## Johnsonian Polytopes

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

### Re: Johnsonian Polytopes

wendy wrote:But none of them got as far as writing them in-line like i do. Coxeter will draw pretty pictures in text, like @-----@-----o. But they stay like that, they are not reduced to x3x3o or anything, despite what Klitzing says. (rk thought coxeter invented my notation )

Sorry for that old posts of mine, which might be read this way. I wasn't intending to do so. "x3x3o" clearly differs from "@-----@-----o". And it is correct that Coxeter usually did use graphical advices here only. (So I do not know whether he, in his late years, adopted your linearization somewhere for inline display.)

I just felt that the transliteration (@ -> x, o -> o, '   ' -> ' ', ' . ' -> 3, ' 4 ' -> 4, etc.) was too immediate to be mentioned as additional Input to be associated to someone special. Neither so that of the snub nodes (O -> s). In fact, it is rather the introduction of your special link symbols A, B, C, E, and G, which deserve that honour. Those finally made the true transition from mere transliterations of already linear graphical advices into texttype, as those now do allow for bifurcation nodes being represented that way too.

Also it is your honour to introduce different edge lengths, eg. |x| = 1, |q| = sqrt(2), |f| = (1+sqrt(5))/2 etc. Formerly those only have been called unspecifiedly as 'varieties'.

In fact, even so you sometimes add ':' or 'z' for loop transliterations, it was only my enhencement by introducing virtual nodes '*a', '*b', '*c', etc. (refering to already before mentioned real nodes, reading from the left to the right, to be counted alphabetically) which made possible to decompose any arbitrarily nested Dynkin symbol into a complete linearization. (E.g. '*a' within x3o3o3/2*a re-refers the first node, making thus a loop or triangular Dynkin graph. The virtual node '*b' in x3o3o3o3/2*b accordingly would re-refer the one but leftmost real node (here the first 'o'), closing the final part into a loop, i.e. representing a Loop'n'Tail Dynkin diagram. But it becomes possible this way as well to provide e.g. a complete tetrahedral diagram (no links being omitted) in your attempted for linearized way!)

--- rk
Klitzing
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### Re: Johnsonian Polytopes

Coxeter does not use an ascii-form in line. It's like a little picture, like you would draw in a figure. It is never reduced to @ signs etc. It's only us who type this stuff in that write it like this.

A Schläfli symbol is best thought of as a kind of 'continued fraction'. It works something like that. The closest mathematical idiom to it is something like {3,3,5} = fn3(fn3(fn5(1))). There's nothing there at the punctuation. Also, it is all about what in the CD diagram is branches. You can calculate, eg the vertex-radius from the schl̈äli determinates. You'ld also be hard pressed trying to explain how x3o3o fits into {3,3}.

The sorts of Dynkin symbols that Coxeter used, that Conway uses, and that Tom Ruen keeps sending me, are description of groups. That is, it suffices to suggest a group 3[{4}] (or something), as a square made oute of '3' branches, and a different one 3[{3,3}] for a tetrahedron so arranged. You know, it works for laying out groups, because the first one corresponds, eg to AA=BB=CC=DD=1, AC=CA, BD=DB, ABA=BAB, BCB=CBC, CDC=DCD, DAD=ADA. That's all you need for a Lie group. It would not fly here, because there's nowhere to put your node rubbish on.

The notion i designed was that it was the _nodes_, not the branches, that were really important, and if you can make the group look like a regular group, then you can decorate it like x3x3o5o. So it's really this important mind-shift from branches and Lie groups, to nodes and position polytopes. Once you have a notion, you can but a half-dozen different notations on top of it. You can then stick by stick, translate one to the other. This was part of the design. There is a way difference between o3x3o3o3o3o3o3o3oBx and 1/6B/ (ie \$t_{5_1 1_0} 5_21\$ by TeX-Maths). Can you imagine that as a subscript?

Yes, i am familiar with your '*' notation, because i had to modify it to make it work. In essence, the idea to put the * in front of the node-jump allows you to have *q and not get it confused with q. You can run it onto an other node, and keep it separate, eg o3o3f*b3o works, there is no need for the space, since the * tells you it's a different node (and word), to what the f is on. I don't use it a lot myself, because i view the notation as a single word, but i don't stop other people doing it.

