Give a paper strip one half-twist and it loses a side
Tape the ends of a paper strip together after a single half-twist and you get a surface with just one side and one edge. Mathematicians described it in 1858, but Roman mosaic makers drew it sixteen centuries earlier, and factory machinists used it to make belts last twice as long.
In 1858 the shape was found independently by two Germans, August Ferdinand Möbius and Johann Benedict Listing. Yet it appears in Roman mosaics of the third century, often as coiled ribbon borders that become one-sided whenever the number of coils happens to be odd. One mosaic from Sentinum shows the god Aion holding the zodiac as a band with a single twist, though whether the artist meant anything by it is unknown. Much later, a chain pump illustrated by Ismail al-Jazari in 1206 used the same configuration for its drive chain.
Its strangest property is that it has no consistent notion of clockwise. Slide a flat, lopsided shape once around the strip and it comes back as its own mirror image. It is the simplest surface with this property, and any surface lacking a consistent orientation must contain one. Trace its edge and you cover the whole boundary in one continuous loop, twice as long as the centreline. The strip is also chiral: a left-handed version cannot be stretched into a right-handed one.
Scissors reveal more surprises. Cutting down the middle does not produce two loops but one long band with four half-twists, which is really a cylinder. Cut a third of the way in instead and you get two linked loops, one of them a slimmer Möbius strip. Maps drawn on it can need six colours rather than the four that suffice on a flat plane, and six is always enough.
Practical uses followed. A belt twisted this way wears on both faces, so it lasts twice as long; a written account of the trick dates to 1871. Parisian seamstresses reportedly initiated novices by having them sew one as a collar. The shape also inspired the recycling symbol, artworks by M. C. Escher and Max Bill, the NASCAR Hall of Fame building, dual-track roller coasters, magic tricks by Harry Blackstone Sr. and T. Nelson Downs, and analyses of canons by Johann Sebastian Bach.
Source: Möbius strip
