The Truncated 120-cell


The truncated 120-cell is a uniform polychoron bounded by 120 truncated dodecahedra and 600 tetrahedra.

The truncated
120-cell

It is a relatively simple member of the 120-cell/600-cell family of uniform polytopes, and, as its name implies, can be constructed by truncating the 600 vertices of the 120-cell. Equivalently, it can be constructed by expanding the edges of the 120-cell outwards radially, and filling in the resulting gaps with 600 tetrahedra.

Structure

The Near Side

First Layer

As is customary, we shall explore the structure of the truncated 120-cell by means of its parallel projection into 3D.

Parallel
projection of the truncated 120-cell, showing nearest truncated
dodecahedron

The above image shows the nearest truncated dodecahedron to the 4D viewpoint. For clarity, we have omitted edges and vertices not on this nearest cell, and render all the other cells in a transparent color.

The 20 triangular faces of this nearest truncated dodecahedron are joined to 20 tetrahedra, shown in red below:

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

Second Layer

The 12 decagonal faces of the nearest cell are joined to another 12 truncated dodecahedra:

Parallel
projection of the truncated 120-cell, showing 12 more truncated
dodecahedra

Nestled among these truncated dodecahedra are 20 more tetrahedra, joined to the previous 20 tetrahedra by an edge:

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

Slightly farther out from them are another 30 tetrahedra, straddling each adjacent pair of truncated dodecahedra:

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

The remaining exposed triangular faces of the truncated dodecahedra are joined to another 60 tetrahedra, which come in circles of 5:

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

Third Layer

Over these tetrahedra are a third layer of 20 truncated dodecahedra:

Parallel
projection of the truncated 120-cell, showing 20 more truncated
dodecahedra

Between each adjacent pair of truncated dodecahedra are more pairs of tetrahedra, adding another 60 tetrahedra:

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

The remaining exposed decagonal faces of the previous layer of truncated dodecahedra are where another 12 truncated dodecahedra are joined:

Parallel
projection of the truncated 120-cell, showing 12 more truncated
dodecahedra

Another 60 tetrahedra straddle these 12 truncated dodecahedra and the previous 20 truncated dodecahedra:

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

Finally, the remaining exposed triangular faces of those 20 truncated dodecahedra are peaked by 20 more tetrahedra:

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

These are all the cells on the near side of the truncated 120-cell.

The Equator

Next comes the cells that lie on the equator of the truncated 120-cell. There are 30 truncated dodecahedra that lie on the equator:

Parallel
projection of the truncated 120-cell, showing 30 equatorial truncated
dodecahedra

For the sake of clarity, we've omitted the cells that were previously seen.

There are 60 tetrahedra that lie on the equator:

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

These are all the cells that lie on the equator. The remaining 12 decagonal gaps do not correspond to any equatorial cells; they are where 12 truncated dodecahedra on the near side of the polytope touches their corresponding counterparts on the far side.

Past this point, we reach the far side of the truncated 120-cell, where the arrangement of cells exactly mirrors the arrangement of cells on the near side, as described earlier.

Summary

The following table summarizes the cell counts in the truncated 120-cell:

Region Layer Truncated dodecahedra Tetrahedra
Near side 1 1 20
2 12 20 + 30 + 60 = 110
3 20 + 12 = 32 60 + 60 + 20 = 140
Subtotal 45 270
Equator 30 60
Far side 3 32 140
2 12 110
1 1 20
Subtotal 45 270
Grand total 120 600

Coordinates

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

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

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

where φ=(1+√5)/2 is the Golden Ratio.


Last updated 01 Jan 2013.

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