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).

For a regular -simplex of polytope edge length , let , , be the distances from a point in its affine hull to its vertices. Then

The point need not lie inside the simplex. This identity generalizes the equilateral triangle distance relation to every (Hajja et al. 2017).

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).

See also

Cayley-Menger Determinant, Choquet Simplex, Cross Polytope, Equilateral Triangle, Hypercube, Line Segment, Nerve, Pentatope, Point, Polytope, Regular Polytope, Simple Polyhedron, Simple Polytope, Simplex Graph, Simplex Method, Simplex Simplex Picking, Simplicial Complex, Simplicial Polytope, Spherical Simplex, Tetrahedron, Vertex Figure

Explore with Wolfram|Alpha

References

Bourke, P. "Regular Polytopes (Platonic Solids) in 4D." http://local.wasp.uwa.edu.au/~pbourke/geometry/platonic4d/.Buekenhout, F. and Parker, M. "The Number of Nets of the Regular Convex Polytopes in Dimension ." Disc. Math. 186, 69-94, 1998.Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.Eppstein, D. "Triangles and Simplices." https://ics.uci.edu/~eppstein/junkyard/triangulation.html.Fukuda, K. "What Is Simple/Simplicial Polytope?" Aug. 26, 2004. https://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node10.html.Hajja, M.; Hayajneh, M.; Nguyen, B.; and Shaqaqha, S. "Distances from the Vertices of a Regular Simplex." Results Math. 72, 633-648, 2017. https://doi.org/10.1007/s00025-017-0689-1.Munkres, J. R. "Simplices." §1.1 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 2-7, 1993.Renner, L. E. "Descent Systems for Bruhat Posets." J. Algebraic Combin. 29, 413-435, 2009. https://doi.org/10.1007/s10801-008-0141-4.Stein, P. "A Note on the Volume of a Simplex." Amer. Math. Monthly 73, 299-301, 1966. https://doi.org/10.2307/2315353.

Referenced on Wolfram|Alpha

Simplex

Cite this as:

Weisstein, Eric W. "Simplex." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Simplex.html

Subject classifications