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Consider a two-dimensional tessellation with q regular p-gons at each polygon vertex. In the plane, (1-2/p)pi=(2pi)/q (1) 1/p+1/q=1/2, (2) so (p-2)(q-2)=4 (3) (Ball and ...

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a ...

Define the nome by q=e^(-piK^'(k)/K(k))=e^(ipitau), (1) where K(k) is the complete elliptic integral of the first kind with modulus k, K^'(k)=K(sqrt(1-k^2)) is the ...

A d-repdigit is a number composed of repetition of a single digit (in a given base, generally taken as base 10 unless otherwise specified). For example, the beast number 666 ...

The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane x=0. (The general intersection of a torus ...

The circle H which touches the incircles I, I_A, I_B, and I_C of a circular triangle ABC and its associated triangles. It is either externally tangent to I and internally ...

A Lyapunov function is a scalar function V(y) defined on a region D that is continuous, positive definite, V(y)>0 for all y!=0), and has continuous first-order partial ...

Four circles may be drawn through an arbitrary point P on a torus. The first two circles are obvious: one is in the plane of the torus and the second perpendicular to it. The ...

A positive integer: 1, 2, 3, 4, ... (OEIS A000027), also called a natural number. However, zero (0) is sometimes also included in the list of counting numbers. Due to lack of ...

The cubefree part is that part of a positive integer left after all cubic factors are divided out. For example, the cubefree part of 24=2^3·3 is 3. For n=1, 2, ..., the first ...