If you’re looking to pick a fight, simply ask your friends, “Is Pluto a planet?” Or “Is a hotdog a sandwich?” Or “How many holes does a straw have?” The first two questions will have them arguing yay or nay, while the third yields claims of two, one and even zero.
These questions all hinge on definitions. What is the precise definition of a planet? A sandwich? A hole? We will leave the first two for your friends to argue about. The third, however, can be viewed through a mathematical lens. How have mathematicians — particularly topologists, who study spatial relationships — thought about holes?
In everyday language, we use “hole” in a variety of nonequivalent ways. One is as a cavity, like a pit dug in the ground. Another is as an opening or aperture in an object, like a tunnel through a mountain or the punches in three-ring binder paper. Yet another is as a completely enclosed space, such as an air pocket in Swiss cheese. A topologist would say that all but the first example are holes. But to understand why – and why mathematicians even care about holes in the first place — we have to travel through the history of topology, starting with how it differs from its close kin, geometry.
In geometry, shapes like circles and polyhedra are rigid objects; the tools of the trade are lengths, angles and areas. But in topology, shapes are flexible things, as if made from rubber. A topologist is free to stretch and twist a shape. Even cutting and gluing are allowed, as long as the cut is precisely reglued. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable. If you want a mathematical justification that a T-shirt and a pair of pants are different, you should turn to a topologist, not a geometer. The explanation: They have different numbers of holes.
Leonhard Euler kicked off the topological investigation of shapes in the 18th century. You might think that by then mathematicians knew almost all there was to know about polyhedra. But in 1750, Euler discovered what I consider one of the all-time great theorems: If a polyhedron has F polygonal faces, E edges and V vertices, then V – E + F = 2. For example, a soccer ball has 20 white hexagonal and 12 black pentagonal patches for a total of 32 faces, as well as 90 edges and 60 vertices. And, indeed, 60 – 90 + 32 = 2. This elementary observation has deep connections to many areas of mathematics and yet is simple enough to be taught to kindergartners. But it eluded centuries of geometers like Euclid, Archimedes and Kepler because the result does not depend on geometry. It depends only on the shape itself: It is topological.
Euler implicitly assumed his polyhedra were convex, meaning a line segment joining any two points stayed completely within the polyhedron. Before long, scholars found nonconvex exceptions to Euler’s formula. For instance, in 1813 the Swiss mathematician Simon Lhuilier recognized that if we punch a hole in a polyhedron to make it more donut-shaped, changing its topology, then V – E + F = 0.