The Cantitruncated Tesseract


The cantitruncated tesseract is a uniform polytope in the tesseract family. Its surface consists of 8 great rhombicuboctahedra, 32 triangular prisms, and 16 truncated tetrahedra.

The
cantitruncated tesseract, projected into 3D

The cantitruncated tesseract may be constructed by radially expanding the octagonal ridges of the truncated tesseract outwards, causing the truncated cubes to become great rhombicuboctahedra, the tetrahedra to become truncated tetrahedra, and creating 32 gaps which can be filled in by triangular prisms.

Structure

We shall explore the structure of the cantitruncated tesseract using its parallel projection into 3D, centered on a great rhombicuboctahedron:

Parallel
projection of the cantitruncated tesseract, showing nearest cell

This image shows the nearest cell to the 4D viewpoint. For clarity, we have omitted the edges that don't lie on this cell, and rendered the other cells in a light transparent color.

The 12 square faces of this nearest great rhombicuboctahedron are joined to 12 triangular prisms, shown in red below:

Parallel
projection of the cantitruncated tesseract, showing 12 triangular prisms

The hexagonal faces of the nearest cell are joined to 8 truncated tetrahedra:

Parallel
projection of the cantitruncated tesseract, showing 8 truncated
tetrahedra

These are all the cells that lie on the near side of the cantitruncated tesseract. Past this point, we reach the cells on the “equator”:

Parallel
projection of the cantitruncated tesseract, showing 8 equatorial triangular
prisms

These 8 triangles are actually the projection images of 8 triangular prisms that lie on the equator. They appear foreshortened into triangles because they are being seen from a 90° angle. In 4D, they are perfectly uniform triangular prisms.

For the sake of clarity, we have omitted the cells we saw previously.

There are 6 other cells that lie on the equator, in 3 pairs:

Parallel
projection of the cantitruncated tesseract, showing first pair of equatorial
great rhombicuboctahedra

Parallel
projection of the cantitruncated tesseract, showing second pair of equatorial
great rhombicuboctahedra

Parallel
projection of the cantitruncated tesseract, showing third pair of equatorial
great rhombicuboctahedra

These cells appear flattened into octagons, but that is only because they are being seen from a 90° angle. In 4D, they are perfectly uniform great rhombicuboctahedra.

Here are all of the equatorial cells together:

Parallel
projection of the cantitruncated tesseract, showing 6 equatorial great
rhombicuboctahedra

Past this point, we reach the far side of the cantitruncated tesseract, where the arrangement of cells exactly mirrors the arrangement on the near side, that we have seen previously.

In summary, the following table shows the cell counts in the various parts of the cantitruncated tesseract:

Region Great rhombicuboctahedra Triangular
		prisms Truncated tetrahedra
Near side 1 12 8
Equator 6 8 0
Far side 1 12 8
Total 8 32 16
Grand total 56 cells

Coordinates

The Cartesian coordinates of the cantitruncated tesseract, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:


Last updated 13 Feb 2016.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid XHTML 1.1! Proud to be Microsoft-free