The Runcinated 24-cell
The runcinated 24-cell is a uniform polychoron bounded by of 48 octahedra and 192 triangular prisms. It has 576 edges and 144 vertices. It may be constructed by expanding the 24 cells of a 24-cell outwards radially and filling in the resulting gaps with triangular prisms and 24 more octahedra.
We shall explore the structure of the runcinated 24-cell by means of its parallel projection into 3D, centered on an octahedral cell.
The above image shows the nearest octahedral cell. Its 6 vertices are shared with 6 other octahedral cells, shown below:
The 48 octahedra of the runcinated 24-cell correspond with two sets of 24 octahedra each: one corresponding with the cells of one 24-cell, and the other corresponding with the cells of the dual 24-cell. The nearest octahedron (yellow) corresponds with one of the cells of the first 24-cell, and the other 6 (red) with the cells of its dual. To emphasize this correspondence, we shall color all octahedra from the first 24-cell yellow, and all octahedra from the dual 24-cell red.
There are 8 more octahedra from the first set that share vertices with these red octahedra:
All 15 of these octahedra lie on the “northern hemisphere” of the runcinated 24-cell. At the equator there are 6 more octahedra of the first set:
And 12 more octahedra from the second set:
These octahedra appear as squares and rhombuses because they are being viewed from a 90° angle. They are regular octahedra in 4D.
This makes a total of 18 octahedra on the equator. Past this point, there are another 15 octahedra in the “southern hemisphere” of the runcinated 24-cell, in an arrangement that exactly mirrors the northern hemisphere octahedra. In total, this makes 15 + 18 + 15 = 48 octahedra.
The Triangular Prisms
Filling the gaps between the 48 octahedra that we have seen are 192 triangular prisms. We'll explore them using the same parallel projection, starting with the following 8 triangular prisms which share a face with the nearest octahedron:
Straddling these triangular prisms are another 12 triangular prisms:
Another 12 triangular prisms in complementary orientation lie over them:
Over these are another 24 triangular prisms, which close up the octahedral gaps filled by the 6 red octahedral cells we saw earlier:
Straddling these triangular prisms are another 24 that begin to close up the gaps occupied by the 8 yellow octahedral cells seen earlier:
These are all the triangular prisms in the northern hemisphere of the runcinated 24-cell. The 8 triangular gaps looking into the octahedral holes are filled in by 8 triangular prisms on the equator, foreshortened to triangles because they are viewed from a 90° angle:
To reduce clutter, we omit the other cells that we have seen so far, except for the nearest octahedron, left in for reference.
The edges of these 8 triangular images are actually foreshortened square faces of the triangular prisms. These square faces are joined to another 24 triangular prisms on the equator:
The remaining gaps on the equator are octahedral cells, as we have previously seen. We show all the equatorial cells together below:
In summary, we have 8+12+12+24+24=80 triangular prisms in the northern hemisphere of the runcinated 24-cell, and 8+24=32 triangular prisms on its equator. The southern hemisphere exactly mirrors the structure of the northern hemisphere (in reverse order), having another 80 triangular prisms. Thus, there is a total of 80+32+80=192 triangular prisms.
The perspective projection of the runcinated 24-cell, centered on an octahedron, is shown below:
For clarity, we have omitted cells that lie on the far side of the runcinated 24-cell, including the equatorial cells because of the perspective projection.
The Cartesian coordinates of the runcinated 24-cell, centered on the origin and with edge length 2, are all permutations of coordinates and changes of sign of the following two points:
- (0, 0, √2, 2+√2)
- (1, 1, 1+√2, 1+√2)
The runcinated 24-cell has a higher degree of symmetry than the 24-cell, because its 48 octahedral cells correspond with both the vertices and cells of a 24-cell, and its 192 triangular prisms correspond with both the faces and edges of the 24-cell, making these two pairs of element types (cells/vertices and faces/edges) interchangeable, thus doubling the number of symmetries. A few other members of the 24-cell family of uniform polychora exhibit this extended symmetry, including the bitruncated 24-cell and the omnitruncated 24-cell.