Likewise, Richard's lace towers with F=ff and G=f+v or whatever, is perfectly in line with the original notion. In essence, as long as you can tell that a node has started or a branch has started, then the notation works. I sometimes write eg 'A5', meaning a A-branch marked 5, but because there it's made of the same stuff other branches are, it works.

The pashion that descends from Jonathan's pictographs (eg o5oxo), can be a bit confusing, because you don't know if there is one or three nodes. oo4ox is doubly so (at o3o4o3x vs o4o || o4x).
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### Re: Johnsonian Polytopes

I just ?re?discovered a CRF. It is possible to regularly truncate the snub 24-cell, resulting in a polytope with 96 tridiminished icosahedra, 120 truncated tetrahedra and 24 truncated icosahedra. It seems to coincide with a maximal diminihing of the truncated 600-cell.
I think a rectification is also possible, resulting in a polytope with 96 tridiminished icosahedra, 120 octahedra and 24 icosidodecahedra. This one seems to concide with a maximal diminishing of a rectified 600-cell.
Are these already known? wouldn't surprise me if they are.
A quick investigation makes me suppose a similar maximal diminishing is also possible for the runcinated 120-cell, which can also be derived by some Kepler/Stott operation on the 24-cell, and for the cantitruncated 600-cell.
As I read Mathieu's article about special cuts of the 600-cell, (http://arxiv.org/abs/0708.3443I think maximal cuts are the most important to determine in order to be able to list all the possible diminishings.
(about that article, I have discovered another 22-diminishing of the 600-cell, bringing my number to 5, but I am still curious about nine to be possible)
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student91
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### Re: Johnsonian Polytopes

Hei "student91",
those seem rather interesting! Indeed very nice 4D CRFs. - I for one have never come across them, perhaps Wendy?

I've to admit, that I came from investigating the regulars, then the convex uniforms, and thereafter, driven by HedronDude, lots of non-convex uniforms. Subsequently, after having elaborated the general "alternated faceting" operation (then allowing for any mix of s, x, and o node symbols onto any Dynkin graph) as well as my segmentotopes, I came to consider scaliform figures as well. Note that those were formerly called "weakly uniform". This is because they too ask for unit edges only, and likewise for a transient action of the symmetry group on the set of vertices. They only do no longer ask for the hierarchical part of the definition, i.e. the subelements are no longer required to be uniforms themself. Instead those might be any CRF or even (non-C)RF. - That is, multiform figures, i.e. ones with differing vertex types, have never been considered systematically (except from rather ad hoc and quite specific restrictions - deltahedra come up to mind). - Sure, quickfur's search for CRFs readily reaches far into that realm! But still we have no (manageable) means for any systematical research on them. All known ones only emerged more or less from a specific, individual quest.

It was clear, that both, truncation and rectification are more general operations, not only restricted in application to regulars. But, on the other hand, in such more general applications, they usually provide figures with non-equal edge lengths. Well, alternated faceting (without relaxation) suffers the same problem. So we might consider applying an afterwards relaxation process there too. - But I never did such a research. Even so, your 2 figures not even need for such afterwards to be applied relaxations, they in fact already come out directly as unit-edged! Thus quite an interesting find!

Btw., I already managed to fiddle out their incidence matrices too!
Here they come:

