A sphere clearly already has a uniform geometric structure: its geometry looks the same no matter where you’re standing on the surface. A doughnut surface, by contrast, is anything but uniform: a region on the outer edge of the doughnut curves in a way that’s reminiscent of a sphere, while a region on the inner ring of the doughnut curves more like the surface of a saddle.
No matter how you try to place a torus in space — no matter how much stretching and distorting you do — there’s no way to make its geometry look the same at every point. Some parts will curve like a sphere and some like a saddle, and some parts may be flat.
Nevertheless, it’s possible to equip the torus with an abstract geometric structure that is identical at every point: simply declare that on each small patch of the torus, distances and angles are to be measured by taking the corresponding measurements on the square from which, as we’ve seen, the torus can be built. It’s impossible to build a physical torus inside ordinary space whose lengths and angles match this abstract rule, but this definition of lengths and angles is internally consistent. Since the square has ordinary flat (Euclidean) geometry, we say that the torus can be equipped with a Euclidean structure. A torus with this geometry is akin to a video game in which, when a creature exits the screen on the right, it reappears on the left, and when it exits at the top of the screen, it reappears at the bottom.
If we try to do the same thing for the double torus, however, we hit a snag. Recall that we can build a double torus by gluing the edges of an octagon. If we declare that geometry on the double torus shall mimic geometry on the octagon, we run into a problem at the octagon’s corners. After the octagon has been glued up into a double torus, the corner points are all glued together to form a single point on the double torus. Eight octagon corners meet up at that point, each corner contributing 135 degrees of angle measure, for a total of 1080 degrees, instead of the usual 360 degrees.
So if we try to give the double torus the same geometric structure as the octagon, we will end up with a double torus that has ordinary Euclidean geometry everywhere except at one point, where the surface buckles like a floppy hat with a sharp peak. (The corner points are not a problem when we glue a square to make a torus: we glue four 90-degree corners to get a perfect 360 degrees.)
To get a smooth geometric structure at the corner point on the double torus, we would need each of the octagon’s eight corners to contribute 45 degrees instead of 135 degrees. Remarkably, such an octagon does exist, but it lives not in the ordinary Euclidean plane but in another geometric structure called the hyperbolic disk: a third kind of geometry which is as uniform and internally consistent as spherical or Euclidean geometry, but which, because it is harder to visualize, was not even discovered by mathematicians until the early 19th century.
Roughly speaking, hyperbolic geometry is what you get if you declare that all the fish in Figure 3 are the same size. It’s as if Figure 3 is really the image of the hyperbolic disk through a distorted lens that makes the fish near the boundary look much smaller than the fish in the middle. In the real hyperbolic disk that is theoretically on the other side of the lens, the fish are all identical in size.
There’s no way to make a nice, smooth hyperbolic disk in ordinary space so that the fish truly are the same size. But once again, from an abstract point of view, the fish-sizing rule produces a geometry that is internally consistent and looks the same at every point — not when viewed by an outsider looking through the distorted lens, but from the perspective of someone who lives in the hyperbolic disk.
In hyperbolic geometry, the shortest path, or “geodesic,” between two points is the path that travels through the fewest possible fishes to get from one point to the other. Such a path, it turns out, is always a semicircle perpendicular to the boundary of the disk; the semicircles that go through the fishes’ spines are examples. From our distorted outside perspective, such paths look curved, but for an insider, these paths are the “straight lines”: to drive along one of them, you would never have to turn the steering wheel, as Thurston often put it. In contrast with the Euclidean plane, in which parallel lines always stay the same distance apart, in the hyperbolic disk, two lines that don’t intersect can spread apart from each other very quickly.