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.
When I saw this photograph (bottom) today of an Australian metrologist holding a near-perfect silicon sphere (weighing 1.0000000 kg, smooth to the nearest 0.0000000003 meter, and round to within 0.000000050 meters), I immediately thought of the print (top) by one of my favorite artists, Dutch printmaker M.C. Escher (1898-1972). Doing some more poking around, I found the photograph of Escher holding the sphere in his 1935 lithograph. The comparison is appropriate, since many of Escher's artworks were mathematically inspired.
