The Cantellated 120-cell


The cantellated 120-cell is a uniform polychoron of the 120-cell/600-cell family. Its surface consists of 120 rhombicosidodecahedra, 600 octahedra, and 1200 triangular prisms.

The cantellated 120-cell
[Full-size image]

It may be constructed by radially expanding the octahedral cells of the rectified 600-cell outwards, which causes the icosahedral cells to become rhombicosidodecahedra, and filling the gaps with triangular prisms.

Structure

We shall explore the structure of the cantellated 120-cell by using its parallel projection into 3D, centered on one of the 120 rhombicosidodecahedral cells.

First Layer

Parallel
projection of the cantellated 120-cell, showing nearest
rhombicosidodecahedron

The above image shows the closest rhombicosidodecahedral cell to the 4D viewpoint. For clarity, we have omitted edges and vertices that do not lie on this cell, and have rendered all other cells with a light transparent color.

The triangular faces of the nearest rhombicosidodecahedron are joined to 20 octahedra, as shown next:

Parallel
projection of the cantellated 120-cell, showing 20 more octahedra

These octahedral cells are connected to each other by 30 triangular prisms, which touch the central rhombicosidodecahedron's square faces:

Parallel
projection of the cantellated 120-cell, showing 30 more triangular
prisms

Second Layer

Another 20 triangular prisms are connected to the octahedra, the beginnings of a network of triangular prisms and octahedra that permeates the surface of the cantellated 120-cell:

Parallel
projection of the cantellated 120-cell, showing 20 more triangular
prisms

These triangular prisms are capped by 20 more octahedra:

Parallel
projection of the cantellated 120-cell, showing 20 more octahedra

Touching the pentagonal faces of the central cell are 12 other rhombicosidodecahedral cells that fit into the framework of triangular prisms and octahedra:

Parallel
projection of the cantellated 120-cell, showing 12 more
rhombicosidodecahedra

These cells are girded by more triangular prisms and octahedra in the framework:

Parallel
projection of the cantellated 120-cell, showing 60 more triangular prisms and
30 more octahedra

There are 60 more triangular prisms and 30 more octahedra here.

Third Layer

The framework of triangular prisms and octahedra continues over the square and triangular faces of the 12 rhombicosidodecahedra, as shown next:

Parallel
projection of the cantellated 120-cell, showing 120 more triangular prisms and
60 more octahedra

There are 60+60=120 more triangular prisms here, as well as 60 more octahedra. The 60 octahedra, of course, have another 60 triangular prisms attached to them:

Parallel
projection of the cantellated 120-cell, showing yet 60 more octahedra

As is clearly seen, these triangular prisms and octahedra trace out the outlines of more rhombicosidodecahedral cells. We show another 20 of the latter below:

Parallel
projection of the cantellated 120-cell, showing 20 more
rhombicosidodecahedra

The remaining gaps are filled in by another 12 rhombicosidodecahedra:

Parallel
projection of the cantellated 120-cell, showing 12 more
rhombicosidodecahedra

Fourth Layer

The exposed triangular faces of the triangular prisms from the previous layer are capped by 60 more octahedra:

Parallel
projection of the cantellated 120-cell, showing 60 more octahedra

These octahedra are bridged to each other by more triangular prisms and octahedra which gird the rhombicosidodecahedral cells:

Parallel
projection of the cantellated 120-cell, showing 60 more octahedra and 150 more
triangular prisms

There are a total of 60+60+30=150 more triangular prisms and 60 more octahedra here. These 60 octahedra have another 120 triangular prisms connected to them:

Parallel
projection of the cantellated 120-cell, showing 120 more triangular
prisms

Finally, 60 of these 120 triangular prisms are capped by another 20 octahedra:

Parallel
projection of the cantellated 120-cell, showing 20 more octahedra

These are all the cells on the near side of the cantellated 120-cell. Past this point, we reach the cells on the limb, or “equator”, of the polytope.

The Equator

There are 30 rhombicosidodecahedral cells that lie on the equator of the cantellated 120-cell, as shown below:

Parallel
projection of the cantellated 120-cell, showing 30 equatorial
rhombicosidodecahedra

For clarity, we have omitted the other cells that we have seen so far.

These rhombicosidodecahedra appear to be flat, because they are being seen from a 90° angle. In 4D, they are perfectly uniform.

There are also 60 octahedral cells on the equator:

Parallel
projection of the cantellated 120-cell, showing 60 equatorial octahedra

Since these octahedra are being seen from a 90° angle, they appear flattened into rhombuses. Between them are 60 triangular prisms:

Parallel
projection of the cantellated 120-cell, showing 60 equatorial triangular
prisms

These triangular prisms, being seen from a 90° angle, have been foreshortened into rectangles. The edges of the rectangles that touch the rhombicosidodecahedra are the images of one of their square faces.

There are another 20 triangular prisms that sit between each triplet of rhombicosidodecahedra:

Parallel
projection of the cantellated 120-cell, showing 20 more equatorial triangular
prisms

These triangular prisms connect the last 20 octahedra on the near side of the cantellated 120-cell with their counterparts on the far side. Due to their different orientation with respect to the previous triangular prisms, they have been foreshortened into triangles instead.

These are all the cells that lie on the equator of the cantellated 120-cell. The remaining pentagonal gaps are not the images of any cells; they are where 12 of the rhombicosidodecahedral cells on the near side touch their counterparts on the far side.

Past this point, we reach the far side of the polychoron, where the arrangement of cells mirrors the arrangement we've seen on the near side.

Summary

The following table summarizes the cell counts for the cantellated 120-cell:

Region Layer Rhombicosidodecahedra Octahedra Triangular
		prisms
Near side 1 1 20 30
2 12 20 + 30 = 50 20 + 60 = 80
3 20 + 12 = 32 60 120 + 60 = 180
4 0 60 + 60 + 20 = 140 150 + 120 = 270
Subtotal 45 270 560
Equator 30 60 60 + 20 = 80
Far side 4 0 140 270
3 32 60 180
2 12 50 80
1 1 20 30
Subtotal 45 270 560
Grand total 120 600 1200

Coordinates

The coordinates of the cantellated 120-cell, centered on the origin and having edge length 2, are all permutations of coordinates and changes of sign of:

along with all even permutations of coordinates and all changes of sign of:

  • (0, 1, 4+5φ, 1+3φ)
  • (0, φ, φ5, 1+4φ)
  • (0, φ2, 3φ3, 2+φ)
  • (0, φ3, 2+5φ, 3φ2)
  • (1, φ2, 2+5φ, 2φ3)
  • (1, φ2, 4+5φ, 2φ2)
  • (1, 2φ, φ5, φ4)
  • (1, φ4, 2φ3, 3φ2)
  • (φ, φ2, 2φ, 3φ3)
  • (φ, 2φ2, 3+4φ, 3φ2)
  • 2, 2φ, 4+5φ, φ3)
  • 2, 2+φ, 3+4φ, 2φ3)
  • 2, φ3, 4φ2, φ4)
  • 2, φ4, 1+4φ, 2φ3)
  • (2φ, 1+3φ, 3+4φ, φ4)
  • (2+φ, φ3, φ5, 2φ2)
  • 3, 2φ2, 1+3φ, 2+5φ)

where φ=(1+√5)/2 is the Golden Ratio.


Last updated 16 Feb 2016.

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