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As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

The insphere of a solid is a sphere that is tangent to all faces of the solid. An insphere does not always exist, but when it does, its radius r is called the inradius and ...

Line segment picking is the process of picking line segments at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a ...

The number of multisets of length k on n symbols is sometimes termed "n multichoose k," denoted ((n; k)) by analogy with the binomial coefficient (n; k). n multichoose k is ...

The multinomial coefficients (n_1,n_2,...,n_k)!=((n_1+n_2+...+n_k)!)/(n_1!n_2!...n_k!) (1) are the terms in the multinomial series expansion. In other words, the number of ...

A mapping of random number triples to points in spherical coordinates according to theta = 2piX_n (1) phi = piX_(n+1) (2) r = sqrt(X_(n+2)) (3) in order to detect unexpected ...

Consider the average volume of a tetrahedron picked at random inside an octahedron of unit volume. The answer is difficult to compute analytically (Zinani 2003), but the mean ...

A skew polygon such that every two consecutive sides (but no three) belong to a face of a regular polyhedron. Every regular polyhedron can be orthogonally projected onto a ...

The integral kernel in the Poisson integral, given by K(psi)=1/(2pi)(1-|z_0|^2)/(|z_0-e^(ipsi)|^2) (1) for the open unit disk D(0,1). Writing z_0=re^(itheta) and taking ...

A zonohedron which can be derived from the rhombic triacontahedron by removing any one of the zones and bringing together the two pieces into which the remainder of the ...