student5 wrote:[...]

I haven't found anything, guess February is over or something, but I did find out how incredibly lucky CJ4.8.2 was

[...]

February is certainly not over! I've found another CRF with tetrahedral symmetry, sporting 8 J92's and 12 J91's. You guys are gonna love this one... It has two identical halves, which are joined in dual orientation to each other. Here is the lace tower:

- Code: Select all
`x3o3o // tetrahedron`

f3o3x // 4 tetrahedra touching vertices of previous one

o3x3f // midpoints of J91's

f3x3x // square faces of J91's

x3o3F // octahedra above J63's

x3F3o // opposite edge of J91's

f3x3f // f3x vertices of 8 J92's (this is the midpoint)

o3F3x

F3o3x

x3x3f

f3x3o

x3o3f

o3o3x

As you can see, there are two antipodal tetrahedra. EDIT 3: Also, there are no lacing edges between f3x3x and x3o3F, so that part of the tower may be better written as:

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`...`

o3x3f

/ f3x3x

x3o3F |

| x3F3o

| f3x3f |

o3F3x |

| F3o3x

x3x3f /

f3x3o

...

Well let's look at a render to see what's going on here:

Here's a projection centered on one of the antipodal tetrahedra. Around this central tetrahedron (yellow; you can faintly see it buried deep in the middle where the J91's meet) are 6 J91's touching its edges (brown), with 4 J63's (tridiminished icosahedra) and 4 other tetrahedra (green -- but not the ones on the outside) filling in the gaps in between. The 4 triangular faces of each J63 opposite the antipodal tetrahedra are joined to a formation of 4 octahedra, 3 surrounding a central one (central one shown in cyan). The central octahedron also has 3 pentagonal antiprisms surrounding it, lying on top of the J91's. The opposite faces of the 4 tetrahedra surrounding the antipodal tetrahedra are joined to triangular cupola, which in turn are joined to 4 J92's. The central octahedra with the neighbouring pentagonal antiprisms are joined to another 4 J92's on the far side (which are a mirror image, in dual orientation, of the near side 4 J92's).

Another interesting thing: besides the tetrahedra surrounding these antipodal tetrahedra, there are 6 other tetrahedra that connect the J91's on the near side to the J91's on the far side, and together, these 8 tetrahedra lie at the vertices of a 16-cell!

Now, the J92 cells may be a bit hard to see, so here's another render highlighting them:

Now you can see that square pyramids link the J92's to the cyan octahedra. Notice that the projection envelope has cubic symmetry? (I'm using parallel projection.) So the cyan octahedra are joined to another 4 J92's on the far side, in the dual positions of where they are on the near side. Similarly, the J92's on this side are joined to 4 octahedra on the far side -- the far side counterparts of the cyan octahedra.

Now, let's look at the side-view of this CRF, just to show you how closely it approximates 16-cell symmetry.

The top and bottom squares are the images of the aforementioned antipodal tetrahedra (the x3o3o and o3o3x in the lace tower). The brown cells are J91's, which, as you can see, link these tetrahedra to the 8 equatorial tetrahedra. But the equatorial tetrahedra aren't linked by J91's, they are linked by pairs of J92's (sorry, the blue coloring is a bit ambiguous, each blue patch is actually a pair of J92's, not a single one). If you look carefully at the nearest part of the projection (in 3D, not in 4D) to the upper right of the green square (image of an equatorial tetrahedron), between the brown J91 on top and the blue J92 to the right, you can see the outline of a pentagonal antiprism in an oblique orientation.

So there you have it. A CRF with tetrahedral symmetry, having J91 and J92 cells, that almost has 16-cell symmetry.

February is far from over, folks. CRFebruary, that is.

EDIT: uploaded the .def and .off files to D4.10.