The Rectified 5-cell


The rectified 5-cell is a semiregular polychoron bounded by 5 tetrahedra and 5 octahedra. It is obtained by truncating the pentatope at the midpoints of its edges.

Persective
projection of the rectified 5-cell

Structure

The Near Side

We shall explore the structure of the rectified 5-cell by means of its parallel projection into 3D:

Parallel projection
of the rectified 5-cell

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:

Parallel projection
of the rectified 5-cell, showing only nearest octahedron

The 4 surrounding tetrahedral cells that share a face with this cell are shown in the next image:

Parallel projection
of the rectified 5-cell, showing 4 tetrahedra

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:

Parallel projection
of the rectified 5-cell, showing a far-side octahedronParallel projection of the rectified 5-cell,
showing another far-side octahedronParallel projection of the rectified 5-cell, showing 3rd far-side
octahedronParallel projection of the
rectified 5-cell, showing last far-side octahedron

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:

Parallel projection
of the rectified 5-cell, showing 4 tetrahedra

Summary

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.

Coordinates

The Cartesian coordinates of the rectified 5-cell, centered on the origin and having edge length 2, are:

Simpler coordinates can be obtained in 5D as all permutations of coordinates of:

The 4D coordinates are derived by projecting these 5D coordinates back into 4D using a symmetric projection.


Last updated 29 May 2014.

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