## Möbius bandMöbius strip, that has only one surface
and one edge and is therefore of great interest in topology.
In particular, it is the starting point for creating non-orientable
surfaces – those for which the concepts of right and left have
no meaning. If a surface is non-orientable it must contain a Möbius
band, and if a surface contains a Möbius band it must be non-orientable.
The Möbius band also falls into the category of mathematical objects
known as ruled surfaces. ## DiscoveryThe band (or strip) is named after the German mathematician, pioneer of topology, and theoretical astronomer August Ferdinand Möbius (1790-1868), who discovered it in September 1858, although his compatriot and fellow mathematician Johann Benedict Listing (1808-1882) independently devised the same object in July 1858.## Construction, experiments, and propertiesIf you now cut down this line with a pair of scissors, the result is surprising: a single long strip with a full 360° twist in it. This new shape is topologically equivalent to a cylinder (although it can't be manipulated into an ordinary cylinder because it is embedded differently in space). Try experimenting further with the products of your experiments. For example, what happens if you cut down the middle of the long strip you get when you cut down the middle of a Möbius band? What happens if you start off with a (non-Möbius) band that has three half-twists and cut it down the middle, or a third the way from one edge? How many surfaces and edges do your creations have? Is there a pattern to the results so that you can easily predict what will happen to a given object sliced in a particular way? A closely related strange geometrical object is the Klein bottle, which can be produced by gluing two Möbius bands together along their edges; however, this can't be done in ordinary three-dimensional Euclidean space without creating self-intersections. ## Practical applications
Other practical applications of Möbius bands have included half-twisted, continuous-loop recording tapes (to double the playing time), typewriter ribbons, and computer printer cartridges. In the 1960s Sandia Laboratories used Möbius bands in the design of versatile electronic resistors. Also there have been a number of ideas for children's toys based on the shape, including a railway in which a miniature engine traverses a three-dimensional looped track using magnets to hold on while it travels some of the sections upside-down. ## The Möbius band in art and sculptureM. C. Escher famously used the Möbius band as the basis for a number of his lithographs including Möbius Strip II (1963) shown here. American physicist and artist Robert Rathbun Wilson (1914-2000) was also fascinated by the shape. "Möbius Strip" is among several of Wilson's sculptures that grace the grounds of the Fermi National Accelerator Laboratory (Fermilab) in Batavia, Illinois. This 8-foot-diameter, three-dimensional version of the loop with a half-twist stands in the middle of a circular pool atop the Norman F. Ramsey Auditorium in front of the Fermilab's largest building – Wilson Hall.## The Möbius band in psychoanalysisTopology and the nature of the self would seem to be very disparate subjects. Yet the French psychoanalyst Jacques Lacan was able to draw a connection. Lacan argued that we tend to think of the self by analogy with the body: clear outlines, and a clear division between inside and outside. He used topological models such as the Möbius band and self-intersecting surfaces to suggest that things may be a good deal more complex than that. [Thanks to Tony Thwaites, from the University of Queensland, for notification of this.]## Möbius bands, the universe, and everythingIn his bookArt & Physics: Parallel Visions in Space, Time, and Light,
Leonard Shlain remarks on the cosmic symbolism of the Möbius band and
how it was employed by Escher in this context.^{3} Shlain points
to the seemingly distinct natures of space and time which are like the two different sides
of spacetime. "But," says Shlain, the
distinction, like the apparency of opposing surfaces on the Möbius
strip, is an illusion. When followed out to their own ends, both sides meet
in one seamless, endless, recursive continuum." Shlain goes on to argue
that the Möbius band is a visual repudiation of Aristotle's claim that extremes
cannot be brought together via an excluded middle. ## The Möbius band in science fictionIn "A Subway Named Möbius" (1950), A. J. Deutsch imagines a train line that has become so complex and convoluted that one of the trains on it disappears into another dimension. The mathematical connection with a Möbius band is tenuous but the story is nevertheless intriguing and memorable.Martin Gardner's "No-Sided Professor" tells of a mathematician who invents a folding that goes one step beyond Möbius and creates an object with zero sides! Perhaps the most striking of all science fiction stories that use the Möbius theme is Arthur C. Clarke's "The Wall of Darkness" (1946). A mysterious, impenetrable wall runs all the way around a planet, in a universe in which this world and its sun are the only objects. Two men, one with money, the other with building skills, determine to scale the wall and discover what's on the other side. What they find is both Möbian and rather unsettling. ## Other
## References- Gardner, Martin.
*Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleights-of-Mind from Scientific American*. New York: Vintage, 1978. - Pickover, Clifford A.
*The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology*. Thunder's Mouth Press, 2006. - Shlain, Leonard.
*Art & Physics: Parallel Visions in Space, Time, and Light*. New York: William Morrow & Company, 1991.
## Related categories• TOPOLOGY• SPACE AND TIME • MATHEMATICS | |||||||||||||||||||

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