In it and its five sequels, he planted numerous topological seeds that would grow, blossom and bear fruit for decades to come. Henri Poincaré was the next to build on, and greatly expand, the study of topology, when he published the groundbreaking 123-page article “Analysis Situs” in 1895. Although largely forgotten today, Listing is also notable for writing about the Möbius band four years before August Möbius, and for coining the very term topologie. This topological understanding of Euler’s formula - in which the shapes were rubber-like and not rigid - was first presented in an article by Johann Listing in 1861. So for any shape that is topologically a sphere, its Euler number is 2 for a donut-like torus, it’s 0 for a flat disk it’s 1 and so on. If we make a polyhedron out of clay, mark the edges with a Sharpie, and roll it into a ball, the faces and edges become curved but their number doesn’t change. Moreover, all that matters is the topology of the object. Today, we would imagine these shapes as hollow shells. So Euler’s number ( V – E + F) actually derives from the two-dimensional surface of the polyhedron. Interestingly, while Euler and Lhuilier imagined their polyhedra as solid, Euler’s formula is computed using only zero-dimensional vertices, one-dimensional edges and two-dimensional faces. 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. Before long, scholars found nonconvex exceptions to Euler’s formula. It depends only on the shape itself: It is topological.Įuler implicitly assumed his polyhedra were convex, meaning a line segment joining any two points stayed completely within the polyhedron. But it eluded centuries of geometers like Euclid, Archimedes and Kepler because the result does not depend on geometry. This elementary observation has deep connections to many areas of mathematics and yet is simple enough to be taught to kindergartners. 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. 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. You might think that by then mathematicians knew almost all there was to know about polyhedra. Leonhard Euler kicked off the topological investigation of shapes in the 18th century. The explanation: They have different numbers of holes. 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. A sphere and a cube are distinct geometric objects, but to a topologist, they’re indistinguishable. Even cutting and gluing are allowed, as long as the cut is precisely reglued.
No two ways to slice it free#
A topologist is free to stretch and twist a shape. But in topology, shapes are flexible things, as if made from rubber. In geometry, shapes like circles and polyhedra are rigid objects the tools of the trade are lengths, angles and areas. 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. A topologist would say that all but the first example are holes. Yet another is as a completely enclosed space, such as an air pocket in Swiss cheese. Another is as an opening or aperture in an object, like a tunnel through a mountain or the punches in three-ring binder paper. One is as a cavity, like a pit dug in the ground. In everyday language, we use “hole” in a variety of nonequivalent ways. How have mathematicians - particularly topologists, who study spatial relationships - thought about holes? The third, however, can be viewed through a mathematical lens. What is the precise definition of a planet? A sandwich? A hole? We will leave the first two for your friends to argue about. These questions all hinge on definitions. 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.
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