The Great Curve

Illustration by Kamran Samimi Readers of H.P. Lovecraft know non-Euclidean geometry as the basis of the architecture of the nightmare corpse city of R’lyeh, but to mathematicians, non-Euclidean geometry is simply another way of dividing up angles, planes, and shapes, and totally doesn’t cause anyone to go insane. We’re Lovecraft readers, not mathematicians, so we called up John C. Stillwell, a professor of mathematics at the University of San Francisco, to learn something about non-Euclidean geometry. Most people have a rough idea of what Euclidean geometry is. It’s the geometry of flat surfaces such as a blackboard or a table. The typical features of these surfaces are parallel lines, triangles whose angles add up to 180 degrees, and rectangles—figures whose angles are all right angles. It’s a kind of geometry that’s orderly and very simple, and we take many of its properties for granted, such as the ability to make scale drawings—plans of a house, say, that are smaller than the actual house, but exactly the same shape. There are many other types of geometry, however. Perhaps the one that people can understand most easily is the geometry of the sphere, because we live on a sphere. Geometry is different on a sphere. There are “lines” on a sphere; namely, the great circles, such as the equator. These lines are straight from the viewpoint of creatures living on the sphere, but they behave differently from lines in the plane. The angles of a spherical triangle add up to more than 180 degrees, and lines are not infinite—they come back to where they started. Also, the shape of a triangle depends on how big it is; the bigger the triangle, the bigger the sum of its angles. So that’s one kind of geometry you might call non-Euclidean. Non-Euclidean geometry arises because of curvature. The convex kind of curvature exemplified by spheres is called positive curvature, but there’s also negative curvature, which is the curvature that you have on a saddle-shaped surface. If you imagine trying to join a whole lot of identical saddle-shaped surfaces together to form an infinite surface, you find that it gets crinkly and crumpled; it doesn’t fit easily in Euclidean three-dimensional space. But in principle, such surfaces can exist. Hyperbolic geometry is the geometry of an infinite surface of constant negative curvature. It is what people usually mean when they speak of non-Euclidean geometry. Hyperbolic geometry is closer to Euclidean than spherical geometry is, because in hyperbolic geometry the lines are infinitely long. The crucial difference is that every line can have many parallels, and some of them are lines that get closer and closer together but never meet. In Euclidean geometry, parallels just stay the same distance apart. The property of having many parallels causes hyperbolic geometry to diverge from Euclidean geometry in other ways, too. In particular, the angle sum of a triangle is always less than 180 degrees, and the circumference of a circle grows exponentially with the radius (instead of being proportional to radius, as in the plane). Thanks to exponential growth, the circumference of a modest-size circle in the hyperbolic plane can be huge. This is why a surface of constant negative curvature becomes crinkly as it grows larger. Euclidean three-dimensional space is not a good environment for exponential growth. Sadly, this is bad news for the human population, which also tends to grow exponentially. If we live in a Euclidean three-dimensional space—which seems to be very nearly the case—population growth will always be curbed by geometry.