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An $ n $- dimensional polytope (cf. Polyhedron) that is the convex hull of $ n + 1 $ points (the vertices of the simplex) which do not lie in any $ ( n - 1) $- dimensional plane. When $ n = 0, 1, 2, $ or 3, the simplex is a point, an interval, a triangle, or a tetrahedron, respectively. The faces of a simplex are simplices of lower dimension. Two simplices of the same dimension are affinely equivalent. Every point of a simplex corresponds to a unique way of distributing a unit mass among its vertices in such a way that the centre of gravity is at the given point. This is used to introduce barycentric coordinates in a simplex and also serves as a method of generalizing the notion of a simplex to the infinite-dimensional case (see Choquet simplex; Simplex (abstract)). A simplex can be ascribed one of two orientations, which then induces a specific orientation on each of its $ ( n - 1) $- dimensional faces.



[a1] B. Grünbaum, "Convex polytopes" , Wiley (1967)
[a2] P. McMullen, G.C. Shephard, "Convex polytopes and the upper bound conjecture" , Cambridge Univ. Press (1971)
How to Cite This Entry:
Simplex. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article