## Uniform polyterons and 120-cell family

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

### Uniform polyterons and 120-cell family

Can polychoron which is in uniform 120-cell family be used in non-prismatic polyteron?

In 4D, there's no polychoron whose all of cells are icosahedron, but there are some non-prismatic uniform polychora whose some of cells are icosahedron.
Likewise, I think that polychoron in uniform 120-cell family can be used in polyteron.

Is that impossible? If that exists, Could you give some examples of that?
hy.dodec
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### Re: Uniform polyterons and 120-cell family

Look at the circumradius of the simplest unit of the 120-cell family, the 600-cell, which has a circumradius of phi = (1+sqrt(5))/2 ~ 1.618. A regular polyteron based on the 600-cell pyramid would not exist because its circumradius must be less than 1 (which isn't).

This rules out the existence of regular polytera based on the 120-cell family. Since in 5D the only regular shapes are the simplex, hypercube, and orthoplex, there is never a choice for a uniform non-prismatic polyteron with 120-cell derived terons. But one can derive two uniforms with 120-cell prism symmetry (order 28800) by putting two sidtaxhis and gadtaxhis along with pentachora. They share the same vertex layout as the 120-cell prism and are considered prismatic polytera. You can find them on hedrondude's miscellaneous page in his uniform polytera page.

Keep in mind that the existence of a particular object sometimes depends on circumradius, especially when finding vertex figures.
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### Re: Uniform polyterons and 120-cell family

The main point here is that when you use a 4D polytope within 5D, you will have left just a single perpendicular dimension for continuation.
Thus you are left to consider lace prisms or lace towers, here esp. with 120-cell family across symmetry.
This then is quite similar to 3D, where you can have the pentagonal pyramid, the pentagonal prism, the pentagonal antiprism, the pentagonal cupola, and even the pentagonal rotunda - to enlist just some.
That is, the mere prism itself is not the only solution, neither here nor there!
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### Re: Uniform polyterons and 120-cell family

I think the OP wasn't looking for a uniform solution, which is (probably?) impossible.

Also, I think lace prisms are kinda cheating, because they are ultimately just derivations / elaborations of prisms.

My guess is that the OP is looking for a polyteron where 120-cell family polychora serve as facets that don't lie in parallel 4-planes. Now obviously if you relax the uniform requirement you'll always be able to find something, so there ought to be some restriction on what other facets there can be. A loose restriction is being CRF, but that would be hard to find a solution for since we don't know the full set of 4D CRFs yet. A tighter restriction is that all facets must be uniform, though the overall polyteron may not be. I'm not sure there's a solution in that case, but maybe it's possible.

An even tighter restriction is if we add the requirement that the overall polyteron must have some symmetry group higher than prismatic symmetry (or even asymmetrical, as would be in the CRF case in general). I'm not 100% sure if this is possible, but perhaps there is a way to do this? E.g., take a uniform polyteron that has 16-cell facets, substitute that with snub 24-cells, then cap them with pyramids to make 600-cells, then expand / modify the rest of the polyteron accordingly so that the remaining gaps can be closed up with uniform (or CRF?) pieces.
quickfur
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### Re: Uniform polyterons and 120-cell family

The trouble with {5,3,3} and dependent figures, is that there is a lace tower that contains fourteen of these figures, but it's completely flat in 5d.

So while something like oxo3ooo5oox&#xt is a solid in four dimensions, the lacing of increasing-size {3,3,5} things is perfectly flat. Kind of like oxxoo6ooxxo&#xt. That is, both the hexagonal pyramid and cupola are flat things.

The same happens with the 4_21. I found the first fifty rings of this, to comfirm the theory of number-occupations in octonions.
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