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MÖBIUS BANDS
A Möbius Band has only one side and one edge. You can make one by taking a strip of paper, putting half a twist in it and sticking the ends together. You can prove that it has only one side by drawing a line along the centre of the strip until the line meets up with where you started. You will find that the line is on both sides of the original strip of paper. When the paper is cut along the drawn line it does not fall into two pieces, as might be expected, but forms a large twisted loop.
Möbius bands made from strips of paper, are great fun to play with. The disadvantage of the paper versions is that once they have been cut the original is lost. We came up with the idea of making them from double-ended zips so that hey can be ‘cut’ apart and put back together again.
August Ferdinand Möbius published a description of a Möbius band in 1865 and he explained its special properties. However, Johann Benedict Listing published a paper in 1861 (four years before Möbius) where he described the properties of the Möbius band. Both studied in Germany under Carl Friedrich Gauss, who was the greatest mathematician of his day, but they were not there at the same time. Some scientists argue that the bands should be named Listing-Möbius.

In 1836 Listing  gave the name topology to the study of objects such as the Möbius Band. This is still the name we use today but it didn’t come into common usage until the twentieth century.

There is documentary evidence that Möbius also looked at bands with more than one half twist. There does not seem to be any evidence that Listing studied these but, realistically, it seems unlikely that he would have stopped at putting just one half twist in his strip of paper.
Möbius band with one half twist, cut along the centre.
Möbius band with two half twists, cut along the centre.
Möbius band with one half twist, cut one third of the way from the edge.
Möbius cross with one straight band and one with a half twist, each cut along the centre.
A Klein bottle made from two mirror-image Möbius bands.
The correct German spelling of the name is Möbius.
In English it is often written as Mobius or Moebius.
Other Möbius things to see:

YouTube video of some of these bands being zipped and unzipped.

Crochet Möbius scarves, headbands and bracelets.

Click pictures for enlargements