In four dimensional geometry, the truncated 5cell, truncated pentatope or truncated 4simplex is a uniform polychoron (4dimensional polytope) bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. Each vertex is surrounded by 3 truncated tetrahedra and one tetrahedron; the vertex figure is an elongated tetrahedron. Construction The truncated 5cell may be constructed from the 5cell by truncating its vertices at 1/3 the edge length. This truncates the 5 tetrahedral cells into truncated tetrahedra, and introduces 5 new tetrahedral cells positioned on the original vertices. Structure The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces. Projections The tetrahedronfirst parallel projection of the truncated 5cell into 3dimensional space has the following structure: * The projection envelope is a truncated tetrahedron. This layout of cells in projection is analogous to the layout of faces in the facefirst projection of the truncated tetrahedron into 2dimensional space. The truncated 5cell is the 4dimensional analogue of the truncated tetrahedron. Imagesstereographic projection Alternate names * Truncated pentatope
The Cartesian coordinates for the vertices of an origincentered truncated 5cell having edge length 2 are:
The vertices of the truncated 5cell can be more simply positioned in 5space as the 20 permutations of: (0,0,0,1,2) This construction is from the positive orthant facet of the truncated pentacross. Related uniform polychora The Truncated pentachoron is one 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/" 