Code: Select all
`trunc( sadi )288   *   * |   1   2   2   0   0   0 |  2  1   2   2   1  0  0 |  1  1  2  1  "head" of {5}  * 288   * |   0   0   2   1   1   0 |  0  0   2   1   2  1  0 |  2  0  1  1  "arms" of {5}  *   * 288 |   0   0   0   1   1   2 |  0  0   2   0   2  1  1 |  2  0  1  1  "legs" of {5}------------+-------------------------+-------------------------+------------  2   0   0 | 144   *   *   *   *   * |  2  0   2   0   0  0  0 |  1  1  2  0  2   0   0 |   * 288   *   *   *   * |  1  1   0   1   0  0  0 |  0  1  1  1  1   1   0 |   *   * 576   *   *   * |  0  0   1   1   1  0  0 |  1  0  1  1  0   1   1 |   *   *   * 288   *   * |  0  0   0   0   2  1  0 |  2  0  0  1  0   1   1 |   *   *   *   * 288   * |  0  0   2   0   0  1  0 |  2  0  1  0  0   0   2 |   *   *   *   *   * 288 |  0  0   1   0   1  0  1 |  1  0  1  1------------+-------------------------+-------------------------+------------  6   0   0 |   3   3   0   0   0   0 | 96  *   *   *   *  *  * |  0  1  1  0  3   0   0 |   0   3   0   0   0   0 |  * 96   *   *   *  *  * |  0  1  0  1  2   2   2 |   1   0   2   0   2   1 |  *  * 288   *   *  *  * |  1  0  1  0  2   1   0 |   0   1   2   0   0   0 |  *  *   * 288   *  *  * |  0  0  1  1  1   2   2 |   0   0   2   2   0   1 |  *  *   *   * 288  *  * |  1  0  0  1  0   3   3 |   0   0   0   3   3   0 |  *  *   *   *   * 96  * |  2  0  0  0  0   0   3 |   0   0   0   0   0   3 |  *  *   *   *   *  * 96 |  0  0  1  1------------+-------------------------+-------------------------+------------ 12  24  24 |   6   0  24  24  24  12 |  0  0  12   0  12  8  0 | 24  *  *  *  ti (pyrit sym) 12   0   0 |   6  12   0   0   0   0 |  4  4   0   0   0  0  0 |  * 24  *  *  tut (full sym)  6   3   3 |   3   3   6   0   3   3 |  1  0   3   3   0  0  1 |  *  * 96  *  tut (axial sym)  3   3   3 |   0   3   6   3   0   3 |  0  1   0   3   3  0  1 |  *  *  * 96  teddi`

Code: Select all
`rect( sadi )144   * |   4   4   0   0 |  2  2   2   4   2  0  0 |  1  1  2  2  "head" of {5}  * 288 |   0   2   2   2 |  0  0   2   1   4  1  1 |  2  0  1  2  "arms" + "legs" of {5}--------+-----------------+-------------------------+------------  2   0 | 288   *   *   * |  1  1   0   1   0  0  0 |  0  1  1  1  1   1 |   * 576   *   * |  0  0   1   1   1  0  0 |  1  0  1  1  0   2 |   *   * 288   * |  0  0   0   0   2  1  0 |  2  0  0  1  0   2 |   *   *   * 288 |  0  0   1   0   1  0  1 |  1  0  1  1--------+-----------------+-------------------------+------------  3   0 |   3   0   0   0 | 96  *   *   *   *  *  * |  0  1  1  0  3   0 |   3   0   0   0 |  * 96   *   *   *  *  * |  0  1  0  1  1   2 |   0   2   0   1 |  *  * 288   *   *  *  * |  1  0  1  0  2   1 |   1   2   0   0 |  *  *   * 288   *  *  * |  0  0  1  1  1   4 |   0   2   2   1 |  *  *   *   * 288  *  * |  1  0  0  1  0   3 |   0   0   3   0 |  *  *   *   *   * 96  * |  2  0  0  0  0   3 |   0   0   0   3 |  *  *   *   *   *  * 96 |  0  0  1  1--------+-----------------+-------------------------+------------  6  24 |   0  24  24  12 |  0  0  12   0  12  8  0 | 24  *  *  *  id (pyrit sym)  6   0 |  12   0   0   0 |  4  4   0   0   0  0  0 |  * 24  *  *  oct (tet sym)  3   3 |   3   6   0   3 |  1  0   3   3   0  0  1 |  *  * 96  *  oct (3-ap sym)  3   6 |   3   6   3   3 |  0  1   0   3   3  0  1 |  *  *  * 96  teddi`

These matrices then esp. readily point out the triformness resp. biformness of your found figures.

--- rk

PS: Could you detail a bit more on the other figures, you conjectured?
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### Re: Johnsonian Polytopes

Klitzing wrote:Hei "student91",
those seem rather interesting! Indeed very nice 4D CRFs. - I for one have never come across them, perhaps Wendy?

[...]
--- rk

PS: Could you detail a bit more on the other figures, you conjectured?

student91 wrote:A quick investigation makes me suppose a similar maximal diminishing is also possible for the runcinated 120-cell, which can also be derived by some Kepler/Stott operation on the 24-cell,

this one is based on the operation that makes a x5o3o3x from a o5o3o3x, i.e. pulling the facets apart. I think such a operation can be applied to the snub 24-cell as well. This would make the x3o5o become x3o3x. Note that this can also be seen as doing this to a 600-cell, and then removing 24 x5o3o's, making x3o5x-gaps.
And for the cantitruncated 120-cell

(I ment the cantellated 120-cell x5o3x3o )
I'm not sure about this one. you should look what is done to get x5o3x3o from o5o3o3x, and then apply this operation to a snub 24-cell, from a 600-cell point of view. As I said, I'm not sure, But it may be possible that some J83's occur.

