Relative of the grand antiprism?

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Relative of the grand antiprism?

Postby username5243 » Mon Mar 16, 2020 8:14 pm

So I was thinking about the grand antiprism (gap) lately, and I thought of a potential CRF that I'm not sure has been discussed here before.

Gap can be constructed from ex (x3o3o5o, 600-cell) by chopping of 2 orthogonal cycles of 10 vertices. My idea was to do something similar starting with sidpixhi (x3o3o5x). That is, remove the vertices of 20 dodecahedra in 2 rings of 10 from sidpixhi.

The chopped off bits here will be segmentochoron doe || srid (dodecahedron || small rhombicosidodecahedron). however, just like in gap itself, the caps intersect, so will be diminished into pabidrids (parabidiminished srids) instead. Therefore, this figure will have 20 pabidrids, connected in two rings of 10 by their decagonal faces. The 100 remaining does should still be present and remain intact. There will also be an arrangement of tets, presumably similar to those of gap, plus some number of pips and trips I haven't worked out yet.

Will this construction produce a valid CRF? If so, what should it be called?
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Re: Relative of the grand antiprism?

Postby URL » Sat Apr 25, 2020 5:07 pm

After digitally building it, with help of quickfur's pages for the runcinated 120-cell and the grand antiprism, I can confirm that this is in fact a valid CRF. It has 300 tetrahedra, 700 triangular prisms, 500 pentagonal prisms, 100 dodecahedra, and 20 parabidiminished rhombicosidodecahedra as cells. There's no polychoron with 1620 cells on Klitzing's CRF list, so yours is probably new.

That said, I'm not sure diminishings have even been widely considered on this forum. And for good reason: there's way too many of them. You can remove two orthogonal rings of dodecahedra from a runcinated 120-cell, and that's cool, but you can also remove lots and lots of other subsets of dodecahedra. (The question is only which). So, I'm not sure if this diminishing is of particular interest (it might be worth looking at its symmetry).

For now, I've dubbed your polychoron the "icosadiminished runcinated hecatonicosachoron", or "icosadiminished runcinated 120-cell". Seems only appropriate. I've uploaded a frontal render and an OFF file with its vertices here https://drive.google.com/open?id=1MI91w ... gSe8b3_SCT, and I've added this model to my larger collection of GeoGebra models here https://drive.google.com/open?id=1nQZ-Q ... djF82T9MVy.
Last edited by URL on Sun Apr 26, 2020 3:54 am, edited 1 time in total.
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Re: Relative of the grand antiprism?

Postby URL » Sat Apr 25, 2020 5:43 pm

Actually, it might be worth pointing out. There are about 300 million diminishings of the 600-cell by non-adjacent vertices, and each corresponds to a diminishing of the runcinated 120-cell (instead of removing icosahedral pyramids, remove dodecahedra atop rhombicosidodecahedra). There are a further unknown amount of diminishings of the 600-cell that include adjacent vertices. I'm not sure if all of these are in correspondence to the analogous ones of the runcinated 120-cell. If they are, then I guess that you've found the analog to the grand antiprism.
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Re: Relative of the grand antiprism?

Postby quickfur » Tue Apr 28, 2020 12:02 am

I was working on maximally-diminished CRFs for a while, but stopped at the 120-cell/600-cell family because there were just too many of them that it was completely unmanageable.

I did find some neat things, though, like various 24-cell-family uniforms with diminishings at the "meta" positions that produce various interesting shapes with duoprismatic symmetry. I think there are pictures of these in the various threads here and also on the CRF polychoron discovery project pages on the wiki.

Years ago I also found various "deep" cuts of various 120-cell family uniforms that produce interesting shapes, including one with two orthogonal cycles of five x5x3x's each, with pentagonal rotunda cells in between, among other things.

And don't forget the 600-cell wedges, many of which may have analogues with deep-cut uniforms that might yield interesting shapes.

Actually, just the 600-cell itself produces very interesting shapes; I've posted here before about deep cuts of the 600-cell that produces rings of dodecahedra with icosahedra in the orthogonal plane, etc.. A lot of these probably have analogues in the 120-cell family uniforms, so that's an area worth looking at if you're into this sort of thing. With 300 million CRFs arising from non-adjacent diminishing of the 600-cell alone, if you count adjacent diminishings there will be a lot more of them, possibly an order of magnitude more or even beyond, so there's literally an endless space of possibilities there to experiment with.

There's also the duoprism family that's somewhat related to the grand antiprism: so far I've discovered augmentations of up to the 10,20-duoprism (sporting elongated pentagonal bicupola cells); one area that I still haven't managed to break into is whether there's some way to Stott-expand one of these duoprisms and insert cells in a way that closes it up in a CRF way. I've also been thinking about non-uniform but CRF analogues of the grand antiprism that extends its pattern of tetrahedra (or tetrahedral-like networks) sandwiched between two orthogonal rings of cells. Might it be possible to have, say, a hexagonal/square antiprisms in a double ring with appropriately-shaped CRF cells bridging them? Research so far as still inconclusive. Plenty to explore here.
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