The Cantitruncated 120-cell


The cantitruncated 120-cell is a beautiful uniform polytope from the 120-cell family of uniform polychora.

Perspective projection of the
cantitruncated 120-cell
[Full-size image]

It is bounded by 120 great rhombicosidodecahedra, 1200 triangular prisms, and 600 truncated tetrahedra. The above image shows its perspective projection into 3D, centered on a great rhombicosi­dodecahedron. For clarity, only facets facing the 4D viewpoint are included; facets lying on the far side of the polytope have been culled.

The cantitruncated 120-cell may be constructed by expanding the decagonal ridges of the truncated 120-cell outwards radially, which causes the truncated dodecahedra to turn into great rhombicosi­dodecahedra, the tetrahedra to turn into truncated tetrahedra, and introduces gaps which can be filled in by triangular prisms.

Structure

We shall explore the structure of the cantitruncated 120-cell using its parallel projections into 3D, centered on a great rhombicosi­dodecahedron.

First Layer

Parallel
projection of the cantitruncated 120-cell, showing nearest great
rhombicosidodecahedron

The above image shows the nearest cell to the 4D viewpoint, a great rhombicosidodecahedron. For clarity, we have rendered all the other cells in a light transparent color.

Second Layer

Surrounding this nearest cell are 30 triangular prisms that touch this cell at its square faces:

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

There are also 20 truncated tetrahedra that touch the nearest cell at its hexagonal faces:

Parallel
projection of the cantitruncated 120-cell, showing 20 truncated
tetrahedra

These two sets of cells form an alternating pattern around each decagonal face of the nearest cell. This pattern is extended by another 20 triangular prisms that sit atop the truncated tetrahedra:

Parallel
projection of the cantitruncated 120-cell, showing another 20 triangular
prisms

And continues with 20 truncated tetrahedra on the other end:

Parallel
projection of the cantitruncated 120-cell, showing another 20 truncated
tetrahedra

These alternating cells form a framework into which 12 great rhombicosidodecahedra, joined to the decagonal faces of the nearest cell, are fitted:

Parallel
projection of the cantitruncated 120-cell, showing 3 of another 12 great
rhombicosidodecahedra

Parallel
projection of the cantitruncated 120-cell, showing 6 of 12 great
rhombicosidodecahedra

Parallel
projection of the cantitruncated 120-cell, showing 9 of 12 great
rhombicosidodecahedra

Parallel
projection of the cantitruncated 120-cell, showing 12 of 12 great
rhombicosidodecahedra

Third Layer

The alternating pattern of triangular prisms and truncated tetrahedra continues with 60 more triangular prisms joined to the visible truncated tetrahedra:

Parallel
projection of the cantitruncated 120-cell, showing 60 more triangular
prisms

And 30 more truncated tetrahedra bridging these triangular prisms:

Parallel
projection of the cantitruncated 120-cell, showing 30 more truncated
tetrahedra

And yet another 60 triangular prisms extending from these:

Parallel
projection of the cantitruncated 120-cell, showing yet 60 more triangular
prisms

Which lead to 60 more truncated tetrahedra:

Parallel
projection of the cantitruncated 120-cell, showing yet 60 more truncated
tetrahedra

Which are, in turn, bridged by another 60 triangular prisms:

Parallel
projection of the cantitruncated 120-cell, showing yet another 60 triangular
prisms

Into this network of alternating triangular prisms and truncated tetrahedra another 20 great rhombicosidodecahedra are fitted:

Parallel
projection of the cantitruncated 120-cell, showing another 20 great
rhombicosidodecahedra

Fourth Layer

The network of alternating truncated tetrahedra and triangular prisms continues with 60 more triangular prisms nestled between adjacent pairs of great rhombicosidodecahedra:

Parallel
projection of the cantitruncated 120-cell, showing another 60 triangular
prisms

They are, of course, topped by 60 truncated tetrahedra, continuing in the pervasive pattern we've seen so far:

Parallel
projection of the cantitruncated 120-cell, showing another 60 truncated
tetrahedra

These truncated tetrahedra, as expected, are bridged by 30 more triangular prisms:

Parallel
projection of the cantitruncated 120-cell, showing another 30 triangular
prisms

The other triangular faces of the truncated tetrahedra are also joined to more triangular prisms, 120 in total:

Parallel
projection of the cantitruncated 120-cell, showing another 120 triangular
prisms

These triangular prisms meet at another 60 truncated tetrahedra:

Parallel
projection of the cantitruncated 120-cell, showing another 60 truncated
tetrahedra

Emanating from these truncated tetrahedra are 60 more triangular prisms:

Parallel
projection of the cantitruncated 120-cell, showing another 60 triangular
prisms

These triangular prisms converge on 20 truncated tetrahedra:

Parallel
projection of the cantitruncated 120-cell, showing another 20 truncated
tetrahedra

The obvious bowl-shaped depressions are filled in by 12 more great rhombicosidodecahedra:

Parallel
projection of the cantitruncated 120-cell, showing another 12 great
rhombicosidodecahedra

Lying against these great rhombicosidodecahedra and the previous truncated tetrahedra are another 60 triangular prisms:

Parallel
projection of the cantitruncated 120-cell, showing another 60 triangular
prisms

These are all the cells that lie on the near side of the cantitruncated 120-cell. The pattern of cells continues past this point on its limb, or “equator”.

