Portions of this entry contributed by Margherita Barile

German mathematician and astronomer who, because of his lack of self-assertion, was relegated to a low-profile job in an observatory. On his own, he did celestial mechanics. He also studied geometry, into which he introduced homogeneous coordinates for the first time. He was also an innovator in projective geometry and an early investigator of topology.

Möbius studied at the University of Leipzig, where he switched from law to mathematics, physics, and astronomy. He joined Gauss at the observatory in Göttingen, and later Pfaff in Halle. Throughout his academic career, he bore the title of astronomer and taught mechanics. He also wrote a Textbook of Statics (1837) and studied systems of lenses. His most famous contributions, however, are in the field of pure mathematics. While Möbius was not the inventor of the one-sided so-called Möbius strip , which is actually a discovery of Johann Benedict Listing, he did introduce the notion of orientability, which allowed him to put a minus sign in front of lengths, areas, and volumes. Furthermore, the well-known strip which carries his name is not the only one-sided surface that he considered; he described a whole class of polyhedra with this property, which he called extraordinary. They all have volume zero, and violate Euler's polyhedral formula. The smallest has 10 triangular faces, 15 edges, and 6 vertices. The notion of the dual polyhedron is also due to Möbius.

Möbius's idea to use barycentric coordinates to identify points in the projective plane attracted the attention of the whole mathematical community. By creating the first significant example of homogeneous coordinates, he showed the way to transfer Descartes' analytic approach to the context of projective geometry. This was the starting point of Plücker's investigations, which defined coordinates for lines and planes as well.

The new algebraic tools developed by Möbius in Der barycentrische Calcul (1827) included a formula for the cross-ratio, and provided general solutions to various fundamental problems, such as determining a conic section passing through given points (or tangent to given lines). One of the most intriguing numerical results is the theorem according to which the probability that five points randomly chosen in the projective plane lie on a hyperbola is infinitely greater than the probability that they lie on an ellipse: the ratio arising from the computations is . The abstract formulation of the duality principle and the algebraic characterization of affine transformations are also part of Möbius new unifying vision of geometry which Gauss classified among the most revolutionary intuitions in the history of mathematics. In fact, Gauss placed the barycentric calculus beside his own theory of congruences, literal calculus, differential calculus and Lagrange's calculus of variations.

August Ferdinand was not the only member of the Möbius family to become a famous scientist. His grandson Paul, a neurologist, is remembered for his controversial theories on the structure of the human brain. According to one of his conclusions, the center of mathematical reasoning was located at the left corner of the forehead (1900). From him we also know that his grandfather used to regard mathematics as something poetic.

The German historian of mathematics Moritz Cantor tells us about Möbius' habits in everyday life. Before going out for a walk, he recite the German formula "3S und Gut" composed of the initial letters of the objects that he absolutely did not want to forget: Schlüssel (key), Schirm (umbrella), Sacktuch (handkerchief), Geld (money), Uhr (watch), Taschenbuch (notebook).

Additional biographies: MacTutor (St. Andrews), Bonn