Just realized, by looking at x5o3o3x, that this one can be diminished in a similar way the grand antiprism is made by diminishig a 600-cell. again, this can also be seen as grand antiprism -> bigyrodecadiminished x5o3o3x. It might even be possible that a diminishing similar to 600-cell => bi-icositetradiminished 600-cell is possible for the x5o3o3x.
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student91
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### Re: Johnsonian Polytopes

Klitzing wrote:Hei "student91",
those seem rather interesting! Indeed very nice 4D CRFs. - I for one have never come across them, perhaps Wendy?

I have not come across them either. Good job! The only CRFs I know related to the snub 24-cell is the "runcinated" snub 24-cell, a scaliform that I think either Olshevsky or Johnson himself discovered (or maybe it was Bowers?), made by (pseudo-)Stott expansion of the icosahedral cells and inserting pentagonal prisms between them, and modifying the other cells appropriately.

I've to admit, that I came from investigating the regulars, then the convex uniforms, and thereafter, driven by HedronDude, lots of non-convex uniforms. Subsequently, after having elaborated the general "alternated faceting" operation (then allowing for any mix of s, x, and o node symbols onto any Dynkin graph) as well as my segmentotopes, I came to consider scaliform figures as well. Note that those were formerly called "weakly uniform". This is because they too ask for unit edges only, and likewise for a transient action of the symmetry group on the set of vertices. They only do no longer ask for the hierarchical part of the definition, i.e. the subelements are no longer required to be uniforms themself. Instead those might be any CRF or even (non-C)RF. - That is, multiform figures, i.e. ones with differing vertex types, have never been considered systematically (except from rather ad hoc and quite specific restrictions - deltahedra come up to mind). - Sure, quickfur's search for CRFs readily reaches far into that realm! But still we have no (manageable) means for any systematical research on them. All known ones only emerged more or less from a specific, individual quest.

Well, I think most people start their quest for polytopes by investigating the regulars, since those are the most tractible and manageable esp. in very high dimensions (since the number of surtopes grow exponentially with dimension, and having all elements equivalent greatly reduce the amount of information you need to process!). The uniforms are a natural further investigation, being directly derived from the regulars by various means, mostly Stott expansion. Personally, I was never interested in the Johnson solids or CRFs, except perhaps for various simple pyramids and the like -- until Keiji (re)discovered the bicupolic rings / bicupola wedges, and I had a hand in verifying them. That sparked a cursory interest in me to look for other similar figures, which quickly blossomed into a full-fledged search for any and all 4D CRFs (in the process reigniting interest in the Johnsons that serve as the basis for CRF construction).

As for systematic searches: I agree there is as yet no general research scheme for finding CRFs, because we don't yet know what classes exist in 4D! But, following Johnson's pattern in 3D, I have categorized them roughly into:
1. The monostratic CRFs: includes Klitzing's segmentochora, but also non-orbiform monostratics like prisms of the Johnson solids
2. Cups: bi-/tri-/etc.-stratic CRFs, that roughly have the shape of a "cup" (shallow truncation of a 3-sphere): this includes stacking the monostratics (by analogy with Johnson's elongated cupolae/pyramids/bicupolae/bipyramids). Quite a number of these overlap with sections of the uniform polychora (below), though the extent is yet to be determined as both categories are not fully enumerated yet.
3. Combinations of the preceding two categories (cutting-n-pasting CRFs from those categories).
4. Diminished uniform polychora: since there are too many of these to individually enumerate (esp. in the 600-cell family!), I have embarked on an ongoing quest to find all maximal diminishings -- i.e., diminishings that cannot be diminished further without becoming non-CRF (or changing edge length -- so alternation is not considered here even though that may be a possible future direction)). These could be considered somewhat as "fundamental" CRFs that we can glue together to build the bigger ones. This way, we avoid needing to wade through the combinatorial explosion of, e.g., the 600-cell diminishings / cuttings.
5. Augmented uniform polychora: again, there are too many individual augmentations due to the shallowness of augmenting shapes like the pentagonal prism pyramid (which causes a combinatorial explosion of 5,n-duoprism augmentations up to the 5,20-duoprism), so a future research direction for me is to find all maximally augmented uniforms: those that cannot be augmented further without becoming non-convex. There is much room for further research here, since only the duoprisms so far have been systematically considered, and even then just for simple pyramid augments -- stacked augments have not been fully considered, neither have augmentations with things other than pyramids and cupolae.
6. Crown jewels: a catch-all category for any unusual CRFs that cannot be immediately derived from the uniforms or the monostratics. Unfortunately, no known method of research is available here: besides cube||icosahedron, which is already included in the monostratics category, we don't know of any crown jewels, so there's really not enough information to know how to even approach the problem (besides brute-force search, which is infeasible due to the huge number of CRFs produced by the 600-cell family -- even a computer search may take a long time to find anything new).

