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. The pyramid product of an m-dimensional simplex and an n-dimensional simplex is an m+n+1 dimensional simplex.

Every simplex can be made regular. As such, the simplexes comprise one of the three infinite families of regular polytopes 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 simplexes 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.

In total, an n-simplex has 2n-1 elements, including the nullitope and excluding the bulk of the polytope.

Examples
Excluding the degenerate nullitope and the point, the simplexes up to 10D are the following:

Vertex coordinates
Coordinates for the vertices of an n-simplex of edge length 1, centered at the origin, are given by:
 * $$\left(-\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\sqrt{\frac{1}{2n(n+1)}}\right),$$
 * $$\left(\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\sqrt{\frac{1}{2n(n+1)}}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\sqrt{\frac{1}{2n(n+1)}}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,\ldots,\,-\sqrt{\frac{1}{2n(n+1)}}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,\ldots,\,-\sqrt{\frac{1}{2n(n+1)}}\right),$$
 * $$\ldots$$
 * $$\left(0,\,0,\,0,\,0,\,\ldots,\,\sqrt{\frac{n}{2(n+1)}}\right).$$

Alternatively, there are simpler coordinates but the simplex is not centered at the origin:
 * all permutations of $$\left(\frac{\sqrt{2}}{2},\,0,\,0,\,...,\,0\right)$$
 * $$(a,\,a,\,a,\,...,\,a)$$ where $$a = \frac{\left(1-\sqrt{n+1}\right)\sqrt{2}}{2n}$$.

Much simpler coordinates can be given in n+1 dimensions, as all permutations of
 * ($\sqrt{2}$/2, 0, 0, ..., 0).

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).