The Pentagonorhombic Trisnub Trisoctachoron


The Pentagonorhombic Trisnub Trisoctachoron (D4.11), also known by its Bowers' Acronym pretasto, is a CRF polychoron with demitesseractic symmetry that has a surface containing 144 cells (24 bilunabirotundae (J91), 8 cuboctahedra, 40 octahedra, 32 tridiminished icosahedra (J63), and 40 tetrahedra), 592 polygons (96 pentagons, 48 squares, and 448 triangles), 624 edges, and 176 vertices. It was discovered on 12 Mar 2014.

D4.11

Its primary point of interest lies in its 24 bilunabirotunda (J91) cells, which are transitive under its demitesseractic symmetry and therefore correspond with the icosahedra of the snub 24-cell. It can be constructed from the snub 24-cell using a kaleido-faceting operation followed by a partial Stott expansion that restores convexity.

These J91 cells can be augmented with J91 pseudopyramids, whose pentagonal pyramid cells will lie in the same hyperplanes as the J63 cells, thus augmenting them into icosahedra. The resulting polychoron is CRF, has demitesseractic symmetry, and sports 32 icosahedral cells.

Structure

We shall explore the structure of D4.11 using its parallel projections into 3D.

The Near Side

Parallel projection of
D4.11, showing nearest tetrahedron

The above image shows the tetrahedral cell nearest to the 4D viewpoint. For the sake of clarity, we render all other cells in a light transparent color.

The edges of this tetrahedron are joined to 6 bilunabirotundae (J91):

Parallel projection of
D4.11, showing 2/6 bilunabirotundae

Parallel projection of
D4.11, showing 4/6 bilunabirotundae

Parallel projection of
D4.11, showing 6/6 bilunabirotundae

Note the alternating orientation of these pairs of J91 cells, which indicates the underlying demitesseractic symmetry of the polytope as a whole.

The vertices of the nearest tetrahedron are joined to 4 other tetrahedra:

Parallel projection of
D4.11, showing 4 more tetrahedra

For clarity, we omit the J91 cells here.

These tetrahedra each belong to a cluster of cells that contain 4 tetrahedra surrounding a cuboctahedron. The corresponding cuboctahedra are shown next:

Parallel projection of
D4.11, showing 4 cuboctahedra

Three of the other triangular faces of these cuboctahedra are joined to other tetrahedra, for a total of 12 more tetrahedra:

Parallel projection of
D4.11, showing 12 more tetrahedra

These latest tetrahedra lie at a sharper angle to the 4D viewpoint, and therefore appear flattened by foreshortening. They are, of course, perfectly regular in 4D.

These clusters that each consist of cuboctahedron and 4 tetrahedra are symmetric to each other with respect to the underlying demitesseractic symmetry. There are 8 such groups in the polychoron; these 4 lie on the near side, facing the current 4D viewpoint, and the other 4 lie on the far side.

The tetrahedra directly touching these cuboctahedra are distinguished from the nearest tetrahedron in the center of the projection. We may call them tetrahedra of the second kind, whereas the tetrahedron in the center of the projection, which is part of a set of 8 equivalent tetrahedra, can be termed tetrahedra of the first kind. As we shall see, tetrahedra of the first kind only share edges with the J91 cells; whereas tetrahedra of the second kind share faces with the J91 cells.

Coming back to the nearest tetrahedron, its 4 faces are joined to 4 tridiminished icosahedra (J63):

Parallel projection of
D4.11, showing 4 J63's

Again, for clarity, we omit the other cells seen so far, except for the central tetrahedron itself, for reference.

The pentagonal faces of these J63 cells are joined to the bilunabirotundae we saw earlier. Here they are for reference:

Parallel projection of
D4.11, showing 4 J63's with J91's

The exposed triangular faces of the J63 cells are joined to clusters of octahedra, each having 5 octahedra: one central octahedron surrounded by 4 others in tetrahedral formation. The next image shows the central octahedra:

Parallel projection of
D4.11, showing 4 central octahedra

The lateral octahedra surrounding these central octahedra are shown next:

Parallel projection of
D4.11, showing 12 lateral octahedra

The remaining exposed faces of the central octahedra are joined to more J63 cells. Together with the J63 cells we saw earlier, these J63 cells surround each central octahedron in tetrahedral formation, in dual orientation to the lateral octahedra.

