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For every ring containing p spheres, there exists a ring of q spheres, each touching each of the p spheres, where 1/p+1/q=1/2, (1) which can also be written (p-2)(q-2)=4. (2) ...

Let a triangle have sides of length a, b, and c and let the angles opposite these sides be denoted A, B, and C. The law of tangents states ...

A representation of a Lie algebra g is a linear transformation psi:g->M(V), where M(V) is the set of all linear transformations of a vector space V. In particular, if V=R^n, ...

Machin's formula is given by 1/4pi=4cot^(-1)5-cot^(-1)239. There are a whole class of Machin-like formulas with various numbers of terms (although only four such formulas ...

A map is a way of associating unique objects to every element in a given set. So a map f:A|->B from A to B is a function f such that for every a in A, there is a unique ...

The orthographic projection is a projection from infinity that preserves neither area nor angle. It is given by x = cosphisin(lambda-lambda_0) (1) y = ...

The secant numbers S_k, also called the zig numbers or the Euler numbers E_n^*=|E_(2n)| numbers than can be defined either in terms of a generating function given as the ...

The semiperimeter on a figure is defined as s=1/2p, (1) where p is the perimeter. The semiperimeter of polygons appears in unexpected ways in the computation of their areas. ...

A solvable Lie group is a Lie group G which is connected and whose Lie algebra g is a solvable Lie algebra. That is, the Lie algebra commutator series ...

The square wave, also called a pulse train, or pulse wave, is a periodic waveform consisting of instantaneous transitions between two levels. The square wave is sometimes ...