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A module M over a unit ring R is called flat iff the tensor product functor - tensor _RM (or, equivalently, the tensor product functor M tensor _R-) is an exact functor. For ...

A volume element is the differential element dV whose volume integral over some range in a given coordinate system gives the volume of a solid, V=intintint_(G)dxdydz. (1) In ...

The 16-cell beta_4 is the finite regular four-dimensional cross polytope with Schläfli symbol {3,3,4}. It is also known as the hyperoctahedron (Buekenhout and Parker 1998) or ...

The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other ...

The Wallis formula follows from the infinite product representation of the sine sinx=xproduct_(n=1)^infty(1-(x^2)/(pi^2n^2)). (1) Taking x=pi/2 gives ...

A heart-shaped surface given by the sextic equation (x^2+9/4y^2+z^2-1)^3-x^2z^3-9/(80)y^2z^3=0 (Taubin 1993, 1994). The figures above show a ray-traced rendering (left) and ...

The hyperbolic octahedron is a hyperbolic version of the Euclidean octahedron, which is a special case of the astroidal ellipsoid with a=b=c=1. It is given by the parametric ...

Consider a reference triangle DeltaABC and externally inscribe a square on the side BC. Now join the new vertices S_(AB) and S_(AC) of this square with the vertex A, marking ...

In 1803, Malfatti posed the problem of determining the three circular columns of marble of possibly different sizes which, when carved out of a right triangular prism, would ...

There are two versions of the moat-crossing problem, one geometric and one algebraic. The geometric moat problems asks for the widest moat Rapunzel can cross to escape if she ...