The Cantellated 5-cell


The cantellated 5-cell is a uniform polychoron bounded by 5 octahedra, 5 cuboctahedra, and 10 triangular prisms.

Perspective
projection of the cantellated 5-cell

Structure

The cantellated 5-cell has a relatively simple structure. We first explore the layout of its cells in its parallel projection into 3D, centered on an octahedral cell:

Parallel projection
of the cantellated 5-cell, showing nearest octahedron

This image shows the nearest octahedral cell to the 4D viewpoint. Four of its 8 faces are joined to 4 triangular prisms, as shown in the next image:

Parallel projection
of the cantellated 5-cell, adding 4 triangular prisms

The other 4 faces are joined to 4 cuboctahedra:

Parallel projection
of the cantellated 5-cell, showing the first of 4 cuboctahedra Parallel projection of the cantellated
5-cell, showing the second of 4 cuboctahedra Parallel projection of the cantellated
5-cell, showing the third of 4 cuboctahedra Parallel projection of the cantellated
5-cell, showing the fourth of 4 cuboctahedra

These cuboctahedra look flattened, because they are being viewed from an angle. They are perfectly uniform cuboctahedra in 4D.

These are all the cells that lie on the near side of the cantellated 5-cell. On the far side, a 5th cuboctahedron lies at the center of the projection, being the cell farthest away from the 4D viewpoint:

Parallel projection
of the cantellated 5-cell, showing antipodal cuboctahedron

Its square faces are joined to 6 triangular prisms:

Parallel projection
of the cantellated 5-cell, showing triangular prisms surrounding antipodal
cell

Four of its 8 triangular faces are joined to the 4 cuboctahedra on the near side of the polytope; the other four are joined to 4 more octahedra:

Parallel projection
of the cantellated 5-cell, showing 4 octahedra

These octahedra appear flattened because they are being viewed from an angle; in 4D they are perfectly regular octahedra.

In total, there are 1 octahedron, 4 triangular prisms, and 4 cuboctahedra on the near side of the cantellated 5-cell. On the far side, there are 1 cuboctahedron, 6 triangular prisms, and 4 octahedra. Thus, in total, there are 5 octahedra, 5 cuboctahedra, and 10 triangular prisms.

Perspective Projections

The following image shows the perspective projection of the cantellated 5-cell into 3D. The 4D viewpoint is looking at an octahedral cell:

Perspective
projection of the cantellated 5-cell

The nearest cell is shown in cyan, and the triangular prisms are shown in transparent yellow.

The next image shows the perspective projection with the 4D viewpoint looking at a cuboctahedral cell:

Perspective
projection of the cantellated 5-cell

The cuboctahedral cell lies in the interior of this projection, surrounded by a layer of triangular prisms and octahedra. The octahedra are almost flat, being foreshortened by a viewing angle close to 90°. They are, of course, perfectly regular in 4D.

Coordinates

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

  • (4/√10, 0, 0, ±2)
  • (4/√10, 0, ±3/√3, ±1)
  • (4/√10, −4/√6, 2/√3, 0)
  • (4/√10, 4/√6, −2/√3, 0)
  • (4/√10, −4/√6, −1/√3, ±1)
  • (4/√10, 4/√6, 1/√3, ±1)
  • (−6/√10, −2/√6, −2/√3, 0)
  • (−6/√10, 2/√6, 2/√3, 0)
  • (−1/√10, −1/√6, −4/√3, 0)
  • (−1/√10, −5/√6, −2/√3, 0)
  • (−1/√10, 3/√6, 0, ±2)
  • (−1/√10, −1/√6, 2/√3, ±2)
  • (−1/√10, −5/√6, 1/√3, ±1)
  • (−1/√10, 3/√6, ±3/√3, ±1)
  • (−6/√10, −2/√6, 1/√3, ±1)
  • (−6/√10, 2/√6, −1/√3, ±1)

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 13 Feb 2016.

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