Parallel projection of
D4.11, showing 12 more J63 cells

Next, we add the cells from the clusters of cuboctahedra and tetrahedra that we saw earlier:

Parallel projection of
D4.11, showing cuboctahedra and tetrahedra clusters

These are all the cells that lie on the near side of the polytope.

The Equator

Now we come to the equator of the polychoron. There are 6 tetrahedra that lie on the equator:

Parallel projection of
D4.11, showing equatorial tetrahedra

These tetrahedra lie at a 90° angle to the 4D viewpoint, and therefore appear foreshortened into squares. They are, of course, regular tetrahedra in 4D. They are all tetrahedra of the first kind, as defined earlier.

There are also 12 bilunabirotundae (J91) on the equator:

Parallel projection of
D4.11, showing equatorial J91 cells

These J91 cells have been foreshortened to octagons because they lie at a 90° angle to the 4D viewpoint.

The 8 remaining triangular gaps above are not the images of any cell; they are where the cuboctahedra on one side of the polytope touch the lateral octahedra on the other side.

The Far Side

Now we come to the far side of the polytope. It has the same arrangement as the near side of the polytope, except in complementary orientation, due to its demitesseractic symmetry.

Parallel projection of
D4.11, showing antipodal tetrahedron

Here is the farthest tetrahedron, antipodal to the first tetrahedron we saw earlier. Notice how its orientation is dual to the orientation of the first tetrahedron.

Here are the J91 cells on the far side of the polychoron, sharing edges with the farthest tetrahedron:

Parallel projection of
D4.11, showing 2/6 far side J91 cells

Parallel projection of
D4.11, showing 4/6 far side J91 cells

Parallel projection of
D4.11, showing 6/6 far side J91 cells

They are all in the analogous positions as the J91 cells on the near side, except with a different orientation.

The faces of the farthest tetrahedron are shared with 4 J63 cells:

Parallel projection of
D4.11, showing 4 far side J63 cells

The triangular faces of these J63 cells opposite the tetrahedron are joined to 4 central octahedra, parts of the clusters of 5 octahedra that correspond with the analogous clusters on the near side of the polytope.

Parallel projection of
D4.11, showing 4 far side central octahedra

Here are the lateral octahedra surrounding these central octahedra, 16 of them in total:

Parallel projection of
D4.11, showing 12 far side lateral octahedra

Straddling the edges of the J63 cells, and touching the vertices of the farthest tetrahedron, are 4 more tetrahedra of the second kind:

Parallel projection of
D4.11, showing 4 far side tetrahedra of the 2nd kind

Their triangular faces opposite the vertices of the farthest tetrahedron are joined to 4 cuboctahedra:

Parallel projection of
D4.11, showing 4 cuboctahedra

These cells are all filling the gaps between the J91 cells, of course. The next image shows them together:

Parallel projection of
D4.11, showing 6 J91 cells again

The remaining exposed triangular faces of these J91 cells are joined to 12 more tetrahedra of the second kind:

Parallel projection of
D4.11, showing 12 more tetrahedra of the 2nd kind

Finally, between the lateral octahedra and the exposed pentagonal faces of the J91 cells are 12 more J63 cells:

Parallel projection of
D4.11, showing 12 more J63 cells

These are all the cells that lie on the far side of the polychoron.

Summary

The following table summarizes the cell counts in various regions of D4.11:

Region Layer Bilunabirotundae Cuboctahedra Octahedra Tridiminished icosahedra Tetrahedra
Near side 1 6 - - - 1
2 - - - - 4
3 - 4 - - 12
4 - - 4 4 -
5 - - 16 12 -
Subtotal 6 4 20 16 17
Equator 12 - - - 6
Far side 1 6 - - - 1
2 - - 4 4 -
3 - - 16 - -
4 - 4 - - 4
5 - - - 12 12
Subtotal 6 4 20 16 17
Grand total 24 8 40 32 40
144 cells

Coordinates

The coordinates of the pentagonorhombic trisnub trisoctachoron, centered on the origin with edge length 2, are all permutations of coordinates and even changes of sign of:

where φ=(1+√5)/2 is the Golden Ratio.


Last updated 14 May 2019.

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