In four dimensional geometry, the truncated 5-cell, truncated pentatope or truncated 4-simplex is a uniform polychoron (4-dimensional 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.
The truncated 5-cell may be constructed from the 5-cell 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.
The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.
The tetrahedron-first parallel projection of the truncated 5-cell into 3-dimensional 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 face-first projection of the truncated tetrahedron into 2-dimensional space. The truncated 5-cell is the 4-dimensional analogue of the truncated tetrahedron.
* Truncated pentatope
The Cartesian coordinates for the vertices of an origin-centered truncated 5-cell having edge length 2 are:
The vertices of the truncated 5-cell can be more simply positioned in 5-space as the 20 permutations of:
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