Simplex

A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedron to dimensions. The boundary of a -simplex has 0-faces (polytope vertices), 1-faces (polytope edges), and -faces, where is a binomial coefficient. The simplex is so-named because it represents the simplest possible polytope in any given space.

A regular -dimensional simplex can be denoted using the Schläfli symbol .

The content (i.e., hypervolume) of a simplex can be computed using the Cayley-Menger determinant.

In one dimension, the regular simplex is the line segment . In two dimensions, the regular simplex is the convex hull of the equilateral triangle. In three dimensions, the regular simplex is the convex hull of the tetrahedron. The regular simplex in four dimensions (the regular pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The regular simplex in dimensions with is denoted .

If , , ..., are points in such that , ..., are linearly independent, then the convex hull of these points is an -simplex.

When the are represented as column vectors, the content of this simplex is

Omitting the absolute value gives the oriented content, whose sign depends on the order of the polytope vertices (Stein 1966).

The above figures show the skeletons for the -simplexes with to 7. Note that the graph of an -simplex is the complete graph of graph vertices.

The -simplex has graph spectrum (Cvetkovic et al. 1998, p. 72; Buekenhout and Parker 1998).