Topology

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Topology is sometimes called "rubber-sheet geometry" because if a shape is drawn on a rubber sheet (like a piece of a balloon), then all of the shapes you can make by stretching or twisting—but never tearing—the sheet are considered to be topologically the same.

Topological properties are based on elastic motions rather than rigid motions like rotations or inversions. Mathematicians are interested in the qualities of figures that remain unchanged even when they are stretched and distorted. The qualities that are unchanged by such transformations are said to be topologically invariant because they do not vary, or change, when stretched.

As an example, the figure shows a triangle, a square, a rough outline of the United States, and a ring. The first three shapes are topologically equivalent; we can stretch and pull the boundary of the square until it becomes a circle or the U.S. shape. But no matter how much we pull or stretch this basic outline we cannot make it look like a ring.

Since a triangle is topologically the same as a square, and a sphere is the same as a cone, the idea of angle, length, perimeter, area, and volume play no role in topology. What remains the same is the number of boundaries that a shape has. A triangle has an inside and an outside separated by a closed boundary line. Every possible distortion of a triangle will also have an inside and an outside separated by a boundary. The ring, on the other hand, has two boundaries forming an inside region separated from two disconnected outside regions. No matter how you transform a ring it will always have two boundaries, one inside region, and two outside regions.

It can be quite challenging and surprising to discover whether two shapes are topologically the same. For example, a soda bottle is the same as a soup bowl, which is the same as a dinner plate. A coffee cup and a donut are topologically the same. But a coffee cup is topologically different from a soup bowl because the hole in the cup's handle does not occur in the bowl.

Because topology treats shapes so differently from the way we are accustomed to thinking about them, some of the most interesting objects studied in topology may seem very strange. One of the most well known objects is called the Möbius Strip, named for the German mathematician August Ferdinand Möbius who first studied it. This curious object is a two-dimensional surface that has only one side. A Möbius Strip can be easily constructed by taking the two ends of a long, rectangular strip of paper, giving one end a half twist, and gluing the two ends together. The Klein Bottle, which theoretically results from sewing together two Möbius Strips along their single edge, is a bottle with only one side. In our three-dimensional world, it is impossible to construct because a true Klein Bottle can exist only in four dimensions.

The concepts of sidedness, boundaries, and invariants have been generalized by topologists to higher dimensions. Although difficult to visualize, topologists will talk about surfaces in four, five, and even higher dimensions. While much of the study of topology is theoretical, it has deep connections to relativity theory and modern physics which also imagine our universe as having more than three dimensions.

A soprano and a baritone can sing the same song even though one sings high notes and the other low notes. Shifting a song up or down the musical scale changes some qualities of the music but not the pattern of notes that creates the song. Similarly, topology deals with variations that occur without changing the underlying "melody" of the shape.

Dimensions; Mathematics, Impossible; MÖbius, August Ferdinand.

Chinn, W.G. and Steenrod, N.E. First Concepts of Topology. New York: Random House, 1966.

"Beginnings of Topology." In Math Forum. August 98. <http://mathforum.com/~isaac/pro blems/bridges1.html>.

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