condition for the orbit of a point to be a regular simplex is that the matrices form a basis for the non-trivial irreducible real representations of , and the vector being rotated is not stabilized by any of them. In practical terms, for even this means that every matrix is , there is an equality of sets and, for every , the entries of upon which acts are not both zero. For example, when , one possible matrix is Applying this to the vector results in the simplex whose vertices are each of which has distance √5 from the others. When is odd, the condition means that exactly one of the diagonal blocks is , equal to , and acts upon a non-zero entry of ; while the remaining diagonal blocks, say , are , there is an equality of sets and each diagonal block acts upon a pair of entries of which are not both zero. So, for example, when , the matrix can be For the vector , the resulting simplex has vertices each of which has distance 2 from the others. Geometric properties Volume The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is where each column of the n × n determinant is a vector that points from vertex to another vertex . This formula is particularly useful when is the origin. The expression employs a Gram determinant and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions, e.g., a triangle in . A more symmetric way to compute the volume of an n-simplex in is Another common way of computing the volume of the simplex is via the Cayley–Menger determinant, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions. Without the 1/n! it is the formula for the volume of an n-parallelotope. This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis of . Given a permutation of , call a list of vertices a n-path if (so there are n! n-paths and does not depend on the permutation). The following assertions hold: 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