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.

Construction

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.

Structure

The truncated tetrahedra are joined to each other at their hexagonal faces, and to the tetrahedra at their triangular faces.

Projections

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.
* One of the truncated tetrahedral cells project onto the entire envelope.
* One of the tetrahedral cells project onto a tetrahedron lying at the center of the envelope.
* Four flattened tetrahedra are joined to the triangular faces of the envelope, and connected to the central tetrahedron via 4 radial edges. These are the images of the remaining 4 tetrahedral cells.
* Between the central tetrahedron and the 4 hexagonal faces of the envelope are 4 irregular truncated tetrahedral volumes, which are the images of the 4 remaining trucated tetrahedral cells.

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.

Imagesstereographic projection

Alternate names

* Truncated pentatope
* Truncated 4-simplex
* Tip (Jonathan Bowers: for truncated pentachoron)

Coordinates

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:

(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.

Name	5-cell	truncated 5-cell	rectified 5-cell	cantellated 5-cell	bitruncated 5-cell	cantitruncated 5-cell	runcinated 5-cell	runcitruncated 5-cell	omnitruncated 5-cell
Schläfli symbol
Coxeter-Dynkin diagram
Schlegel diagram
Coxeter plane projection

Images: Robert Webb's Great Stella software

Geometry