Simplex

a method of two-way communication wherein, at each communication station, transmission alternates with reception.

in mathematics, the simplest convex polyhedron of some given dimension n. When n = 3, we have a three-dimensional simplex, which is a tetrahedron; the tetrahedron may be irregular. A two-dimensional simplex is a triangle, a one-dimensional simplex is a line segment, and a zero-dimensional simplex is a point.

An n-dimensional simplex has n + 1 vertices, which do not belong to any (n - 1)-dimensional subspace of the Euclidean space (of dimension at least n) in which the simplex lies. Conversely, any n + 1 points of a Euclidean m-dimensional space R^m, m ≥ n, that do not lie in a subspace of dimension less than n uniquely determine an n-dimensional simplex with vertices at the given points e₀, e₁, • • •, e_n. This simplex can be defined as the convex closure of the set of the given n + 1 points—that is, as the intersection of all convex polyhedra of R^m that contain the points.

If a system of Cartesian coordinates x₁, x₂, • • •, x_m is defined in R^m such that the vertex e_i, i = 0, 1, • • •, n, has the coordinates , then the simplex with the vertices e₀, e₁, • • •, e_n consists of all points of R^m whose coordinates are of the form

where μ⁽⁰⁾, μ⁽¹⁾ •••, μ⁽ⁿ⁾ are arbitrary nonnegative numbers whose sum is 1. By analogy with the case where n ≤ 3, we can say that all points of a simplex with given vertices are obtained if we place arbitrary nonnegative masses (not all of which are zero) at the vertices and determine the center of gravity of these masses. It should be noted that the requirement that the sum of the masses be equal to 1 eliminates only the case where all the masses are zero.

Any r + 1 vertices, 0 ≤ r ≤ n − 1, selected from the given n + 1 vertices of an n-dimensional simplex determine an r-dimensional simplex, which is called a face of the original simplex. The zero-dimensional faces of a simplex are its vertices; the one-dimensional faces are called its edges.