The Omnitruncated 24-cell
The following image shows its perspective projection into 3D, centered on a great rhombicuboctahedron:
For clarity, cells lying on the far side of the polytope have been omitted.
We will explore the structure of the omnitruncated 24-cell by means of its parallel projection centered on a great rhombicuboctahedron.
The above image shows the nearest great rhombicuboctahedron to the 4D viewpoint. The hexagonal faces of this cell are joined to 8 hexagonal prisms, as shown in the next image:
The square faces of the nearest cell are joined to 12 other hexagonal prisms. The next image includes these as well:
The remaining octagonal faces of the nearest cell are joined to 6 other great rhombicuboctahedra, in a skewed orientation. These are shown below:
Straddling these cells are 12 more hexagonal prisms, lying on top of the blue hexagonal prisms. These are shown next:
In addition, the hexagonal faces of these great rhombicuboctahedra are joined to 24 more hexagonal prisms. We show them below:
Next, the square faces of these great rhombicuboctahedra are joined to yet another 24 hexagonal prisms. These are added below:
Finally, the octagonal faces of the great rhombicuboctahedra, and the hexagonal and square faces of the hexagonal prisms we have seen so far, form 8 gaps into which 8 more great rhombicuboctahedral cells may be fitted. The following image shows these cells fitted in.
This makes a total of 1+6+8 = 15 great rhombicuboctahedral cells, and 8+12+12+24+24 = 80 hexagonal prisms. These are the cells that lie on the “northern hemisphere” of the omnitruncated 24-cell.
Now we proceed to the “equator” of the omnitruncated 24-cell, where there are another 18 great rhombicuboctahedral cells. The next image shows 12 of these cells:
The following image shows the other 6 great rhombicuboctahedral cells:
There are 32 hexagonal prisms filling the remaining gaps. Eight of these are shown in the next image:
Finally, the remaining hexagonal prisms are shown in the following image:
These are all the cells that lie on the equator of the omnitruncated 24-cell. There are 50 of them in total.
On the far side of the omnitruncated 24-cell are the “southern hemisphere” cells, which are laid out in a structure that exactly mirrors the northern hemisphere cells. Therefore, there are 15 + 80 + 50 + 80 + 15 = 240 cells in total, with 48 great rhombicuboctahedra and 192 hexagonal prisms.
The coordinates of the vertices of the omnitruncated 24-cell are all permutations of coordinates and changes of sign of the following base points:
- (1, 1+√2, 1+2√2, 5+3√2)
- (1, 3+√2, 3+2√2, 3+3√2)
- (2, 2+√2, 2+2√2, 4+3√2)