Simplex

In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent, which means u 1 − u 0 , … , u k − u 0 {displaystyle u_{1}-u_{0},dots ,u_{k}-u_{0}} are linearly independent.Then, the simplex determined by them is the set of points In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions.Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent, which means u 1 − u 0 , … , u k − u 0 {displaystyle u_{1}-u_{0},dots ,u_{k}-u_{0}} are linearly independent.Then, the simplex determined by them is the set of points For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and a 4-simplex is a 5-cell. A single point may be considered a 0-simplex, and a line segment may be considered a 1-simplex. A simplex may be defined as the smallest convex set containing the given vertices. A regular simplex is a simplex that is also a regular polytope. A regular n-simplex may be constructed from a regular (n − 1)-simplex by connecting a new vertex to all original vertices by the common edge length. The standard simplex or probability simplex is the simplex formed from the k + 1 standard unit vectors, or In topology and combinatorics, it is common to “glue together” simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word “simplex” simply means any finite set of vertices. The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them 'prime confines'. Henri Poincaré, writing about algebraic topology in 1900, called them 'generalized tetrahedra'.In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ('simplest') and then with the same Latin adjective in the normal form simplex ('simple'). The regular simplex family is the first of three regular polytope families, labeled by Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the infinite tessellation of hypercubes, he labeled as δn. The convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient ( n + 1 m + 1 ) {displaystyle { binom {n+1}{m+1}}} . Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex; see simplical complex for more detail. The number of 1-faces (edges) of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the (n − 1)th tetrahedron number, the number of 3-faces of the n-simplex is the (n − 2)th 5-cell number, and so on.