Simplex

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

LineSegmentEquilateralTriangleTetrahedronPentatope

In one dimension, the simplex is the line segment [-1,1]. In two dimensions, the simplex {3} is the convex hull of the equilateral triangle. In three dimensions, the simplex {3,3} is the convex hull of the tetrahedron. The simplex in four dimensions (the 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 graphs 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).

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