1963
Möbius strip
The Möbius strip has several curious properties. A line drawn starting from the seam down the middle meets back at the seam but at the other side. If continued the line meets the starting point, and is double the length of the original strip. This single continuous curve demonstrates that the Möbius strip has only one boundary.
The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858.
Many mathematical concepts are named after him, including the Möbius plane, the Möbius transformations, important in projective geometry, and the Möbius transform of number theory. His interest in number theory led to the important Möbius function μ(n) and the Möbius inversion formula. In Euclidean geometry, he systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.
The Möbius strip is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-oriented. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. The strip was immortalized by M.C. Escher (1898–1972).
Escher was once quoted as saying: “In 1960 I was exhorted by an English mathematician (whose name I do not call to mind) to make a print of a Möbius strip. At that time I scarcely knew what it was”.1 He responded to this challenge by producing two images that became famous: Möbius Strip I and Möbius Strip II. In the first of these woodcuts, which seems to depict three snakes biting each others’ tails, Escher invites us to follow the line of the snakes. What we discover, to our surprise, is that the three reptiles are all on the same surface. Even though they appear to be following two distinct orbits.
In the second woodcut, Möbius Strip II, we see nine ants all crawling in the same direction. This time Escher asks us to follow their path and confirm that it is indeed a path without end, because no matter which starting point you choose, you always end up at the same point. The ants appear to be crawling on two separate sides of a single surface, but ultimately each of them travels the entire length of the surface on which they are crawling. In both these images the paths are endless.
Swiss artist Max Bill (1908–1994) was a pioneer in sculpting Möbius strips. Starting in the 1930s, he created a variety of “endless ribbons” out of paper, metal, granite, and other materials. When Bill first made a Möbius strip, in 1935 in Zurich, he thought he had invented a completely new shape.
The artist had been invited to craft a piece of sculpture to hang above a fireplace in which everything was to be electric. The idea was to add some sort of dynamic element to increase the attractiveness of an electric fireplace that would need to glow without a mesmerizing dance of flames.
One possibility was a sculpture that would rotate from the upward flow of hot air. Bill’s design experiments included twisting paper strips into different configurations.
Over the years, Bill nonetheless became a strong advocate of using mathematics as a framework for art. As a sculptor, Bill firmly believed that geometry is the principal mechanism by which we try to understand our physical surroundings and learn to appraise relations and interactions between objects in space.
Mathematical art is best defined as “the building up of significant patterns from ever-changing relations, rhythms, and proportions of abstract forms, each one of which, having its own causality, is tantamount to a law unto itself,” he insisted.
References
- Clifford A. Pickover (March 2005). The Möbius Strip : Dr. August Möbius’s Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. Thunder’s Mouth Press. ISBN 1–56025–826–8.
- Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide Fourth Edition, Gerald Farin, September 1996.
- Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, 1976