The Möbius Strip

For those of you who don't know what a Möbius strip is, the instructions are simple.

Take a long strip of paper and tape the ends together, but in a special way. Give the paper a half-twist before taping it so that A is matched with D and B is matched with C. 

The finished Möbius strip looks something like this:

The Möbius strip, or Möbius band, is named after August Ferdinand Möbius, a nineteenth century German mathematician and astronomer.

The Möbius Strip has some very special properties. For example, because you taped the back side of the paper to the front side, the Möbius Strip is a one sided object. Also, because you taped the top edge to the bottom edge, it is a one-edged object.

If you aren't convinced try these tests:

(1) Start midway between the "edges" of a Möbius Strip and draw a line down its centre; continue the line until you return to your starting point. Did you ever cross an edge?

(2) Next, hold the edge of a Möbius Strip against the tip of a felt-tipped highlighter pen. Colour the edge of the Möbius Strip by holding the highlighter still and just rotating the Möbius Strip around. Were you able to colour the entire edge?

(3) Now, with scissors cut the Möbius Strip along the centre line that you drew. Then draw a centre line around the resulting band, and cut along it. Did you predict what would happen?

If you want to investigate further try making similar strips but with a full twist, or one and half twists and try the same tests.

Giant Möbius Strips have been used as conveyor belts (to make them last longer, since "each side" gets the same amount of wear) and as continuous-loop recording tapes (to double the playing time).

The famous artist, M. C. Escher, used mathematical themes in some of his work, including a Möbius parade of ants.

 

 

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