Monday, June 1, 2009

Möbius strip

Behold the Möbius strip. It was discovered independently by German mathematicians August Ferdinand Möbius (1790-1868) and Johann Benedict Listing (1808-1882), and was the subject of this etching by Dutch graphic artist M.C. Escher (1898-1972). To make one, you simply take a strip of paper, give it a half twist, and fasten it together. Why would you want to make one? To demonstrate its curious properties:
  • It has only a single surface: you can draw a line down the center that will meet without lifting the pencil (see this link for an animation of Escher's etching).
  • It has only one edge: you can run the edge against a felt-tip pen without leaving the paper.
  • It is chiral, meaning that it cannot be a mirror image of itself, since it must be twisted to the left or right.
  • If you cut it along the center line, you will end up with two strips - each with two full twists - that are wound around each other.
The Möbius strip can be described by parametrization and cylindrical polar coordinates and the concept of a fiber bundle, but equations aside, its most practical purpose may be as a conveyor belt, because the entire surface area of the belt gets the same amount of wear.

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