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More Than Just a Plane Game

When topologists started to explore the world of geometric forms in higher dimensions, they found that neither the intuition nor the vocabulary of ordinary geometry was sufficient to describe and classify the new forms they discovered. They introduced "manifold" as a general term for a certain common type of higher-dimensional geometric object.

In everyday parlance, a manifold is a pipe or chamber bristling with subsidiary tubes. Its mathematical manifestation encompasses surfaces that appear flat, or Euclidean, locally but on a larger scale may bend and twist into exotic and intricate forms.

A circle, although it curves through two dimensions, is an example of a one-dimensional manifold, or one-manifold. A close-up view reveals that any small segment of the circle is practically indistinguishable from a straight line.

Similarly, a sphere's two-dimensional surface, even though it curves through three dimensions, is an example of a two-manifold. Seen locally, the surface appears flat. Roughly speaking, Earth's surface is an example of such a manifold.

One way to think about two-manifolds is in terms of gluing together the sides of a rubbery many-sided, or polygonal, figure. Consider a creature moving about on a rectangle. It may find that when it moves off the top edge of the rectangle, it reappears at the bottom. When it moves off the right edge, it reappears at the left.

Many video and computer games operate on the same principle to keep a figure on the screen. Researchers often use a similar strategy in computer simulations of physical phenomena to circumvent the special effects that may happen at edges or walls.

Such movements can occur if, in some sense, the top of the rectangle is glued to the bottom and the right side to the left. The resulting surface is a two-manifold called a torus. The first gluing creates a tube, and the second gluing connects the two ends of the tube to form a doughnutlike ring.

 Gluing one pair of opposite sides of a rectangle creates a cylinder. Joining the remaining pair of sides turns the cylinder into a doughnut, or torus.

Introducing a twist or two produces more exotic shapes. For example, joining two sides of a rectangle after giving it a half twist produces the one-sided, one-edged surface called a Möbius strip. Connecting the remaining sides (without a twist) creates a bizarre form known as a Klein bottle, named after the German mathematician Felix Klein (1849–1925).

 Putting in a twist (so that the arrows match) before gluing together a pair of opposite sides of a rectangle produces a Möbius strip. Carefully gluing together the remaining pair of sides results in a Klein bottle. In this particular representation, the Klein bottle (with a segment removed to reveal its figure-eight cross section) can be seen as a figure eight that moves in a circle as it changes its orientation.

The gluing idea can also be applied to the description of three-dimensional manifolds. For instance, one can try to imagine gluing together the opposite faces of a flexible cube to produce a so-called three-torus.

It's like attaching the front wall of a room to the back wall, the left wall to the right wall, and the floor to the ceiling. The result is a room that has to bend around and join itself in the fourth dimension. A person living in such a house would probably find its geometry quite disconcerting—stepping through a door and ending up on the opposite side of the room, passing through the ceiling to come up through the floor, disappearing at one wall and reappearing at another.

To provide a hint of what such a space would feel like, Jeff Weeks, a freelance geometer based in Canton, N.Y., has created a Web site (http://www.northnet.org/weeks/TorusGames/TorusGames.html ) where visitors can play games such as tic-tac-toe and chess and solve mazes and other puzzles on rectangular or square boards that are actually connected behind the scenes to form a torus or Klein bottle.

Beware, however. Tic-tac-toe on a Klein bottle can be really tricky, and that surface is just a two-manifold! Imagine how challenging, confusing, and disorienting it would be to play typical board, or planar, games on a three-torus or some other three-manifold.

Three-manifolds may serve as models of a universe that has a finite volume yet no boundary of any sort. "It's quite possible that the real universe is flat, finite, and has no edges or boundaries," Weeks notes. "What seem to be distant galaxies may be images of our own galaxy. Some of the light that enters our telescopes may be light which left our galaxy billions of years ago and has made a complete trip around the universe."

References:

Peterson, I. 1998. Circles in the sky. Science News 153(Feb. 21):123.

______. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.

Thurston, W.P., and J.R. Weeks. 1984. The mathematics of three-dimensional manifolds. Scientific American (July):108.

Weeks, J.R. 1985. The Shape of Space. New York: Marcel Dekker.

Try playing some games (tic-tac-toe, maze, crossword, word search, jigsaw, and chess) in a finite space at http://www.northnet.org/weeks/TorusGames/TorusGames.html. Additional information about such geometry games and related software is available at http://www.geom.umn.edu/docs/forum/weeks_software/.

Information about the Geometry Center's The Shape of Space video is available at http://www.geom.umn.edu/video/sos/.

Illustrations created using Mathematica 3.0 (http://www.wolfram.com).