The Equator

There are 30 great rhombicosidodecahedra lying on the equator of the cantitruncated 120-cell, shown below:

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

For clarity, we have omitted the cells on the near side that we have seen before.

These cells appear flattened into irregular dodecagons because they are at a 90° angle to the 4D viewpoint. In 4D, they are perfectly uniform great rhombicosidodecahedra.

The tiny triangular gaps nestled between these cells are where 20 triangular prisms are located, forming the vertices of a virtual dodecahedron:

Parallel
projection of the cantitruncated 120-cell, showing 12 equatorial triangular
prisms

We have omitted the previous great rhombicosidodecahedra so that the triangular prisms are more easily discernible. They have been foreshortened into triangles because they are seen from a 90° angle. In 4D, they are perfectly uniform triangular prisms.

Here are both sets of cells together:

Parallel
projection of the cantitruncated 120-cell, showing 12 equatorial triangular
prisms with previous great rhombicosidodecahedra

These aren't the only triangular prisms on the equator; there are 60 others, also touching the great rhombicosidodecahedra, but in a different orientation:

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

These triangular prisms have been foreshortened into rectangles rather than triangles, because they are oriented differently from the previous triangular prisms, even though both are being seen at a 90° angle.

They alternate with 60 truncated tetrahedra, shown next:

Parallel
projection of the cantitruncated 120-cell, showing 60 equatorial truncated
tetrahedra

These truncated tetrahedra appear as irregular pentagons due to their being seen from a 90° angle, but in reality they are perfectly uniform truncated tetrahedra.

These are all the cells that lie on the equator of the cantitruncated 120-cell. Past this point, we get to its far side. The remaining 12 decagonal gaps are not the images of any cells; they are where 12 great rhombicosidodecahedral cells on the near side touch their counterparts on the far side. The arrangement of cells on the far side mirrors that of the near side, repeating in reverse order until it ultimately converges on the antipodal great rhombicosi­dodeca­hedron.

Summary

The cell counts of each layer of the cantitruncated 120-cell are summarized in the following table:

Region Layer Great rhombicosidodecahedra Triangular
		prisms Truncated tetrahedra
Near side 1 1 0 0
2 12 30 + 20 = 50 20 + 20 = 40
3 20 60 + 60 + 60 = 180 30 + 60 = 90
4 12 60 + 30 + 120 + 60 + 60 = 330 60 + 60 + 20 = 140
Subtotal 45 560 270
Equator 30 20 + 60 = 80 60
Far side 4 12 330 140
3 20 180 90
2 12 50 40
1 1 0 0
Subtotal 45 560 270
Grand total 120 1200 600

Coordinates

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

as well as even permutations of coordinates and all changes of sign of:

  • (0, 1, 6+7φ, 3+7φ)
  • (0, 1, 6+9φ, 1+5φ)
  • (0, φ2, φ6, 2+7φ)
  • (0, φ2, 7+8φ, 2+5φ)
  • (0, 2φ, 6+8φ, 2+6φ)
  • (1, 4+5φ, 2φ4, 5φ2)
  • (1, 2, 5+7φ, 4+7φ)
  • (1, 2+φ, 4φ3, 3+7φ)
  • (1, φ3, 7+8φ, 3+4φ)
  • (1, φ3, 5+10φ, 3+2φ)
  • (1, 3φ, 6+8φ, φ5)
  • (1, φ4, 4φ3, 5φ2)
  • (2, φ2, 5+10φ, φ4)
  • (2, φ3, 6+9φ, 3φ2)
  • (2, 2φ2, 4φ3, 2φ4)
  • (2, 1+3φ, φ6, 4+5φ)
  • 2, φ5, 3+7φ, 5φ2)
  • 2, 3φ, 5+10φ, 2φ2)
  • 2, 3φ2, 5+7φ, 5φ2)
  • (2φ, 2+φ, 1+3φ, 5+10φ)
  • (2φ, 3+4φ, 4+7φ, 5φ2)
  • (2+φ, 3φ, 6+9φ, φ4)
  • (2+φ, 3+2φ, 5+7φ, 2φ4)
  • (2+φ, φ4, 6+7φ, 4+5φ)
  • 3, φ5, 2+7φ, 2φ4)
  • 3, 2+6φ, 3+7φ, 4+5φ)
  • (3φ, 2+5φ, 4+7φ, 4+5φ)
  • (3φ, 3+4φ, 3+7φ, 2φ4)
  • (2φ2, 1+3φ, 7+8φ, φ4)
  • (2φ2, 3+2φ, 4+7φ, 3+7φ)
  • (2φ2, 1+4φ, 6+7φ, φ5)
  • (1+3φ, 1+5φ, 5+7φ, φ5)
  • (3+2φ, φ4, 6+8φ, 3φ2)
  • (3+2φ, 1+4φ, φ6, 3+4φ)
  • 4, 1+4φ, 5+7φ, 2+6φ)
  • 4, 1+5φ, 4φ3, 3+4φ)
  • (1+4φ, 3φ2, 2+5φ, 4φ3)

where φ=(1+√5)/2 is the Golden Ratio.


Last updated 16 Feb 2016.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid XHTML 1.1! Proud to be Microsoft-free