The third, however, can be viewed through a mathematical lens. How have mathematicians — particularly topologists, who study spatial relationships — thought about holes? 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. 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. 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. 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.
Today, we would imagine these shapes as hollow shells. Moreover, all that matters is the topology of the object. Each surface has its own Euler number.
Around the same time, Bernhard Riemann was studying surfaces that arose in his study of complex numbers. He observed that one way of counting holes was by seeing how many times the object could be cut without producing two pieces.
For a surface with boundary, such as a straw with its two boundary circles, each cut must begin and end on a boundary. So, according to Riemann, because a straw can be cut only once — from end to end — it has exactly one hole. If the surface does not have a boundary, like a torus, the first cut must begin and end at the same point. A hollow torus can be cut twice — once around the tube and then along the resulting cylinder — so by this definition, it has two holes.
In it and its five sequels, he planted numerous topological seeds that would grow, blossom and bear fruit for decades to come. The modern definition of homology is quite involved, but it is roughly a means of associating to each shape a certain mathematical object.
From this object we can extract simpler information about the shape, like its Betti numbers or its Euler number. The rules are simple: The loops can slip and slide around, and can even cross themselves, but they cannot leave the surface.
On some surfaces, like a circular disk or a sphere, any loop can shrink down to a single point. Such spaces have trivial homology. But other surfaces, like a straw or a torus, have loops that wrap around their holes.
These have nontrivial homology. The torus shows us how to visualize Betti numbers. We can produce infinitely many nontrivial loops on one, and they can wind, double back and wrap around multiple times before ending at their starting point. Eppstein, D. Evans, J. ACM 10 , and , Francis, G. A Topological Picturebook. Gardner, M. New York: Scribner's, Gemignani, M. Elementary Topology. Gray, A. Greever, J. Theory and Examples of Point-Set Topology.
Heitzig, J. Hirsch, M. Differential Topology. Hocking, J. Kahn, D. Kelley, J. General Topology. Kinsey, L. Topology of Surfaces. Kleitman, D. Lietzmann, W. Visual Topology. London: Chatto and Windus, Lipschutz, S. Theory and Problems of General Topology. New York: Schaum, Mendelson, B. Introduction to Topology. Munkres, J. Elementary Differential Topology.
Topology: A First Course, 2nd ed. Oliver, D. Praslov, V. Providence, RI: Amer. Rayburn, M. Renteln, P. Seifert, H. A Textbook of Topology. Shafaat, A. Shakhmatv, D. Sloane, N. Steen, L. Counterexamples in Topology. Thurston, W. Three-Dimensional Geometry and Topology, Vol. Tucker, A. Open Problems in Topology. New York: Elsevier, Veblen, O. Analysis Situs, 2nd ed. New York: Amer. Weisstein, E. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions.
Practice online or make a printable study sheet. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. MathWorld Book. Wolfram Web Resources ». Created, developed, and nurtured by Eric Weisstein at Wolfram Research. Wolfram Alpha » Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.
Step-by-step Solutions » Walk through homework problems step-by-step from beginning to end. Wolfram Problem Generator » Unlimited random practice problems and answers with built-in Step-by-step solutions. Wolfram Education Portal » Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.
0コメント