Simplex

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

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

In one dimension, the simplex is the line segment . In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron. The simplex in four dimensions (the 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.

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

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