Simplex

The simplex is one of the three infinite families of regular polytopes that exist in every dimension (the other two are the hypercube and the orthoplex). They are the simplest possible non-degenerate polytope in each respective dimension and are self-dual. The D+1-dimensional simplex can be constructed as the pyramid of the D-dimensional simplex. In even dimensions greater than 2, simplexes can be thought of as step prisms, such as the pentachoron (5-2 step prism) and the heptapeton (7-2-3 step prism). In odd dimensions greater than 1, simplexes can be thought of as disphenoids made out from a (D-1)/2 dimensional simplex, such as the tetrahedron (digonal disphenoid) and the hexateron (triangular disphenoid).

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 $\sqrt{(D+1)/(2^{D})}$/D! (where ! means factorial).
 * The angle between two facet hyperplanes is acos(1/d).