The Truncated 5-cell

The truncated 5-cell is a polychoron formed by truncating a pentachoron at one third of its edge length, so that its triangular faces become hexagons. New triangular faces form in place of the original vertices, forming new tetrahedral cells, and the original tetrahedral cells become truncated tetrahedra.


The following image shows the parallel projection of the truncated 5-cell into 3D:

Parallel projection
of truncated 5-cell

The five tetrahedral cells are rendered in yellow, and are clearly visible here. Four of them appear to be flattened, due to foreshortening. They are regular tetrahedra in 4D.

The following images show where the truncated tetrahedral cells are:

First truncated
tetrahedron Second truncated tetrahedron Third truncated tetrahedron Fourth truncated tetrahedron Fifth truncated tetrahedron

Four of these cells also appear flattened due to foreshortening, but they are uniform truncated tetrahedra in 4D.

Perspective Projection

The following animation shows the perspective projection of the truncated 5-cell rotating in the ZW plane.

Truncated 5-cell
rotating in ZW plane

The red tetrahedron and one of the yellow tetrahedra appear to be turning inside-out and engulfing each other, but this is merely an artifact of the perspective projection. In 4D, they neither deform nor touch each other.


The Cartesian coordinates of the truncated 5-cell, centered on the origin and having edge length 2, are:

  • (3/√10, −1/√6, 2/√3, ±2)
  • (3/√10, −1/√6, −4/√3, 0)
  • (3/√10, 3/√6, 0, ±2)
  • (3/√10, 3/√6, ±3/√3, ±1)
  • (3/√10, −5/√6, 1/√3, ±1)
  • (3/√10, −5/√6, −2/√3, 0)
  • (−2/√10, 2/√6, 2/√3, ±2)
  • (−2/√10, 2/√6, −4/√3, 0)
  • (−2/√10, −6/√6, 0, 0)
  • (−7/√10, 1/√6, 1/√3, ±1)
  • (−7/√10, 1/√6, −2/√3, 0)
  • (−7/√10, −3/√6, 0, 0)

Simpler coordinates can be obtained in 5D as all permutations of coordinates of:

The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection.

Last updated 13 Mar 2019.

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