Hellenica World

Truncated 5-cell

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.
* 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)


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.

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
Coxeter plane projection

Images: Robert Webb's Great Stella software

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