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Given a smooth manifold M with an open cover U_i, a partition of unity subject to the cover U_i is a collection of smooth, nonnegative functions psi_i, such that the support ...

A dissection fallacy is an apparent paradox arising when two plane figures with different areas seem to be composed by the same finite set of parts. In order to produce this ...

Geometry

Lehmer (1938) showed that every positive irrational number x has a unique infinite continued cotangent representation of the form x=cot[sum_(k=0)^infty(-1)^kcot^(-1)b_k], (1) ...

E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant zeta(3) given by zeta(3) approx 10+zeta(16)-sqrt(96), (1) which is good to 6 digits. M. ...

The q-digamma function psi_q(z), also denoted psi_q^((0))(z), is defined as psi_q(z)=1/(Gamma_q(z))(partialGamma_q(z))/(partialz), (1) where Gamma_q(z) is the q-gamma ...

There are at least 15 classes of convex pentagonal tilings, as illustrated above. The first five were discovered during investigations of German mathematician Karl Reinhardt ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron ...

Minkowski's question mark function is the function y=?(x) defined by Minkowski for the purpose of mapping the quadratic surds in the open interval (0,1) into the rational ...