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A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to n dimensions. The boundary of a k-simplex has k+1 0-faces (polytope vertices),k(k+1)/2 1-faces (polytope edges), and (k+1; i+1) i-faces, where (n; k) is a binomial coefficient. The simplex is so-named because it represents the simplest possible polytope in any given space.

A regular n-dimensional simplex can be denoted using the Schläfli symbol {3,...,3_()_(n-1)}.

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 [-1,1]. In two dimensions, the regular simplex {3} is the convex hull of the equilateral triangle. In three dimensions, the regular simplex {3,3} is the convex hull of thetetrahedron. The regular simplex in four dimensions (the regular pentatope) is a regular tetrahedron ABCD in which a point E along the fourth dimension through the center of ABCD is chosen so that EA=EB=EC=ED=AB. The regular simplex in n dimensions with n>=5 is denoted alpha_n.

If p_0, p_1, ..., p_n are n+1 points in R^n such that p_1-p_0, ..., p_n-p_0 are linearly independent, then the convex hull of these points is an n-simplex.

SimplexGraphs

The above figures show the skeletons for the n-simplexes with n=2 to 7. Note that the graph of an n-simplex is the complete graph of n+1 vertices.

The n-simplex has graph spectrum n^1(-1)^n (Cvetkovic et al. 1998, p. 72; Buekenhout and Parker 1998).

See also

Cayley-Menger Determinant, Cross Polytope, Equilateral Triangle, Hypercube, Line Segment, Nerve, Pentatope,Point, Polytope, Simplex Method, Simplex Graph, Simplex Simplex Picking, Simplicial Complex, Spherical Simplex, Tetrahedron

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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<=4." 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." http://www.ics.uci.edu/~eppstein/junkyard/triangulation.html.Munkres, J. R. "Simplices." §1.1 in Elements of Algebraic Topology. New York: Perseus Books Pub.,pp. 2-7, 1993.

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Simplex

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

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