The Bitruncated 5-cell

The bitruncated 5-cell is a cell-transitive polychoron formed by truncating a pentachoron at halfway to the depth that would yield a dual pentachoron. It is bounded by 10 truncated tetrahedra in two groups of 5, with the groups corresponding to the cells of a pentachoron and its dual, respectively.


The following image shows the cell-first perspective projection of the bitruncated 5-cell into 3D:

projection of the bitruncated 5-cell into 3D

For clarity, we have omitted cells that lie on the far side of the polytope. The nearest cell to the 4D viewpoint is a truncated tetrahedron, shown below:

projection of the bitruncated 5-cell, nearest cell shown

Surrounding this cell are 4 other truncated tetrahedra, as shown below:

neighbouring cell Second neighbouring cell Third neighbouring cell Fourth neighbouring cell

These cells look flattened because of foreshortening by the perspective projection. They are actually all uniform truncated tetrahedra. They are joined to each other by triangular faces.

The triangular faces of these 4 cells are connected to the antipodal truncated tetrahedron lying on the opposite side of the polychoron:

projection of the bitruncated 5-cell, farthest cell shown

As can be seen, the antipodal truncated tetrahedron lies in a dual orientation to the nearest truncated tetrahedron.

The triangular faces of the nearest cell, on the other hand, are connected to the four cells surrounding the far-side cell, shown below:

First cell
surrounding farthest cell Second cell surrounding farthest
cell Third cell surrounding farthest
cell Fourth cell surrounding farthest

For reference, we have included the farthest cells in these images.

Comparing these images with the earlier ones, we see that these cells are joined to the four cells surrounding the nearest cell by their hexagonal faces.

Altogether, these are the 10 cells that bound the bitruncated 5-cell.


The bitruncated 5-cell is one of the cell-transitive uniform polychora that aren't regular. Besides the n,n-duoprisms, the other such polychoron is the bitruncated 24-cell.


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

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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