Editing Möbius strip (section)

==History==
{{multiple image|total_width=480
|image1=Aion mosaic Glyptothek Munich W504.jpg|caption1=Mosaic from ancient [[Sentinum]] depicting [[Aion (deity)|Aion]] holding a Möbius strip
|image2=Al-Jazari Automata 1205.jpg|caption2=[[Chain pump]] with a Möbius drive chain, by [[Ismail al-Jazari]] (1206)}}
The discovery of the Möbius strip as a mathematical object is attributed independently to the German mathematicians [[Johann Benedict Listing]] and [[August Ferdinand Möbius]] in {{nowrap|1858.{{r|pickover}}}} However, it had been known long before, both as a physical object and in artistic depictions; in particular, it can be seen in several Roman mosaics from the {{nowrap|third century CE.{{r|roman|ancient}}}} In many cases these merely depict coiled ribbons as boundaries. When the number of coils is odd, these ribbons are Möbius strips, but for an even number of coils they are topologically equivalent to [[Annulus (mathematics)|untwisted rings]]. Therefore, whether the ribbon is a Möbius strip may be coincidental, rather than a deliberate choice. In at least one case, a ribbon with different colors on different sides was drawn with an odd number of coils, forcing its artist to make a clumsy fix at the point where the colors did not {{nowrap|match up.{{r|roman}}}} Another mosaic from the town of [[Sentinum]] (depicted) shows the [[zodiac]], held by the god [[Aion (deity)|Aion]], as a band with only a single twist. There is no clear evidence that the one-sidedness of this visual representation of celestial time was intentional; it could have been chosen merely as a way to make all of the signs of the zodiac appear on the visible side of the strip. Some other ancient depictions of the [[ourobouros]] or of [[lemniscate|figure-eight]]-shaped decorations are also alleged to depict Möbius strips, but whether they were intended to depict flat strips of any type is {{nowrap|unclear.{{r|ancient}}}}

Independently of the mathematical tradition, machinists have long known that [[Belt (mechanical)|mechanical belt]]s wear half as quickly when they form Möbius strips, because they use the entire surface of the belt rather than only the inner surface of an untwisted belt.{{r|roman}} Additionally, such a belt may be less prone to curling from side to side. An early written description of this technique dates to 1871, which is after the first mathematical publications regarding the Möbius strip. Much earlier, an image of a [[chain pump]] in a work of [[Ismail al-Jazari]] from 1206 depicts a Möbius strip configuration for its drive {{nowrap|chain.{{r|ancient}}}} Another use of this surface was made by seamstresses in Paris (at an unspecified date): they initiated novices by requiring them to stitch a Möbius strip as a collar onto a {{nowrap|garment.{{r|roman}}}}