The Rectified 24-cell
The rectified 24-cell is a uniform polychoron bounded by 48 cells (24 cuboctahedra, 24 cubes), 240 polygons (96 triangles, 144 squares), 288 edges, 96 vertices. It coincides with the cantellated 16-cell, due to the coincidence of the 24-cell with the rectified 16-cell.
The rectified 24-cell, as its name implies, is constructed by rectifying the 24-cell: truncating it to the midpoints of its edges. This operation reduces its octahedral cells into cuboctahedra, and introduces 24 cubical cells.
Structure
We shall explore the structure of the rectified 24-cell by looking at its cube-first parallel projection into 3D.
The above image shows the cubical cell nearest to the 4D viewpoint. For clarity, we render the rest of the cells in a transparent color and omit edges and vertices not on this cube.
This cube is surrounded by 6 cuboctahedra, as shown next as 3 opposing pairs:
These cuboctahedra appear to be slightly flattened, because they lie at a 45° angle to the 4D viewpoint. In 4D, they are perfectly uniform. The next image shows all of them together:
In addition to these cuboctahedra, there are 8 cubes that touch the original cube's vertices. These are shown below:
As with the cuboctahedra, these cubes are seen at an angle, and thus appear foreshortened. They are perfectly regular in 4D. The following image shows them and the cuboctahedra together:
These 9 cubes (including the first one) and 6 cuboctahedra lie on the “northern hemisphere” of the rectified 24-cell. Past this point, there are a number of cells on its “equator”:
For the sake of clarity, we omit the cells previously seen. These 6 squares are actually 6 cubes; they have been foreshortened into squares because they are seen at a 90° angle.
There are 12 cuboctahedra on the equator; these fill up the remaining gaps on the equator:
Just like the cubes, these cuboctahedra are seen at a 90° angle, so they have been foreshortened into hexagons. In reality, they are perfectly uniform cuboctahedra. As can be clearly seen here, the projection envelope of the rectified 24-cell under the current cube-centered projection is a truncated rhombic dodecahedron.
After these equatorial cells, we have the cells in the “southern hemisphere” of the rectified 24-cell. These cells exactly mirror the arrangement as the northern hemisphere cells. Thus, in the northern hemisphere we have 1+8=9 cubes, on the equator we have 6 cubes, and in the southern hemisphere we have another 9 cubes, for a total of 24 cubes. There are 6 cuboctahedra in the northern hemisphere, 12 on the equator, and another 6 in the southern hemisphere, also totalling 24 cuboctahedra. In total, there are 48 cells.
The Cantellated 16-cell
Due to the coincidence of the 24-cell as the rectified 16-cell, the rectified 24-cell is the same as the cantellated 16-cell. The symmetry of the cantellated 16-cell can be clearly seen when we examine the projections of the rectified 24-cell centered on a cuboctahedron:
This is the closest cuboctahedron to the 4D viewpoint. Its 6 square faces are joined to 6 cubes:
The triangular faces of the nearest cuboctahedron are joined to another 8 cuboctahedra, shown below in two sets of four:
Here are all 8 cuboctahedra together:
And here are the 6 cubes added into the mix:
These are all the cells that lie on the near side of the rectified 24-cell from this 4D viewpoint. On the limb of the rectified 24-cell are 6 more cuboctahedra, as shown below:
For clarity, we omit the cells seen before. These cuboctahedra have been foreshortened into squares, because they are seen at a 90° angle. Joining these cuboctahedra are 12 cubes:
These cubes are perfectly regular in 4D, but they appear flattened into rectangles because they are being seen from a 90° angle.
The remaining triangular gaps are not the images of any cells; they are the triangular face where the 8 cuboctahedra from the near side touch their corresponding counterparts on the far side of the rectified 24-cell. The structure of cells on the far side exactly mirrors the near side.
In total, there are 1+8=9 cuboctahedra on the near side, 6 cuboctahedra on the limb, and another 9 cuboctahedra on the far side, giving a total of 24 cuboctahedra. There are 6 cubes on the near side, 12 on the limb, and another 6 cubes on the far side; also totalling 24 cubes.
Coordinates
The coordinates of the rectified 24-cell, which is the same as the cantellated 16-cell, are all permutations of coordinate and changes of sign of:
- (0, √2, √2, 2√2)
These coordinates give a rectified 24-cell centered on the origin with edge length 2.
