The Moebius Strip

The Moebius Strip

You’ve probably heard the expression, “There are two sides to everything”. But are there? You can find out by making this strip.

Things you’ll need: several strips of paper 25cm long and 2cm wide, scissors, a pen, and tape.

1. To make the Moebius strip, you need to half-twist the strip of paper and tape the ends together. Now you have the Moebius strip.

2. Make some more Mobius strips. Then, cut one-half in the middle of the Moebius Strip with scissors as shown below. Oops! What happened?

3. Now cut a new strip one-third (1/3) just like the picture above but cut it 1/3.

The Möbius strip, Möbius band, Mobius or Moebius, is a surface with only one side and 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.

An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist and then joining the ends of the strip to form a loop. However, the Möbius strip is not a surface of only one exact size and shape, such as the half-twisted paper strip depicted in the illustration. Rather, mathematicians refer to the closed Möbius band as any surface that is homeomorphic to this strip. Its boundary is a simple closed curve, i.e., homeomorphic to a circle. This allows for a very wide variety of geometric versions of the Möbius band as surfaces each having a definite size and shape. For example, any rectangle can be glued to itself (by identifying one edge with the opposite edge after a reversal of orientation) to make a Möbius band. Some of these can be smoothly modeled in Euclidean space, and others cannot.

On 2. the Mobius strip will turn into a normal paper strip. On 3. the strip will turn into 2 paper bands!