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A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered ...

The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron A_4 (Holden 1971, p. 55). It is Wenninger dual W_(12). It is composed of 30 ...

The small triambic icosahedron is the dual polyhedron of the small ditrigonal icosidodecahedron with Maeder index 30 (Maeder 1997), Weinninger index 70 (Wenninger 1971, p. ...

Given a function f(x)=f_0(x), write f_1=f^'(x) and define the Sturm functions by f_n(x)=-{f_(n-2)(x)-f_(n-1)(x)[(f_(n-2)(x))/(f_(n-1)(x))]}, (1) where [P(x)/Q(x)] is a ...

The total number of contravariant and covariant indices of a tensor. The rank R of a tensor is independent of the number of dimensions N of the underlying space. An intuitive ...

There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British definition is a quadrilateral with two sides ...

Consecutive number sequences are sequences constructed by concatenating numbers of a given type. Many of these sequences were considered by Smarandache and so are sometimes ...

The resistance distance between vertices i and j of a graph G is defined as the effective resistance between the two vertices (as when a battery is attached across them) when ...

A Mersenne prime is a Mersenne number, i.e., a number of the form M_n=2^n-1, that is prime. In order for M_n to be prime, n must itself be prime. This is true since for ...

The Alexander polynomial is a knot invariant discovered in 1923 by J. W. Alexander (Alexander 1928). The Alexander polynomial remained the only known knot polynomial until ...