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twice, forming three self-intersections. (The Ousalm vessel is named after the mythical
bird that goes around in ever decreasing circles until it disappears up its own end.) If
the Ouslam vessel is cut vertically, it falls apart into two three-twist Mobius bands.
Normally, a Klein bottle falls apart into two one-twist Mobius bands when cut, but
Bennet and Stewart show that you can also cut a Klein bottle along a different curve to
get just one Mobius band. Glassblower Alan Bennett can be contacted at Hi-Q Glass,
2 Mill Lane, Greenfield, Bedford, UK MK45 5DF.
Concluding Remarks
1. See note 1 in Chapter 5.
Appendix A
1. Solution to the random walk question. Mathematical theoreticians tell us that
the answer is one—infinite likelihood of return for a 1-D random walk. If the ant
were placed at the origin of a two-space universe (a plane), and then executed an
infinite random walk by taking a random step north, south, east, or west, the probability
that the random walk will eventually take the ant back to the origin is also
one—infinite likelihood. Our 3-D world is special: 3-D space is the first Euclidean
space in which it is possible for the ant to get hopelessly lost. The ant, executing an
infinite random walk in a three-space universe, will eventually come back to the
origin with a 0.34 or 34 percent probability. In higher dimensions, the chances of
returning are even slimmer, about l/(2n) for large dimensions n. The l/(2») probability
is the same as the probability that the ant would return to its starting point on
its second step. If the ant does not make it home in early attempts, it is probably
lost in space forever. Some of you may enjoy writing computer programs that simulate
ant walks in confined hypervolumes and making comparisons of the probability
of return. By "confined" I mean that the "walls" of the space are reflecting so
that when the ant hits them, the ant is, for example, reflected back. Other kinds of
confinement are possible. You can read more about higher dimensional walks in
Asimov, D. (1995) There's no space like home. The Sciences. September/October
35(5): 20-25.
2. Solution to the Rubik's tesseract question. The total number of positions of Rubik's
tesseract is 1.76 X 10 I2 °, far greater than a billion! The total number of positions of
Rubik's cube is 4.33 X 10 19 . If either the cube or the tesseract changed positions every
second since the beginning of the universe, they would still be turning today and not
have exhibited every possible configuration. The mathematics of Rubik's tesseract are
discussed in Velleman, D. (1992) Rubik's tesseract, Mathematics Magazine. February
65(1): 27-36.