, making the backdrop too far away to see.
But unlike the torus, a spherical universe can be detected through purely local measurements. Spherical shapes differ from infinite Euclidean space not just in their global topology but also in their fine-grained geometry. For example, because straight lines in spherical geometry are great circles, triangles are puffier than their Euclidean counterparts, and their angles add up to more than 180 degrees:
In fact, measuring cosmic triangles is a primary way cosmologists test whether the universe is curved. For each hot or cold spot in the cosmic microwave background, its diameter across and its distance from the Earth are known, forming the three sides of a triangle. We can measure the angle the spot subtends in the night sky one of the three angles of the triangle. Then we can check whether the combination of side lengths and angle measure is a good fit for flat, spherical or hyperbolic geometry (in which the angles of a triangle add up to less than 180 degrees).
Most such tests, along with other curvature measurements, suggest that the universe is either flat or very close to flat. However, one research team recently argued that certain data from the Planck space telescopes 2018 release point instead to a spherical universe, although other researchers have countered that this evidence is most likely a statistical fluke.
Unlike the sphere, which curves in on itself, hyperbolic geometry opens outward. Its the geometry of floppy hats, coral reefs and saddles. The basic model of hyperbolic geometry is an infinite expanse, just like flat Euclidean space. But because hyperbolic geometry expands outward much more quickly than flat geometry does, theres no way to fit even a two-dimensional hyperbolic plane inside ordinary Euclidean space unless were willing to distort its geometry. Here, for example, is a distorted view of the hyperbolic plane known as the Poincaré disk:
From our perspective, the triangles near the boundary circle look much smaller than the ones near the center, but from the perspective of hyperbolic geometry all the triangles are the same size. If we tried to actually make the triangles the same size maybe by using stretchy material for our disk and inflating each triangle in turn, working outward from the center our disk would start to resemble a floppy hat and would buckle more and more as we worked our way outward. As we approached the boundary, this buckling would grow out of control.
From the point of view of hyperbolic geometry, the boundary circle is infinitely far from any interior point, since you have to cross infinitely many triangles to get there. So the hyperbolic plane stretches out to infinity in all directions, just like the Euclidean plane. But in terms of the local geometry, life in the hyperbolic plane is very different from what were used to.
In ordinary Euclidean geometry, the circumference of a circle is directly proportional to its radius, but in hyperbolic geometry, the circumference grows exponentially compared to the radius. We can see that exponential pileup in the masses of triangles near the boundary of the hyperbolic disk.
Because of this feature, mathematicians like to say that its easy to get lost in hyperbolic space. If your friend walks away from you in ordinary Euclidean space, theyll start looking smaller, but slowly, because your visual circle isnt growing so fast. But in hyperbolic space, your visual circle is growing exponentially, so your friend will soon appear to shrink to an exponentially small speck. If you havent tracked your friends route carefully, it will be nearly impossible to find your way to them later.
And in hyperbolic geometry, the angles of a triangle sum to less than 180 degrees for example, the triangles in our tiling of the Poincaré disk have angles that sum to 165 degrees:
The sides of these triangles dont look straight, but thats because were looking at hyperbolic geometry through a distorted lens. To an inhabitant of the Poincaré disk these curves are the straight lines, because the quickest way to get from point A to point B is to take a shortcut toward the center:
Theres a natural way to make a three-dimensional analogue to the Poincaré disk simply make a three-dimensional ball and fill it with three-dimensional shapes that grow smaller as they approach the boundary sphere, like the triangles in the Poincaré disk. And just as with flat and spherical geometries, we can make an assortment of other three-dimensional hyperbolic spaces by cutting out a suitable chunk of the three-dimensional hyperbolic ball and gluing together its faces.
Is Our Universe Hyperbolic?
Hyperbolic geometry, with its narrow triangles and exponentially growing circles, doesnt feel as if it fits the geometry of the space around us. And indeed, as weve already seen, so far most cosmological measurements seem to favor a flat universe.
But we cant rule out the possibility that we live in either a spherical or a hyperbolic world, because small pieces of both of these worlds look nearly flat. For example, small triangles in spherical geometry have angles that sum to only slightly more than 180 degrees, and small triangles in hyperbolic geometry have angles that sum to only slightly less than 180 degrees.
Thats why early people thought the Earth was flat on the scales they were able to observe, the curvature of the Earth was too minuscule to detect. The larger the spherical or hyperbolic shape, the flatter each small piece of it is, so if our universe is an extremely large spherical or hyperbolic shape, the part we can observe may be so close to being flat that its curvature can only be detected by uber-precise instruments we have yet to invent.