In geometry, the pentachoron is a fourdimensional object bounded by 5 tetrahedral cells. It is also known as the 5cell, pentatope, or hyperpyramid. It is a 4simplex, the simplest possible convex regular 4polytope (fourdimensional analogue of a polyhedron), and is analogous to the tetrahedron in three dimensions and the triangle in two dimensions. The regular pentachoron is bounded by regular tetrahedra, and is one of the six regular convex polychora, represented by Schläfli symbol {3,3,3}.
The pentachoron is selfdual, and its vertex figure is a tetrahedron. Its maximal intersection with 3dimensional space is the triangular prism. Its dihedral angle is cos−1(1/4), or approximately 75.52°. Construction The pentachoron can be constructed from a tetrahedron by adding a 5th vertex such that it is equidistant from all the other vertices of the tetrahedron. (The pentachoron is essentially a 4dimensional pyramid with a tetrahedral base.) The Cartesian coordinates of the vertices of an origincentered regular pentachoron having edge length 2 are: The vertices of a 4simplex (with edge √2) can be more simply constructed on a hyperplane in 5space, as permutations of (0,0,0,0,1) or (0,1,1,1,1); in these positions it is a facet of, respectively, the 5orthoplex or the rectified penteract. Projections Alternative names * 5cell Related uniform polychora The pentachoron (5cell) is the simplest of 9 uniform polychora constructed from the [3,3,3] Coxeter group.
Images: Robert Webb's Great Stella software Retrieved from "http://en.wikipedia.org/" 
