The Rectified 5-cell
The Near Side
We shall explore the structure of the rectified 5-cell by means of its parallel projection into 3D:
The above image shows the parallel projection of the rectified 5-cell into 3D, centered on an octahedron. For clarity, we've omitted the cells that lie on the far side of the rectified 5-cell. The nearest octahedron to the 4D viewpoint is shown below:
The 4 surrounding tetrahedral cells that share a face with this cell are shown in the next image:
These tetrahedra appear to be flattened, because they are being viewed from an angle. In reality, they are perfectly regular tetrahedra. They touch 4 of the 8 faces of the nearest octahedron. Together with the octahedral cell, these are all the cells that lie on the near side of the rectified 5-cell.
The Far Side
The other 4 faces of the octahedron are shared with 4 octahedral cells on the far side of the rectified 5-cell, which are shown below:
These octahedra appear flattened, but only because they are being viewed from an angle. They are all perfectly regular octahedra in 4D.
Finally, these 4 octahedra surround the far-side tetrahedron, which is antipodal to the nearest octahedron. This far-side tetrahedron is shown below:
In summary, on the near side of the rectified 5-cell there are an octahedron and 4 tetrahedra. On the far side, there are 4 octahedra and 1 tetrahedron. This makes a total of 5 octahedra and 5 tetrahedra.
The Cartesian coordinates of the rectified 5-cell, centered on the origin and having edge length 2, are:
- (−3/√10, −3/√6, 0, 0)
- (−3/√10, 1/√6, −2/√3, 0)
- (−3/√10, 1/√6, 1/√3, ±1)
- (2/√10, 2/√6, 2/√3, 0)
- (2/√10, −2/√6, −2/√3, 0)
- (2/√10, 2/√6, −1/√3, ±1)
- (2/√10, −2/√6, 1/√3, ±1)
Simpler coordinates can be obtained in 5D as all permutations of coordinates of:
- (0, 0, √2, √2, √2)
The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection.