Great pic (as usual
![Very Happy :D](./images/smilies/grin.gif)
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This is a pretty polychoron with 120 rhombicosidodecahedra, 600 octahedra, and 1200 triangular prisms. It can be made by a Stott expansion on the octahedral cells of the rectified 600-cell. I would say it's the closest analogue to the rhombicosidodecahedron itself, and as such exhibits the possibility of gyrations and diminishings. In particular, one can cut decagonal_prism||pentagon segmentochora from it to produce various diminishings, as well as gluing said segmentochoron back the "wrong way", i.e., gyrated, to form a large number of CRF gyrated cantellated 120-cells. The octahedral cells split into square pyramids. The number of possible diminishings/gyrations is the same as the number of non-adjacent pentagons on the 120-cell. I don't know how to accurately count that number, but it has to be quite a big number!
There is (at least) another kind of diminishing possible, that is to cut off rhombicuboctahedron||great_rhombicosidodecahedron.segmentochora. This introduces great rhombicuboctahedron cells to the result, which is still CRF, and so adds another layer of possible CRF diminishings. This type of diminishing can be done in the positions isomorphic to non-adjacent cells on the 120-cell.
This is neither a "great_rhombicosidodecahedron" nor a "great rhombicuboctahedron"! It is a truncated dodecahedron. (Cf. below.)
All diminishings will produce various diminished rhombicosidodecahedral cells, along with some combination of gyrations -- I haven't proved it, but you should be able to get all of the Johnson solids derived from the rhombicosidodecahedron as cells, by the appropriate truncations/gyrations.
So this little pretty thing is a veritable gold mine of CRFs, even more so than the cantellated tesseract!
On that note, it seems that cantellated polytopes in general have this property of having CRF diminishings and gyrations -- at least AFAICT. The cantellated 24-cell is another example that have yet to be fully explored: it shows a lot of promise of generating a large number of CRFs (though not as many as this one generates).
You question what lace prism would be obtained as the rhombi-figure first caps of a small rhombated polychoron (i.e. cantellated one). The answer can be given in a closed form. I just have to introduce some secant lengths first:
let x(n, k) be the length of the secant of the regular n-gon, which connects some vertex with the k-th follower. So, for example we have generally x(n, 0) = 0 and x(n, 1) = 1. Likewise x(n, n-1) = 1 and x(n, n) = 0. More generally we have x(n, k) = sin(k pi/n)/sin(pi/n).
(You even could use rational numbers in the first argument (polygrams), the second clearly remains integral, esp. x(P, numerator(P)) = 0.)
Some special values would be:
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P | x(P,2) x(P,3)
----+--------------
3 | 1 0
4 | sqrt2 1
5 | tau tau
5/2 | 1/tau -1/tau
That desired cap of any o-P-x-Q-o-R-x at the x-Q-o-R-x first direction then would be: x-Q-o-R-x || x(P,3)-Q-x(Q,2)-R-x.
Esp. if you ask the bottom figure to be unit edged too, both x(P,3) and x(Q,2) should be either of length 1 or 0. Therefore Q is forced to be 3, and P might be 3 or 4.
You would be just left with:
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o-3-x-3-o-P-x has caps of the form x-3-o-P-x || o-3-x-P-x (i.e. xo3oxPxx&#x)
o-4-x-3-o-3-x has caps of the form x-3-o-3-x || x-3-x-3-x (i.e. xx3ox3xx&#x)
--- rk