Möbius

Möbius refers to August Ferdinand Möbius (1790–1868), a German mathematician and astronomer. His work spanned geometry, topology, and number theory, and several mathematical constructions bear his name. Notable eponymous concepts include the Möbius strip, the Möbius transformation, and the Möbius function, which have roles in topology, complex analysis, and arithmetic.

The Möbius strip is a non-orientable surface with only one side and one boundary component. It is formed by taking a rectangular strip and joining its ends after giving one end a half-twist. A cut along the middle yields a single longer strip with two twists. The surface is used to illustrate orientation, topology, and geometry, and it serves as a classic example in discussions of non-orientability.

A Möbius transformation (also called a linear fractional transformation) is a map f(z) = (az+b)/(cz+d) with ad − bc ≠ 0 acting on the extended complex plane. It sends circles and lines to circles or lines, preserves cross-ratio, and forms a group under composition. These transformations are central in complex analysis and in the study of conformal mappings, since they describe all automorphisms of the Riemann sphere.

The Möbius function μ(n) is defined for positive integers: μ(1) = 1; μ(n) = (−1)^k if n is a product of k distinct primes; μ(n) = 0 if n has a squared prime factor. It appears in multiplicative number theory and in the Möbius inversion formula, which expresses one arithmetic function in terms of its sums over divisors.