Topology: The Mathematics of Form

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Topology: The Mathematics of Form

Overview

Mathematics began in earliest times as a collection of practical methods for counting and measuring. In the nineteenth century a branch of geometry arose in which sizes and distances were irrelevant. Topology, sometimes called "rubber sheet geometry," deals with properties that remain the same when an object is bent or stretched. These properties include the number of times a curve intersects itself and whether a surface is open or closed.

Background

During the lifetime of the famous Swiss mathematician Leonhard Euler (1707-1783), there were seven bridges in the Prussian town of Königsberg, where two branches of the River Pregel flowed around an island. The townspeople amused themselves by considering whether it was possible to cross all seven bridges in one continuous trip without re-crossing any of them. Eventually the so-called Königsberg Bridge Problem became a well-known puzzle, but no one was able to find a solution.

Euler approached this problem by replacing the land areas by points, or vertices, and the bridges by lines connecting the vertices. An odd vertex is one in which the number of lines connected to it is odd, and an even vertex has an even number of lines connected to it. The problem may then be viewed as a graph, and the object is to traverse the graph without backtracking or lifting the pencil from the paper.

The solution Euler published, in a paper that is considered to be the first publication in the field of topology, was that a graph of this type can be traversed if it has only even vertices. If it has one or two odd vertices, it can be traversed in one trip, but not with a return to the starting point. In general, a graph with 2n odd vertices, where n is any integer, will require n distinct trips. In the Königsberg problem, all four vertices are odd. Therefore, its traversal could not be accomplished in a single trip.

Another contribution Euler made to topology was his famous formula for a polyhedron, or many-sided solid:v-e+f = 2where v is the number of vertices of the polyhedron, e is the number of edges, and f is the number of faces.

Euler published his formula in 1752. In 1813 Antoine-Jean Lhuilier (1750-1840) literally found a hole in it. That is, the formula does not work if there are any holes in the solid. The general formula isv-e+f = 2 - 2gwhere g is the number of holes. This distinction among objects with different numbers of holes was key to the further development of topology.

Impact

Topology holds all objects that can be continuously deformed into one another to be equivalent, regardless of their original shape. To understand distinctness in topology, it is helpful to think of a ball of clay. The rules are that you can stretch or mold it, but you are not allowed to push a hole into it, tear it, or stick unattached edges of it together. You can mold the ball into a cube, and in fact in topology there is no difference between a cube and a sphere.

Suppose you want to make a torus, or doughnut shape, out of your ball of clay. One way might be to flatten it out and push a hole through the center with your thumb. But according to the rules of topology, that isn't a valid deformation. So you try molding your sphere into a cigar shape. So far, so good. Now all you need to do to make a torus is to wrap your cigar shape around and stick the two ends together... but that's not allowed either. A torus is fundamentally different from a sphere.

The term topology, the study of shape, was first used by Johann Benedict Listing (1802-1882). Having introduced the term in correspondence beginning about 1837, he published Vorstudien zur Topologie ten years later. Listing's ideas on topology owed much to Carl Friedrich Gauss (1777-1855). Among Gauss's innovations was the linking number, which determines whether circles are linked and is invariant even when the circles are continously deformed.

Around 1850 Georg Friedrich Bernhard Riemann (1826-1866) was at the University of Göttingen studying for his Ph.D. under Gauss's supervision. Listing was also a professor there. In Riemann's thesis, which received an uncharacteristically enthusiastic response from the rather haughty Gauss, he developed the concept that was later to be called the Riemann surface. The importance of this work was that it introduced topological methods into the theory of complex variables. Complex numbers are those of the form a + bi, where i is the imaginary square root of -1. The Riemann surface is multi-layered, allowing a multi-valued function of a complex variable to be interpreted as a single-valued function.

One of the most famous constructions in topology is the Möbius strip. This was discovered independently by August Frederick Möbius (1790-1868) and Listing in the early 1860s. The Möbius strip is intriguing because at first glance it appears to be a simple ring with inside and outside surfaces. However, it is actually a single-sided surface.

You can construct a Möbius strip yourself by taking a strip of paper and giving it a half-twist before taping the two ends together. Then take a pencil and, beginning anywhere along the strip, follow it around its length, keeping always to the middle of the strip. You will find that, because of the half-twist, you will continuously traverse what appears to be both "sides" of the paper until your pencil arrives back it its starting point, demonstrating that the construction really only has one side. If you cut along the line you have made, you don't get two narrow rings as you might expect; instead, the strip stays in one piece. Repeat the procedure and you get two separate, intertwined strips.

The Klein bottle is analogous to the Möbius strip but in three dimensions. It is a closed surface, but it has no inside or outside. Unfortunately, the Klein bottle does not lend itself to demonstration. Just as the two-dimensional Möbius strip is constructed with a half-twist through the third dimension, constructing a Klein bottle would require manipulation in a fourth spatial dimension.

The topological concept illustrated by the Möbius strip and Klein bottle is that of connectivity. Connectivity was among the ideas formalized by the French mathematician Jules-Henri Poincaré (1854-1912) in 1895 in a series of papers called Analysis situs ("Positional Analysis"). Poincaré was among the first to use the tools of algebra in topology.

One familiar topological problem first posed in the middle of the nineteenth century was not solved until more than 100 years later. Maps are generally drawn so that no two regions sharing a boundary are the same color. Mathematicians were interested in determining the minimum number of colors required to color any map in this way. Three colors were clearly not enough; a map of four regions could easily be drawn in which each region shared a boundary with three others, and that required four colors.

In 1890 it was proven mathematically that five colors would always be sufficient. A map had never been found in which more than four colors were required. Individual proofs had been constructed for four-color mapping of up to about 40 regions by the middle of the twentieth century. Still, a general proof that four colors were always enough remained elusive. The problem was finally solved in 1976 by mathematicians at the University of Illinois, with the aid of more than 1,000 hours of computerized number-crunching.

SHERRI CHASIN CALVO