The Cantitruncated 24-cell


The cantitruncated 24-cell is a uniform polychoron in the 24-cell family. Its surface consists of 24 great rhombicuboctahedra, 24 truncated cubes, and 96 triangular prisms.

The
cantitruncated 24-cell

It may be constructed from the bitruncated 24-cell by radially expanding 24 truncated cubes outwards, causing the other 24 truncated cubes to expand into great rhombicuboctahedra, and filling in the remaining gaps with triangular prisms.

Structure

We shall explore the structure of the cantitruncated 24-cell using its parallel projections into 3D, centered on a great rhombicuboctahedron.

Parallel
projection of the cantitruncated 24-cell, showing nearest cell

The above image shows one of the 24 great rhombicuboctahedral cells. It is the nearest cell to the 4D viewpoint.

Its square faces are joined to 12 triangular prisms, shown next:

Parallel
projection of the cantitruncated 24-cell, showing 12 triangular prisms

The octagonal faces of the nearest cell are joined to 6 truncated cubes:

Parallel
projection of the cantitruncated 24-cell, showing 6 truncated cubes

The hexagonal faces of the nearest cell are joined to 8 other great rhombicuboctahedra:

Parallel
projection of the cantitruncated 24-cell, showing 8 great
rhombicuboctahedra

Here's all these cells together:

Parallel
projection of the cantitruncated 24-cell, showing previous cells
together

There are 24 remaining gaps here, between each truncated cube and a pair of great rhombicuboctahedra. They are where another 24 triangular prisms are fitted:

Parallel
projection of the cantitruncated 24-cell, showing 24 more triangular
prisms

These are all the cells that lie on the near side of the cantitruncated 24-cell. Past this point, we reach its limb, or “equator”, where there are 6 great rhombicuboctahedra:

Parallel
projection of the cantitruncated 24-cell, showing 6 equatorial great
rhombicuboctahedra

For clarity, we have omitted the cells we have seen previously. These great rhombicuboctahedra appear to be flattened into octagons; this is because they are being seen from a 90° angle. In 4D, they are perfectly uniform great rhombicuboctahedra.

There are another 12 truncated cubes on the equator, shown next:

Parallel
projection of the cantitruncated 24-cell, showing 12 equatorial truncated
cubes

As with the previous cells, these truncated cubes also appear to be flattened into octagons; however, in 4D they are perfectly uniform truncated cubes. This flattening effect is merely a foreshortening caused by the projection into 3D.

Another 24 triangular prisms fit into the gaps straddling the truncated cubes and the great rhombicuboctahedra:

Parallel
projection of the cantitruncated 24-cell, showing 24 equatorial triangular
prisms

These triangular prisms have also been foreshortened into rectangles due to their being seen from a 90° angle. In 4D, they are perfectly uniform triangular prisms. The edges of the rectangle that touch the cyan truncated cubes are the images of the triangular prism's triangular faces, and the edge touching the yellow great rhombicuboctahedron is the image of one of the triangular prism's square faces.

These are all the cells that lie on the limb of the cantitruncated 24-cell. The remaining hexagonal gaps are not the images of any cells; they are where 8 great rhombicuboctahedra on the near side of the polychoron touch their counterparts on the far side.

Past this point, we reach the far side of the polytope, where the arrangement of cells perfectly mirrors that of the near side. The following table summarizes the cell counts on the far and near sides and the equator:

Region Great rhombicuboctahedra Truncated cubes Triangular prisms
Near side 1+8=9 6 12+24=36
Equator 6 12 24
Far side 9 6 36
Grand total 24 24 96

Coordinates

The Cartesian coordinates of the cantitruncated 24-cell, centered on the origin with edge length 2, are all permutations of coordinates and changes of sign of:


Last updated 16 Feb 2016.

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