A Twist on the Möbius Band
Researchers work out the shape of a paper strip
A Möbius band is a two-dimensional surface with the puzzling property of having only one side. Despite this mind-bending characteristic, it’s an easy object to make: just take a long strip of paper, give one end a half-twist, and tape the two ends together. Because of the half-twist, the front side of one end of the strip joins with the reverse side of the other end, so that the taped-together band has only one side.
When you do that, precisely what shape do you get? Until very recently, oddly enough, no one had come up with a complete answer. If the material is elastic, describing the shape is straightforward. Imagine holding a circular ring (with no thickness) and placing the midpoint of a stick at any point on it, perpendicular to the plane of the circle. Now move the stick around the circle—either direction will do—and at the same time rotate it smoothly lengthwise so that by the time it returns to its starting place, it has flipped through 180°. The shape it sweeps out in this maneuver is a Möbius band.