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In two dimensions, there are two periodic circle packings for identical circles: square lattice and hexagonal lattice. In 1940, Fejes Tóth proved that the hexagonal lattice ...

Marsaglia (1972) has given a simple method for selecting points with a uniform distribution on the surface of a 4-sphere. This is accomplished by picking two pairs of points ...

A hypersphere S^n is parallelizable if there are n vector fields that are linearly independent at each point. There exist only three parallelizable spheres: S^1, S^3, and S^7 ...

Consider n intersecting circles. The maximal number of regions into which these divide the plane are N(n)=n^2-n+2, giving values for n=1, 2, ... of 2, 4, 8, 14, 22, 32, 44, ...

Let I_A, I_B, and I_C be the vertices of the inner Soddy triangle, and also let E_A, E_B, and E_C be the pairwise contact points of the three tangent circles. Then the lines ...

A coordinate system (mu,nu,psi) given by the coordinate transformation x = (mucospsi)/(mu^2+nu^2) (1) y = (musinpsi)/(mu^2+nu^2) (2) z = nu/(mu^2+nu^2) (3) and defined for ...

A system of curvilinear coordinates for which several different notations are commonly used. In this work (u,v,phi) is used, whereas Arfken (1970) uses (xi,eta,phi) and Moon ...

The Barnes G-function is an analytic continuation of the G-function defined in the construction of the Glaisher-Kinkelin constant G(n)=([Gamma(n)]^(n-1))/(H(n-1)) (1) for ...

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure ...

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of ...