Currently, I'm primarily engaged in systematically searching for maximally-diminished uniforms -- I've covered the 5-cell family (I believe it's complete, though I have no proof of this) and the tesseract family (probably complete), and part of the 24-cell family and a bit of the 120-cell/600-cell family. I'm not sure when I'll finish this category since there are so many diminishings of the 120-cell family! But at some point I'd like to also start systematically considering augmented uniforms, with emphasis on finding maximal augmentations.

In any case, it's clear that there are a lot of 4D CRFs, which is why I'm focusing on finding maximal diminishings / augmentations -- these would serve as the extrema of the set of CRFs, in some sense, from which one could construct the others, so that we can at least identify the major landmarks that demarcate the extent of the set of 4D CRFs, even if what lies between are too numerous to individually enumerate.

[...]
It was clear, that both, truncation and rectification are more general operations, not only restricted in application to regulars. But, on the other hand, in such more general applications, they usually provide figures with non-equal edge lengths. Well, alternated faceting (without relaxation) suffers the same problem. So we might consider applying an afterwards relaxation process there too. - But I never did such a research. Even so, your 2 figures not even need for such afterwards to be applied relaxations, they in fact already come out directly as unit-edged! Thus quite an interesting find!

Indeed, the 120-cell family and its relatives like the snub 24-cell seem to have a penchant for many operations somehow just working out to have unit edge lengths. Some of these things could compete for the crown jewels category, they are so unexpected (like the truncated dodecahedron / icosahedron mega-wedge I discovered recently)!
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:
Klitzing wrote:Hei "student91",
those seem rather interesting! Indeed very nice 4D CRFs. - I for one have never come across them, perhaps Wendy?

I have not come across them either. Good job! The only CRFs I know related to the snub 24-cell is the "runcinated" snub 24-cell, a scaliform that I think either Olshevsky or Johnson himself discovered (or maybe it was Bowers?), made by (pseudo-)Stott expansion of the icosahedral cells and inserting pentagonal prisms between them, and modifying the other cells appropriately.

I think this one is what I (re)discovered in my previous post:
student91 wrote:
student91 wrote:
Klitzing wrote:Hei "student91",
those seem rather interesting! Indeed very nice 4D CRFs. - I for one have never come across them, perhaps Wendy?

[...]
--- rk

PS: Could you detail a bit more on the other figures, you conjectured?

student91 wrote:A quick investigation makes me suppose a similar maximal diminishing is also possible for the runcinated 120-cell, which can also be derived by some Kepler/Stott operation on the 24-cell,

this one is based on the operation that makes a x5o3o3x from a o5o3o3x, i.e. pulling the facets apart. I think such a operation can be applied to the snub 24-cell as well. This would make the x3o5o become x3o3x. Note that this can also be seen as doing this to a 600-cell, and then removing 24 x5o3o's, making x3o5x-gaps.

