Möbius Strip

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).

The Möbius strip has Euler characteristic  (Dodson and Parker 1997, p. 125).

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. M. C. Escher was fond of portraying Möbius strips, and they appear in his woodcuts "Möbius Strip I" and "Möbius Strip II (Red Ants)" (Bool et al. 1982, p. 324; Forty 2003, Plate 70).

A Möbius strip of half-width  with midcircle of radius  and at height  can be represented parametrically by

			(1)
			(2)
			(3)

for  and . In this parametrization, the Möbius strip is therefore a cubic surface with equation

(4)

The illustration above shows interlocked turning gears along the length of a Möbius strip (M. Trott, pers. comm., 2001).

The coefficients of the first fundamental form for this surface are

			(5)
			(6)
			(7)

the second fundamental form coefficients are

			(8)
			(9)
			(10)

the area element is

(11)

and the Gaussian and mean curvatures are

			(12)
			(13)

The perimeter of the Möbius strip is given by integrating the complicated function

(14)

from 0 to , which can unfortunately not be done in closed form. Note that although the surface closes at , this corresponds to the bottom edge connecting with the top edge, as illustrated above, so an additional  must be traversed to comprise the entire arc length of the bounding edge.

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.

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.

The topological result of attaching a Möbius strip to a disk along its boundary is a real projective plane, which cannot be embedded in . However, there are three surfaces that are representations of the projective plane in  with self-intersections, namely the Boy surface, cross-cap, and Roman surface.

Any set of regions on the Möbius strip can be colored using only six colors, as illustrated in Tietze's graph above.

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." https://www.cut-the-knot.org/do_you_know/moebius.shtml.

Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 243, 1976.

Bool, F. H.; Kist, J. R.; Locher, J. L.; and Wierda, F. M. C. Escher: His Life and Complete Graphic Work. New York: Abrams, 1982.

Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.

Dickau, R. "Spinning Möbius Strip Movie." https://mathforum.org/advanced/robertd/moebius.html

Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, pp. 121 and 284, 1997.

Escher, M. C. "Moebius Strip I." Wood engraving and woodcut in red, green, gold and black, printed from 4 blocks. 1961. https://www.mcescher.com/Gallery/recogn-bmp/LW437.jpg.

Escher, M. C. "Moebius Strip II (Red Ants)." Woodcut in red, black and grey-green, printed from 3 blocks. 1963. https://www.mcescher.com/Gallery/recogn-bmp/LW441.jpg.

Forty, S. M.C. Escher. Cobham, England: TAJ Books, 2003.

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.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 10, 1984.

Geometry Center. "The Möbius Band." https://www.geom.umn.edu/zoo/features/mobius/.

Gray, A. "The Möbius Strip." §14.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 325-326, 1997.

Henle, M. A Combinatorial Introduction to Topology. New York: Dover, p. 110, 1994.

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

JavaView. "Classic Surfaces from Differential Geometry: Moebius Strip." https://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_MoebiusStrip.html.

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.

Möbius, A. F. Werke, Vol. 2. p. 519, 1858.

Nordstrand, T. "Moebiusband." https://jalape.no/math/moebtxt.

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.

Pickover, C. A. The Möbius Strip: Dr. August Mobius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. New York: Thunder's Mouth Press, 2006.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 269-274, 1999.

Trott, M. "The Mathematica Guidebooks Additional Material: Rotating Möbius Bands." https://www.mathematicaguidebooks.org/additions.shtml#G_2_01.

Underwood, M. "Mobius Scarf, Klein Bottle, Klein Bottle 'Hat'." https://www.woolworks.org/patterns/klein.txt.

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." https://www.ugcs.caltech.edu/~peterw/portfolio/renderings/

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 152-153 and 164, 1991.