The Cantitruncated 600-cell


The cantitruncated 600-cell is a beautiful uniform polychoron from the 120-cell/600-cell family. It is bounded by 120 truncated icosahedra, 720 pentagonal prisms, and 600 truncated octahedra. It may be constructed from the bitruncated 120-cell by radially expanding the truncated icosahedral cells outwards and filling in the gaps with pentagonal prisms.

The cantitruncated
600-cell
[Full-size image]

The above image shows a perspective projection of the cantitruncated 600-cell into 3D, centered on one of its 120 truncated icosahedral cells. The yellow truncated icosahedron in the center is the nearest cell to the 4D viewpoint. The green cells are 12 other truncated icosahedra separated from this nearest cell by pentagonal prisms, and the cyan cells are another layer of 32 truncated icosahedra. For clarity, we omit the cells that lie on the far side of the polytope, and show the prisms and truncated octahedra only in a light transparent color.

Structure

We shall explore the structure of the cantitruncated 600-cell by means of its parallel projection into 3D, centered on one of its truncated icosahedral cells.

First Layer

Parallel
projection of the cantitruncated 600-cell, showing nearest truncated
icosahedron

The above image shows the nearest truncated icosahedral cell to the 4D viewpoint. For clarity, we omit cells on the far side of the cantitruncated 600-cell and render the remaining cells in a light transparent color.

This truncated icosahedron may be regarded as the “north pole” cell of the cantitruncated 600-cell.

Second Layer

The 12 pentagonal faces of the nearest truncated icosahedron are shared with 12 pentagonal prisms:

Parallel
projection of the cantitruncated 600-cell, showing 12 pentagonal prisms

The 20 hexagonal faces of the truncated icosahedron are shared with 20 truncated octahedra, shown next:

Parallel
projection of the cantitruncated 600-cell, showing 7/20 truncated octahedra Parallel projection of the
cantitruncated 600-cell, showing 14/20 truncated octahedra Parallel projection of the
cantitruncated 600-cell, showing 20 truncated octahedra

Third Layer

The exposed blue pentagonal faces of the pentagonal prisms are joined to another 12 truncated icosahedra:

Parallel
projection of the cantitruncated 600-cell, showing 12 more truncated
icosahedra

These truncated icosahedra are linked by 30 pentagonal prisms, shown below:

Parallel
projection of the cantitruncated 600-cell, showing 30 more pentagonal
prisms

Another 20 truncated octahedra touch the exposed hexagonal faces of the previous group of 20 truncated octahedra:

Parallel
projection of the cantitruncated 600-cell, showing 20 more truncated
octahedra

Fourth Layer

The exposed blue square faces of the pentagonal prisms from the third layer are shared with another 30 truncated octahedra:

Parallel
projection of the cantitruncated 600-cell, showing 30 more truncated
octahedra

Nestled between these truncated octahedra are 60 more pentagonal prisms, in twelve circles of 5 pentagonal prisms each:

Parallel
projection of the cantitruncated 600-cell, showing 60 more pentagonal
prisms

Another 12 pentagonal prisms cap the green truncated icosahedra:

Parallel
projection of the cantitruncated 600-cell, showing 12 more pentagonal
prisms

Overlying these pentagonal prisms are another 60 truncated octahedra:

Parallel
projection of the cantitruncated 600-cell, showing 60 more truncated
octahedra

The bowl-shaped cavities gradually becoming visible here are filled by 20 truncated icosahedra:

Parallel
projection of the cantitruncated 600-cell, showing 20 more truncated
icosahedra

Straddled between these truncated icosahedra are another 30 pentagonal prisms:

Parallel
projection of the cantitruncated 600-cell, showing 30 more pentagonal
prisms

Fifth Layer

The pentagonal circles of truncated octahedra from the previous layer are, of course, where another 12 truncated icosahedral cells lie. These are shown next:

Parallel
projection of the cantitruncated 600-cell, showing 12 more truncated
icosahedra

These cells are bridged to the previous truncated icosahedral cells by 60 pentagonal prisms:

Parallel
projection of the cantitruncated 600-cell, showing 60 more pentagonal
prisms

The remaining exposed hexagonal faces of the truncated octahedral cells from the previous layer are where another layer of 60 truncated octahedra lie:

Parallel
projection of the cantitruncated 600-cell, showing 60 more truncated
octahedra

Yet another 60 truncated octahedra straddle these ones:

Parallel
projection of the cantitruncated 600-cell, showing yet 60 more truncated
octahedra

The obvious pentagonal prismic gaps around these cells are, obviously, filled in by more pentagonal prisms: 120 of them, 60 in twelve circles of 5 above 12 of the truncated icosahedra, and another 60 in 20 groups of three above the other 20 truncated icosahedra. The following image shows the first group of 60:

Parallel
projection of the cantitruncated 600-cell, showing 60 more pentagonal
prisms

The next image adds the second group of 60:

Parallel
projection of the cantitruncated 600-cell, showing another 60 pentagonal
prisms

Finally, the remaining shallow depressions around the pentagonal prisms of the second group are filled in by 20 truncated octahedra:

Parallel
projection of the cantitruncated 600-cell, showing 20 more truncated
octahedra

These are all the cells that lie on the “northern hemisphere” of the cantitruncated 600-cell.

