Great pic (as usual )
This is neither a "great_rhombicosidodecahedron" nor a "great rhombicuboctahedron"! It is a truncated dodecahedron. (Cf. below.)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.
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).
P | x(P,2) x(P,3)
----+--------------
3 | 1 0
4 | sqrt2 1
5 | tau tau
5/2 | 1/tau -1/tau
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)
Klitzing wrote:Great pic (as usual )
This is neither a "great_rhombicosidodecahedron" nor a "great rhombicuboctahedron"! It is a truncated dodecahedron. (Cf. below.)[...]
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.
[...]
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:
- Code: Select all
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
Klitzing wrote:Gyration, i.e. cutting off some cap and glueing it back within a different orientation, OTOH looks like being more an effect of 3D. Here the relevant caps are cupolae. The base has alternating edges which connect to lacing triangles resp. lacing squares. So some rotation (gyration) can interchange those edges. In order to have the same effect for 4D figures, you would need sections, i.e. base polyhedra of the relevant cap, which has a higher symmetry than the cap itself. (Else any symmetry applied to the base, being required for glueing back, would result in an identity operation of the whole cap!) This looks to me rather restrictive in higher dimensions...
Klitzing wrote:Klitzing wrote:Gyration, i.e. cutting off some cap and glueing it back within a different orientation, OTOH looks like being more an effect of 3D. Here the relevant caps are cupolae. The base has alternating edges which connect to lacing triangles resp. lacing squares. So some rotation (gyration) can interchange those edges. In order to have the same effect for 4D figures, you would need sections, i.e. base polyhedra of the relevant cap, which has a higher symmetry than the cap itself. (Else any symmetry applied to the base, being required for glueing back, would result in an identity operation of the whole cap!) This looks to me rather restrictive in higher dimensions...
4D caps with one subsymmetrical base are:
- tet || oct (= tet-cap of rap), but in fact = rap itself --> so no gyration possible
- tet || co (= tet-cap of spid); spid = tet || co || dual tet --> ortho bi-(tet || co) possible
- oct || tut (= oct-cap of srip); wrong way round --> no gyration possible
- co || tut (= co-cap of srip); wrong way round --> no gyration possible
- tut || toe (= tut-cap of prip) --> gyration possible
--- rk
quickfur wrote:You're right, it only works for a few select cases.
Makes me wonder, though, whether it might be possible in 5D if the base of a cut-off segmentoteron is in the shape of a duoprism; then gyration in theory should be possible in two directions. I don't know if there are any actual instances of this among the 5D uniforms, though.
Klitzing wrote:quickfur wrote:You're right, it only works for a few select cases.
Makes me wonder, though, whether it might be possible in 5D if the base of a cut-off segmentoteron is in the shape of a duoprism; then gyration in theory should be possible in two directions. I don't know if there are any actual instances of this among the 5D uniforms, though.
The question is not the symmetry of the section alone.
It rather is that the larger base polytope of the cap shall have a larger symmetry than the cap as a whole (around its orthogonal axis)!
--- rk
Nick wrote:Quickfur, how does one even begin to envision these rotations in one's head? I can kinda do it with a hypercube, but these things are ridiculously complex!
Well we should not only use the total count, but use the internal symmetry as well. So cyclo tetra is unique. But not so para tetra. Therefore I used bi para bi instead. And, as para itself refers to 2 opposite ones, the second bi here is obsolete.
Keiji wrote:[...]
This latest addition also has the most complex imat I've added so far, the dual's edge kinds taking up over half the alphabet, and very nearly filling my 1920x1080p screen as displayed in the explorer. I hope we can find some interesting polytopes between the 14-kind K4.8 and the whopping 43-kind "4D J37"...
quickfur wrote:[...]
There's another "pseudo cantellated tesseract" (or J37 analogue) candidate, which was discovered by Klitzing (I had actually considered its construction before, but I wrongly rejected it by mistakenly assuming that it was the same as the cantellated tesseract itself). This one is the octagyrated x4o3x3o, which has 8 rhombicuboctahedra just like the x4o3x3o, but they are in two rings that are "misaligned" with each other, so that no octahedral cells are formed, but there's a toroidal net of square pyramids that surround two orthogonal rings of rhombicuboctahedra. I'm planning to do that one at some point, too.
Klitzing wrote:quickfur wrote:[...]
There's another "pseudo cantellated tesseract" (or J37 analogue) candidate, which was discovered by Klitzing (I had actually considered its construction before, but I wrongly rejected it by mistakenly assuming that it was the same as the cantellated tesseract itself). This one is the octagyrated x4o3x3o, which has 8 rhombicuboctahedra just like the x4o3x3o, but they are in two rings that are "misaligned" with each other, so that no octahedral cells are formed, but there's a toroidal net of square pyramids that surround two orthogonal rings of rhombicuboctahedra. I'm planning to do that one at some point, too.
In fact there are 2 such!
Both were contained in that post of mine, providing their "patterns" (showing the according augmentations to the unwrapped toroidal surface of squares of the underlying 8,8-duoprism) and their incidence matrices:Both having exactly the same net-count of cells: 8 sircoes + 32 trips + 32 squippies!
- the cyclotetragyrated small rhombitesseract (just gyrating all {4}||ops of one ring - with respect to the uniform small rhombitesseract), and
- the bicyclotetragyrated small rhombitesseract (gyrating all {4}||ops of both rings)
[...]
Keiji wrote:Thanks for the excellent renders and structure explanation as always Quickfur, and thanks for the history Klitzing, as I have been away for too long to catch up otherwise!
I shall not join in on the biparabi- vs. paratetra- debate since I am too ignorant to be taking part in that But as Klitzing has been so kind as to provide the imat I shall endeavor to add it (and its cell, J37) to the Polytope Explorer ASAP.
Edit: Here we go! http://teamikaria.com/hddb/explorer/?n=64
Interestingly, the dual of this shape has two kinds of cells... one is the trigonal bipyramid, and the other is the same as the (unique kind of) cell found in the dual of the BXD! This polytope is essentially an irregular triangular prism, with one of the square faces broken into two triangular faces, creating an extra edge.
This latest addition also has the most complex imat I've added so far, the dual's edge kinds taking up over half the alphabet, and very nearly filling my 1920x1080p screen as displayed in the explorer. I hope we can find some interesting polytopes between the 14-kind K4.8 and the whopping 43-kind "4D J37"...
I(m,m)*I(m,n) = I(n,m)*I(n,n)
Klitzing wrote:Just a question, Keiji:
Why are your "imats" in the polytope explorer triangular ones only?
I usually provide square matrices.
Whereas you just use the subdiagonal entries (the count of the elements of the elements) and finally add what I use for diagonal (the count of elements) in a separate row below.
Further, there is an other useful thing for my square matrices, which is missing in your triangular display:
The IncMat of the dual of some polytope clearly is obtained just by a rotation of the matrix (within the paper plane) about 180°.
(Thus you usually have to set up 2 entries in your "polytope explorer", while the complete info would be contained in a single square matrix. - Which nonetheless could be provided in either orientation, if you would like.)
Keiji wrote:No polytope this month?
[...]
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