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.

For , an -simplex is both a simple polytope and a simplicial polytope. Exactly edges and facets meet at each polytope vertex, and every facet is an -simplex (Fukuda 2004). Its vertex figures are also -simplices (Renner 2009). More generally, the simple condition requires simplices as vertex figures, while the simplicial condition requires simplices as facets. Neither condition requires the whole polytope to be a simplex. The cube is simple but not simplicial, and the octahedron is simplicial but not simple.

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).