The Equator

The “equator” of the cantitruncated 600-cell consists of the cells that lie at 90° to the 4D viewpoint of the parallel projection. Among them are 30 truncated icosahedra, as shown below:

Parallel
projection of the cantitruncated 600-cell, showing 30 equatorial truncated
icosahedra

For clarity's sake, we omit the cells from the northern hemisphere that we have already seen above. Due to the 90° view angle, these cells have been foreshortened into decagons; but in 4D, they are perfectly uniform truncated icosahedra. They are linked to each other via pentagonal prisms:

Parallel
projection of the cantitruncated 600-cell, showing 60 equatorial pentagonal
prisms

There are 60 pentagonal prisms here. They have been foreshortened into rectangles due to the 90° view angle. Another 12 pentagonal prisms lie on the equator, linking 12 truncated icosahedra from the northern hemisphere to their counterparts in the southern hemisphere:

Parallel
projection of the cantitruncated 600-cell, showing another 12 equatorial
pentagonal prisms

Unlike the previous pentagonal prisms, due to their different orientation, these ones have been foreshortened into pentagons. Both are perfectly uniform pentagonal prisms in 4D.

Surrounding these 12 pentagonal prisms are 60 truncated octahedra, shown next:

Parallel
projection of the cantitruncated 600-cell, showing 60 equatorial truncated
octahedra

These are all the cells that lie on the equator of the cantitruncated 600-cell. There are 20 hexagonal gaps remaining: these are not projection images of any cell, but are where 20 truncated octahedra in the northern hemisphere touch their counterparts in the southern hemisphere.

Past this point, we reach the “southern hemisphere”, that is, the far side of the cantitruncated 600-cell. The arrangement of cells exactly mirrors the layers of cells that we have seen in the northern hemisphere, except in reverse order.

Summary

The following table summarizes the cell counts for the various types of cells that form the boundary of the cantitruncated 600-cell:

Region Layer Truncated icosahedra Pentagonal
		prisms Truncated octahedra
Northern hemisphere 1 1 0 0
2 0 12 20
3 12 30 20
4 20 60 + 12 + 30 = 102 30 + 60 = 90
5 12 60 + 120 = 180 60 + 60 + 20 = 140
Subtotal 45 324 270
Equator 30 60 + 12 = 72 60
Southern hemisphere 5 12 180 140
4 20 102 90
3 12 30 20
2 0 12 20
1 1 0 0
Subtotal 45 324 270
Grand total 120 720 600

Coordinates

The Cartesian coordinates of the cantitruncated 600-cell are all permutations of coordinates and changes of sign of:

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

  • (0, 1, 3φ, 3+9φ)
  • (0, 1, 5φ, 5+7φ)
  • (0, 2, 4φ, 4φ3)
  • (0, 4+3φ, φ5, 5+4φ)
  • (0, 3+4φ, 4+5φ, 5+3φ)
  • (1, φ, 6φ2, 1+5φ)
  • (1, 2φ, 3+9φ, 2+φ)
  • (1, 3+φ, 3φ, 4φ3)
  • (1, 3φ, 6φ2, 3φ2)
  • (1, 3+2φ, 3φ3, 5+4φ)
  • (1, φ4, 2φ4, 5+3φ)
  • (φ, 2, φ3, 3+9φ)
  • (φ, 3, 1+3φ, 4φ3)
  • (φ, 3φ2, 2+6φ, 5+4φ)
  • (φ, 2φ3, 3φ3, 5+3φ)
  • (2, 2φ, 6φ2, 2φ3)
  • (2, 3+φ, 3φ3, 4+5φ)
  • (2, 3φ, 5+7φ, 3+2φ)
  • (2, 1+3φ, 5+6φ, 4+3φ)
  • (3, 2φ, 5+7φ, φ4)
  • (3, 2+φ, 2φ4, φ5)
  • (3, φ3, 5+6φ, 3+4φ)
  • (2φ, 3φ2, 1+6φ, 4+5φ)
  • (2φ, 4+3φ, 1+5φ, 3φ3)
  • (2+φ, 3+φ, 2+6φ, 3φ3)
  • (2+φ, 1+3φ, 6φ2, 3+2φ)
  • (2+φ, 4φ, 5+6φ, 3φ2)
  • 3, 3+φ, φ4, 6φ2)
  • 3, 3+2φ, 1+6φ, 3φ3)
  • 3, 3φ2, 5φ, 2φ4)
  • (3+φ, 3φ, 5+6φ, 2φ3)
  • (3φ, 3+2φ, 1+5φ, 2φ4)
  • (3φ, 2φ3, 1+6φ, φ5)
  • (3φ, 1+5φ, 2+6φ, 3+4φ)
  • (1+3φ, φ4, 1+6φ, 2+6φ)
  • (1+3φ, 5φ, 3φ3, 2φ3)
  • (4φ, φ4, 1+5φ, 3φ3)

where φ=(1+√5)/2 is the Golden Ratio.

These coordinates yield a cantitruncated 600-cell with edge length 2, centered on the origin.


Last updated 16 Feb 2016.

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