fivesimplex

The five-simplex, also called the 5-simplex or five-dimensional simplex, is a regular convex polytope in five-dimensional Euclidean space. It can be defined as the convex hull of six points in general position in R^5, or equivalently as the regular convex polytope with six vertices up to congruence.

In a regular 5-simplex, every pair of vertices is connected by an edge of equal length, and all faces are regular simplices of lower dimension. The polytope has 6 vertices, 15 edges, 20 triangular faces, 15 tetrahedral 3-faces, 6 4-dimensional facets, and itself as a 5-dimensional body.

A standard realization places the vertices in the hyperplane sum coordinates = 0 inside R^6 by taking w_i = e_i − (1/6)(1,1,1,1,1,1) for i = 1,...,6. These six points lie at pairwise distance sqrt(2), giving edge length sqrt(2). Thus the 5-simplex is equidistant, inscribed in a sphere, and can be realized in five dimensions as a regular object.

Symmetry and angles: The full symmetry group is the symmetric group S6, of order 720, acting by permuting the six vertices. The dihedral angle between adjacent 4-dimensional facets is arccos(1/5) ≈ 78.463 degrees.

Applications and context: The five-simplex serves as a fundamental building block in higher-dimensional geometry, combinatorics, and topology. It appears in simplicial decompositions and triangulations of five-dimensional spaces and provides a canonical example of a regular polytope in dimension five. Related concepts include n-simplices, regular polytopes, and their symmetry groups.