quickfur wrote:[...]
1. The monostratic CRFs: includes Klitzing's segmentochora, but also non-orbiform monostratics like prisms of the Johnson solids
2. Cups: bi-/tri-/etc.-stratic CRFs, that roughly have the shape of a "cup" (shallow truncation of a 3-sphere): this includes stacking the monostratics (by analogy with Johnson's elongated cupolae/pyramids/bicupolae/bipyramids). Quite a number of these overlap with sections of the uniform polychora (below), though the extent is yet to be determined as both categories are not fully enumerated yet.
3. Combinations of the preceding two categories (cutting-n-pasting CRFs from those categories).
4. Diminished uniform polychora: since there are too many of these to individually enumerate (esp. in the 600-cell family!), I have embarked on an ongoing quest to find all maximal diminishings -- i.e., diminishings that cannot be diminished further without becoming non-CRF (or changing edge length -- so alternation is not considered here even though that may be a possible future direction)). These could be considered somewhat as "fundamental" CRFs that we can glue together to build the bigger ones. This way, we avoid needing to wade through the combinatorial explosion of, e.g., the 600-cell diminishings / cuttings.
5. Augmented uniform polychora: again, there are too many individual augmentations due to the shallowness of augmenting shapes like the pentagonal prism pyramid (which causes a combinatorial explosion of 5,n-duoprism augmentations up to the 5,20-duoprism), so a future research direction for me is to find all maximally augmented uniforms: those that cannot be augmented further without becoming non-convex. There is much room for further research here, since only the duoprisms so far have been systematically considered, and even then just for simple pyramid augments -- stacked augments have not been fully considered, neither have augmentations with things other than pyramids and cupolae.
6. Crown jewels: a catch-all category for any unusual CRFs that cannot be immediately derived from the uniforms or the monostratics. Unfortunately, no known method of research is available here: besides cube||icosahedron, which is already included in the monostratics category, we don't know of any crown jewels, so there's really not enough information to know how to even approach the problem (besides brute-force search, which is infeasible due to the huge number of CRFs produced by the 600-cell family -- even a computer search may take a long time to find anything new).

Currently, I'm primarily engaged in systematically searching for maximally-diminished uniforms -- I've covered the 5-cell family (I believe it's complete, though I have no proof of this) and the tesseract family (probably complete), and part of the 24-cell family and a bit of the 120-cell/600-cell family. I'm not sure when I'll finish this category since there are so many diminishings of the 120-cell family! But at some point I'd like to also start systematically considering augmented uniforms, with emphasis on finding maximal augmentations.

In any case, it's clear that there are a lot of 4D CRFs, which is why I'm focusing on finding maximal diminishings / augmentations -- these would serve as the extrema of the set of CRFs, in some sense, from which one could construct the others, so that we can at least identify the major landmarks that demarcate the extent of the set of 4D CRFs, even if what lies between are too numerous to individually enumerate.

I am using a similar categorisasion of the Johnson-solids in my school project. I myself am mainly interested in finding 4D-"crown jewels", although they seem almost impossible to seek, as you already pointed out.
About the maximal diminishings, did you get my point on the last line of my previous post?:
student91 wrote:[...]
Just realized, by looking at x5o3o3x, that this one can be diminished in a similar way the grand antiprism is made by diminishig a 600-cell. again, this can also be seen as grand antiprism -> bigyrodecadiminished x5o3o3x. It might even be possible that a diminishing similar to 600-cell => bi-icositetradiminished 600-cell is possible for the x5o3o3x.

(I have to say again, your renders are very helpful) I might have been unclear, so I'll try to explain it again.
when comparing x5o3o3x with o5o3o3x, you see the vertices of o5o3o3x are related to the dodecahedra in x5o3o3x. If we delete a dodecahedron in x5o3o3x, you get a x5o3x-gap. Now compare this diminishing with the (vertex)-diminishings of the 600-cell. It's pretty similar

If we diminish two dodecahedra that are next to each other, they make two gaps that look like diminished x5o3x's. This is similar to diminishing two vertices next to each other in the 600-cell,resulting in two gyroelongated pentagonal pyramid gaps.
The same way, you can delete 20 dodecahedra corresponding to the deleted vertices of the grand antiprism, making a nice grand antiprism-like thing.
If you want to go another step further, you can delete all dodecahedra corresponding to the deleted vertices of the bi-icositetradiminished 600-cell, making a polytope with a lot of (48?) J83's.
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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student91
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### Re: Johnsonian Polytopes

student91 wrote:[...]
I am using a similar categorisasion of the Johnson-solids in my school project. I myself am mainly interested in finding 4D-"crown jewels", although they seem almost impossible to seek, as you already pointed out.

