In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space.
A regular simplex is a simplex that is also a regular polytope. A regular k-simplex may be constructed from a regular (k − 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 whose vertices are the k standard unit vectors and the origin, 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 Donald Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the , he labeled as δn.
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.
In layman's terms, an n-simplex is a simple shape (a polygon) that requires n dimensions. Consider a line segment AB as a "shape" in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point E somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.
In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by
The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition.
The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by
The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.
An alternative coordinate system is given by taking the indefinite sum:
This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.
It is also possible to directly write down a particular regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of Rn by e1 through en. Begin with the standard (n − 1)-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular n-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) for some real number α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular n-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for α. Solving this equation shows that there are two choices for the additional vertex:
Either of these, together with the standard basis vectors, yields a regular n-simplex.
The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
A highly symmetric way to construct a regular n-simplex is to use a representation of the cyclic group Zn+1 by orthogonal matrices. This is an n× n orthogonal matrix Q such that Qn+1 = I is the identity matrix, but no lower power of Q is. Applying powers of this matrix to an appropriate vector v will produce the vertices of a regular n-simplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix
where each Qi is orthogonal and either 2 × 2 or 1 ÷ 1. In order for Q to have order n + 1, all of these matrices must have order dividing n + 1. Therefore each Qi is either a 1 × 1 matrix whose only entry is 1 or, if n is odd, −1; or it is a 2 × 2 matrix of the form
where each ωi is an integer between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Qi form a basis for the non-trivial irreducible real representations of Zn+1, and the vector being rotated is not stabilized by any of them.
In practical terms, for n even this means that every matrix Qi is 2 × 2, there is an equality of sets
and, for every Qi, the entries of v upon which Qi acts are not both zero. For example, when n = 4, one possible matrix is
Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are
each of which has distance √5 from the others. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a non-zero entry of v; while the remaining diagonal blocks, say Q1, ..., Q(n − 1) / 2, are 2 × 2, there is an equality of sets
and each diagonal block acts upon a pair of entries of v which are not both zero. So, for example, when n = 3, the matrix can be
The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is
If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping. In particular, the volume of such a simplex is
Finally, the formula at the beginning of this section is obtained by observing that
From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the Pythagorean theorem:
The sum of the squared (n − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)-dimensional volume of the facet opposite of the orthogonal corner.
The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.
In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.
Since all simplices are self-dual, they can form a series of compounds;
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.
A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.
Note that each facet of an n-simplex is an affine (n − 1)-simplex, and thus the boundary of an n-simplex is an affine (n − 1)-chain. Thus, if we denote one positively oriented affine simplex as
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map).
Since classical algebraic geometry allows to talk about polynomial equations, but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is