# 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!

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