I've been attacking the problem from time to time by various means, but so far, I haven't found anything, and there seems to be no specific direction of research that might yield some results besides exhaustive search. And even that is hampered by the difficulty of determining whether n arbitrary 3D CRFs might form a closed shape in 4D. There are many topological possibilities but the unit edge length requirement is a hard problem to solve, because of the non-rigidness of edges of degree ≥4, which means you are potentially dealing with solving a set of quadratic equations, which is known to be intractible in the general case. There may be some way of attacking the specific kind of quadratic equations that arise from geometric assembly of CRF pieces, but so far I haven't found any (and haven't really looked yet -- there are other problems associated with this, such as potential inability to analytically solve these equation systems, and numerical instability in the respective computer algorithms, meaning that if you get some result out of it you still have to verify by hand whether it's actually CRF, and not merely "almost" CRF but the computer thought it was because of roundoff error, for example, some edge lengths may actually be 1.00000000001 instead of exactly 1, but roundoff error may obscure that discrepancy).

About the maximal diminishings, did you get my point on the last line of my previous post?:
student91 wrote:[...]
Just realized, by looking at x5o3o3x, that this one can be diminished in a similar way the grand antiprism is made by diminishig a 600-cell. again, this can also be seen as grand antiprism -> bigyrodecadiminished x5o3o3x. It might even be possible that a diminishing similar to 600-cell => bi-icositetradiminished 600-cell is possible for the x5o3o3x.

(I have to say again, your renders are very helpful) I might have been unclear, so I'll try to explain it again.
when comparing x5o3o3x with o5o3o3x, you see the vertices of o5o3o3x are related to the dodecahedra in x5o3o3x. If we delete a dodecahedron in x5o3o3x, you get a x5o3x-gap. Now compare this diminishing with the (vertex)-diminishings of the 600-cell. It's pretty similar

If we diminish two dodecahedra that are next to each other, they make two gaps that look like diminished x5o3x's. This is similar to diminishing two vertices next to each other in the 600-cell,resulting in two gyroelongated pentagonal pyramid gaps.
The same way, you can delete 20 dodecahedra corresponding to the deleted vertices of the grand antiprism, making a nice grand antiprism-like thing.
If you want to go another step further, you can delete all dodecahedra corresponding to the deleted vertices of the bi-icositetradiminished 600-cell, making a polytope with a lot of (48?) J83's.

In general, a good number of diminishable 600-cell family uniforms can be diminished in the same way as the grand antiprism. And I think the icositetradiminished x5o3o3x may have already been found by Klitzing, as part of a general search for this pattern of diminishing in the 600-cell family. I'll let him confirm that.

One interesting diminishing that I intend to construct at some point is the "swirl-diminished" o5o3x3o: it's possible to delete the top/bottom vertices of each icosahedron in a great circle of icosahedra, to form a ring of alternating pentagonal prisms and antiprisms. Since the vertices of the 600-cell can be partitioned into 12 great circles of 10 vertices each, we can perform this diminishing on 12 great circles on icosahedra in the o5o3x3o. This produces a CRF with swirlprism symmetry, consisting of 12 rings of alternating pentagonal prisms/antiprisms that swirl around each other (according to a subset of the Hopf fibration of the 3-sphere), along with a bunch of square pyramids filling in the gaps between them. This should make a good Polytope of the Month, if I find the time to resume that program.
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### Re: Johnsonian Polytopes

While edges of degree > 3 might be nonrigid, the vertices should provide bigger constraints and limit that.

One 3D crown jewel that we might be able to lift to 4D is bilunabirotunda. It's based on the fact that both icosidodecahedron and rhombicosidodecahedron have decagonal cuts that are deep enough to cross.

If we just straight lift it, we can look at o5x3o3o and x5o3x3o. Both of these have truncated dodecahedral cuts: o5x3o || f5o3o || x5x3o and x5o3x || x5x3o. Could these two cupolas be cut by additional truncated dodecahedral cuts and the pieces then fit together to get a CRF polychoron?
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### Re: Johnsonian Polytopes

quickfur wrote:[...]
In general, a good number of diminishable 600-cell family uniforms can be diminished in the same way as the grand antiprism. And I think the icositetradiminished x5o3o3x may have already been found by Klitzing, as part of a general search for this pattern of diminishing in the 600-cell family. I'll let him confirm that.

