The Octa-augmented Truncated Tesseract

---

The octa-augmented truncated tesseract is a CRF polychoron with 160 vertices, 512 edges, 464 polygons (224 triangles and 240 squares), and 112 cells (8 cuboctahedra, 24 square orthobicupolae (J28), 64 triangular prisms, and 16 tetrahedra).

It may be constructed by augmenting the truncated tesseract with 8 cuboctahedron || truncated cube (K4.129) segmentochora. It can also be obtained via a modified partial Stott expansion of the 24-cell, regarded as an octa-augmented tesseract (rather than according to full 24-cell symmetry). Equivalently, it can be considered as a partial Stott contraction of the cantellated 24-cell.

Structure

We shall explore the octa-augmented truncated tesseract by examining its parallel projections into 3D, centered on one of its cuboctahedral cells:

The above image shows the nearest cuboctahedron to the 4D viewpoint. For the sake of clarity, we have rendered the other cells in a light transparent color. The next images show the 6 square orthobicupolae (J28) that surround this cuboctahedron, touching its square faces:

These square orthobicupolae are somewhat foreshortened, because they lie at a 45° to the 4D viewpoint. However, this is merely an artifact of the projection.

The next image shows the 8 triangular prisms that touch the triangular faces of the cuboctahedron:

For clarity, we have left out the square orthobicupolae in this image. The following image show all of the preceding cells together:

The exposed ends of these triangular prisms are where 8 tetrahedra are attached:

Each of these tetrahedra are joined to 3 more triangular prisms overlying the square faces of the square orthobicupolae, for a total of 24 more triangular prisms:

These are all the cells that lie on the near side of the polytope, the side facing the 4D viewpoint. Past this point, we get to the equator.

There are 6 cuboctahedral cells that lie on the equator of the polychoron:

For clarity, we have omitted the previous cells from the near side.

These cuboctahedra lie at a 90° angle to the 4D viewpoint, and thus have been foreshortened into squares. In 4D, of course, they are perfectly uniform cuboctahedra.

There are 12 square orthobicupolae that lie on the equator, shown next:

Just like the cuboctahedra, these J28 cells have been foreshortened into hexagons. But they are actual, undistorted Johnson solids in 4D.

These are all the cells that lie on the equator of the octa-augmented truncated tesseract. Past this point, we reach the far side of the polychoron, where the arrangement of cells mirror that of the near side shown earlier.

The following table summarizes the cell counts for the different regions of the polytope:

Region	Layer
Near side	1	1	6	8	0
	2	0	0	24	8
Equator		6	12	0	0
Far side	2	0	0	24	8
	1	1	6	8	0
Grand total		8	24	64	16
		112 cells

Coordinates

The Cartesian coordinates of the octa-augmented truncated tesseract, centered on the origin and with edge length 2, are all permutations of coordinates and changes of sign of:

• (1, 1+√2, 1+√2, 1+√2)
• (0, √2, √2, 2+√2)