A Möbius strip is a one-sided surface formed by gluing together two opposite edges of a strip of paper after twisting one of the edges 180 degrees. The Möbius strip was named after the German mathematician and astronomer August Ferdinand Möbius, one of the pioneers of the mathematical field known as topology, which concerns the properties of shape that do not change when an object is stretched or bent without being torn. Möbius investigated the curious properties of the Möbius strip in 1858, in connection with a question about the geometry of polyhedra posed by the Paris Academy. Most historians agree, however, that Möbius was not the first to discover the strip named after him: by coincidence, the German mathematician Johann Benedict Listing also discovered the Möbius strip in 1858, edging out Möbius by a few months.
In spite of the fact that the Möbius strip is formed from a piece of paper that has two sides, the Möbius strip has only one side. If you start drawing a line down the middle of the strip on one side, and continue drawing the line until you come back to where you started, you will find that you have drawn the line on both "sides" of the strip, so that in fact the strip has only one side. Because of this property, Möbius strips have found a use in industry as conveyor belts, since when an ordinary loop is used, one side of the belt eventually gets very worn, while the other side does not get worn at all. The Möbius strip also has only one edge, as can be seen by drawing a line along one of the edges and continuing the line until it returns to its starting point; by that time, the line has run along the entire edge of the strip.
The Möbius strip has several other surprising attributes. If you make a cut all along the center loop of the strip, the result will not be two thinner Möbius strips, but a single long strip with two twists in it. Furthermore, if you cut the Möbius strip into thirds lengthwise, you will end up neither with three short loops nor with one long loop; rather, you will have one short Möbius strip intertwined with a longer loop that has two twists in it.
The Möbius strip is the simplest example of what is known as a non-orientable surface, one on which it is impossible to form a consistent notion of the difference between, say, a right hand and a left hand. Suppose that you draw a right hand on the Möbius strip, using an ink marker that soaks through the paper to the other side. Imagine that you start drawing copies of the right hand along the strip, being careful always to draw a right hand. By the time you get back to the starting place you have gone through a twist and are drawing on the reverse side of the paper, so that it faces the opposite way from the original hand: if you view the new hand from the same vantage point as the original hand, the new hand will look like a left hand. Thus there is no way to define the notion of a right or left hand, so the Möbius strip is non-orientable. What's more, mathematicians have proven that every non-orientable surface contains at least one Möbius strip, so that Möbius strips are the essential building blocks for all non-orientable surfaces.
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