Möbius, August Ferdinand

German Mathematician and Astronomer 1790–1868

An astronomy professor at the University of Leipzig, German mathematician and theoretical astronomer August Ferdinand Möbius made special contributions to theoretical astronomy and analytical geometry. Möbius studied law at the University of Leipzig but soon discovered he disliked the subject, and in his first year decided to follow his interests in mathematics, astronomy, and physics.

Upon graduation in 1815, Möbius was appointed chair of astronomy at the University of Leipzig. However, he was not granted a full professorship, as he was a poor lecturer and was unable to attract many fee-paying students. Yet Möbius's publications spanned a broad range of topics in both astronomy and mathematics. His writings were influenced by his astronomy teacher Karl Mollweide, theoretical astronomer Carl Friedrich Gauss, and mathematician Johann Pfaff, who was Gauss's teacher.

Möbius was also a pioneer within the then-developing mathematical field of topology. He is best known for his description of the curious-looking, one-sided Möbius strip—a loop that has no inside or outside, no top or bottom.

The Möbius Strip

The Möbius strip is a surface that is formed by taking a long, rectangular strip of paper, rotating the ends 180 degrees (a half-twist) with respect to one another, and then gluing the edges together. The result is a one-sided, one-edged Möbius strip. Theoretically, it is a two-dimensional surface with only one side, but it has been constructed in three dimensions.

Strange as it sounds, if a continuous line is drawn along the middle of the loop, the line will eventually end up where it began, thus showing that a Möbius strip has only one side. Even stranger is the result when the Möbius strip is cut along the line down the middle of the loop. Instead of falling apart into two loops, the strip becomes a single, two-sided loop twice as long as the original strip. In contrast, an ordinary paper ring cut in half would give two separate rings, each of them the same length as the original.

Applications of the Möbius Strip

Applications of the Möbius strip concept show up in a variety of settings: monumental sculptures, literature, music, art, magic, science, engineering, synthetic molecules, postage stamps, knitting patterns, skiing acrobatics, and even the recycling symbol.

For example, freestyle skiers have named one of their acrobatic stunts the "Möbius Flip." Author Martin Gardner wrote a humorous short story called "The No-sided Professor" based on the Möbius strip. Artist M.C. Escher included in his many drawings a march of ants on a Möbius strip.

The Möbius strip also has practical applications. A Möbius strip conveyor belt will last twice as long as a normal two-sided belt because twosided belts quickly wear out on one side. Similarly, a Möbius filmstrip that records sound on continuous-loops, or a Möbius tape in a tape recorder, will double the playing time.

Bibliography

Gray, Jeremy. "Möbius' Geometrical Mechanics." In Möbius and His Band: Mathematics and Astronomy in Nineteenth-century Germany, eds. John Fauvel, Raymond Flood, and Robin Wilson. New York: Oxford University Press, 1993.

Stewart, Ian, and John Fauvel. "Möbius' Modern Legacy." In Möbius and his Band: Mathematics and Astronomy in Nineteenth-century Germany, eds. John Fauvel, Raymond Flood, and Robin Wilson. New York: Oxford University Press, 1993.

Internet Resources

Peterson, Ivars. "Möbius and his Band." In Ivar's Peterson's Math Trek. <http://www.maa.org/mathland/mathtrek_7_10_00.html>.