Möbius Strip

A one-sided surface obtained by cutting a band width-wise, giving it a half twist, and re-attaching the two ends. According to Madachy (1979), the B. F.Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts.

A Möbius strip can be represented parametrically by

$$x = [R+s\cos({\textstyle{1\over 2}}\theta)]\cos\theta$$

$$y = [R+s\cos({\textstyle{1\over 2}}\theta)]\sin\theta$$

$$z = s\sin({\textstyle{1\over 2}}\theta),$$

for $s\in [-1,1]$ and $\theta\in [0,2\pi)$. Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called Paradromic Rings (Listing and Tait 1847, Ball and Coxeter 1987) which are summarized in the table below.

half-twists	cuts	divs.	result
1	1	2	1 band, length 2
1	1	3	1 band, length 2
			1 Möbius strip, length 1
1	2	4	2 bands, length 2
1	2	5	2 bands, length 2
			1 Möbius strip, length 1
1	3	6	3 bands, length 2
1	3	7	3 bands, length 2
			1 Möbius strip, length 1
2	1	2	2 bands, length 1
2	2	3	3 bands, length 1
2	3	4	4 bands, length 1

A Torus can be cut into a Möbius strip with an Even number of half-twists, and a Klein Bottle can be cut in half along its length to make two Möbius strips. In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled.

There are three possible Surfaces which can be obtained by sewing a Möbius strip to the edge of a Disk: the Boy Surface, Cross-Cap, and Roman Surface.

The Möbius strip has Euler Characteristic 1, and the Heawood Conjecture therefore shows that any set of regions on it can be colored using only six colors.

See also Boy Surface, Cross-Cap, Map Coloring, Paradromic Rings, Prismatic Ring, Roman Surface

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 127-128, 1987.

Bogomolny, A. ``Möbius Strip.'' http://www.cut-the-knot.com/do_you_know/moebius.html.

Gardner, M. ``Möbius Bands.'' Ch. 9 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 123-136, 1978.

Geometry Center. ``The Klein Bottle.'' http://www.geom.umn.edu/zoo/features/mobius/.

Gray, A. ``The Möbius Strip.'' §12.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 236-238, 1993.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 41-45, 1975.

Kraitchik, M. §8.4.3 in Mathematical Recreations. New York: W. W. Norton, pp. 212-213, 1942.

Listing and Tait. Vorstudien zur Topologie, Göttinger Studien, Pt. 10, 1847.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 7, 1979.

Nordstrand, T. ``Mobiusband.'' http://www.uib.no/people/nfytn/moebtxt.htm.

Pappas, T. ``The Moebius Strip & the Klein Bottle,'' ``A Twist to the Moebius Strip,'' ``The `Double' Moebius Strip.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 207, 1989.

Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, pp. 269-274, 1983.

Wagon, S. ``Rotating Circles to Produce a Torus or Möbius Strip.'' §7.4 in Mathematica in Action. New York: W. H. Freeman, pp. 229-232, 1991.

Wang, P. ``Renderings.'' http://www.ugcs.caltech.edu/~peterw/portfolio/renderings/.