Simplex

A simplex is the simplest possible non-degenerate polytope in each respective dimension. The n-dimensional simplex, or simply n-simplex, consists of n+1 vertices, with each n of them joined in the unique manner by a simplex of the lower dimension. Alternatively, one may construct an n-simplex as the pyramid of the (n–1)-simplex.

Every simplex can be made regular. As such, the simplices comprise one of the three infinite families of regular polytope’s that exist in every dimension, the other two being the hypercubes and the orthoplexes.

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 an (n–1)/2 dimensional simplex, such as the tetrahedron (digonal disphenoid) and the hexateron (triangular disphenoid).

Every simplex is self-dual.

Elements
All of the elements of a simplex are simplices themselves. The number of d-dimensional elements in an n-dimensional simplex is given by the binomial coefficient C(n+1, d+1). This is because any choice of d+1 vertices defines a unique d-dimensional simplex. In particular, an n-dimensional simplex has n+1 vertices and n+1 facets (each shaped like an (n–1)-dimensional simplex), and its vertex figure is also the simplex of the previous dimension.

Examples
The simplices up to 10D are the following.

Measures

 * The circumradius of a regular n-dimensional simplex of unit edge length is given by $\sqrt{n/(2n+2)}$.
 * Its inradius is given by 1/$\sqrt{2n(n+1)}$.
 * Its height from a vertex to the opposite facet is given by $\sqrt{(n+1)/(2n)}$.
 * Its hypervolume is given by $\sqrt{(n+1)/(2^{n})}$/n!.
 * The angle between two facet hyperplanes is acos(1/n).