Simplex

The simplex is one of the three infinite families of regular polytopes that exist in every dimension. They are the simplest possible non-degenerate polytope in each respective dimension. The D+1-dimensional simplex can be constructed as the pyramid of the D-dimensional simplex.

The number of N-dimensional elements in a D-dimensional simplex is given by the coefficient of xn+1 in the full expansion of (x+1)d+1, all of which are simplices of the appropriate dimension. In particular, a D-dimensional simplex has D+1 vertices and D+1 facets (each shaped like a (D-1)-dimensional simplex), with the vertex figure also the simplex of the previous dimension.

The simplices up to 10D are:


 * dyad (1D)
 * Triangle (2D)
 * Tetrahedron (3D)
 * Pentachoron (4D)
 * Hexateron (5D)
 * Heptapeton (6D)
 * Octaexon (7D)
 * Enneazetton (8D)
 * Decayotton (9D)
 * Hendecaxennon (10D)

Measures

 * The circumradius of a D-dimensional simplex of unit edge length is given by $\sqrt{D/(2D+2)}$.
 * The same simplex's inradius can be given by 1/$\sqrt{2D(D+1)}$.
 * Its height from a vertex to the opposite facet is given by $\sqrt{(D+1)/(2D)}$.
 * Its hypervolume is given by $\sqrt{(D+1)/(2Msup>D )}$/D! (where ! means factorial).
 * The angle between two facet hyperplanes is acos(1/d).