What you just said here gave me an interesting idea, although I find it hard to phrase. I'll try anyway:
There is a great similarity between every cuts in the 600-cell family. In fact, the diminishings I've seen so far can all be placed in a number of groups.(e.g. the diminishings of x5o3o3x correspond to the diminishings of the 600-cell, the mesodiminishing of the x3x3o5o corresponds to the "special cuts" of the 600-cell, and so on). Now if we determine the diminishing for one of these, we can say it's applyable to all the others. (e.g. we could state there are the same amound of "special cuts" of the 600-cell as there are "normal" diminishings of the x3x3o5o.)
But you might ask, why do these correspond. I'm still not sure about this, but I think it has something to do with the fundamental domains layout. The fundamental domains in which the "base" of the cut lies, must align a bit like the fundamentall domains of the icosahedral symmetry (seen from a .o3o5o-viewpoint). there are probably more places where this is true. this means that if we determine the maximal cuts for some dept, we can say this is applyable to all polytopes that have such a possiblke diminishing. (e.g. the bi-icositetradiminishing, it can be seen as a maximal diminishing of .o3o5o with dept 1 with single overlap allowed. The 600-cell and x5o3o3x both have such a possible diminishing, meaning these can be bi-icositetradiminished. (they can also be diminished any other way with a dept 1 .o3o5o single overlap cut). if we determine all the general cuts, we only have to say to which polytope it is applied. (in the same way both o5o3x and x3o5x have the same kind of possible diminishings, and thus a comparable naming scheme, while o3x5o has a dept 2 diminishing (making a rotunda))
again, I hope i have been understandable.
One interesting diminishing that I intend to construct at some point is the "swirl-diminished" o5o3x3o: it's possible to delete the top/bottom vertices of each icosahedron in a great circle of icosahedra, to form a ring of alternating pentagonal prisms and antiprisms. Since the vertices of the 600-cell can be partitioned into 12 great circles of 10 vertices each, we can perform this diminishing on 12 great circles on icosahedra in the o5o3x3o. This produces a CRF with swirlprism symmetry, consisting of 12 rings of alternating pentagonal prisms/antiprisms that swirl around each other (according to a subset of the Hopf fibration of the 3-sphere), along with a bunch of square pyramids filling in the gaps between them. This should make a good Polytope of the Month, if I find the time to resume that program.

this one indeed is interesting, do more polytopes have a dept 1 o.o5o-oriented possible diminishing? they too could be "swirl-diminished"
student91
Last edited by student91 on Fri Jan 31, 2014 10:34 pm, edited 2 times in total.
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### Re: Johnsonian Polytopes

Marek14 wrote:While edges of degree > 3 might be nonrigid, the vertices should provide bigger constraints and limit that.

One 3D crown jewel that we might be able to lift to 4D is bilunabirotunda. It's based on the fact that both icosidodecahedron and rhombicosidodecahedron have decagonal cuts that are deep enough to cross.

If we just straight lift it, we can look at o5x3o3o and x5o3x3o. Both of these have truncated dodecahedral cuts: o5x3o || f5o3o || x5x3o and x5o3x || x5x3o. Could these two cupolas be cut by additional truncated dodecahedral cuts and the pieces then fit together to get a CRF polychoron?

I didn't understand the part where you said the bilunabirotunda is based on the fact both the o3x5o and the x5o3x have decagonal cuts that are deep enough to cross. I see these diminishings are possible, but I don't see how the bilunabirotunda is based on this . could you explain this to me?
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student91
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### Re: Johnsonian Polytopes

student91 wrote:
Marek14 wrote:While edges of degree > 3 might be nonrigid, the vertices should provide bigger constraints and limit that.

One 3D crown jewel that we might be able to lift to 4D is bilunabirotunda. It's based on the fact that both icosidodecahedron and rhombicosidodecahedron have decagonal cuts that are deep enough to cross.

If we just straight lift it, we can look at o5x3o3o and x5o3x3o. Both of these have truncated dodecahedral cuts: o5x3o || f5o3o || x5x3o and x5o3x || x5x3o. Could these two cupolas be cut by additional truncated dodecahedral cuts and the pieces then fit together to get a CRF polychoron?

I didn't understand the part where you said the bilunabirotunda is based on the fact both the o3x5o and the x5o3x have decagonal cuts that are deep enough to cross. I see these diminishings are possible, but I don't see how the bilunabirotunda is based on this . could you explain this to me?

Well, you can create the "luna" by cutting two decagons into x5o3x to get a 2-triangles + 1-square surface. It's not a Johnson solid by itself because it's bordered by partial decagons, but these partial decagons are the same which appear through similar construction on o5x3o, so they can be fused and, eventually, closed into bilunabirotunda.
Marek14
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### Re: Johnsonian Polytopes

Marek14 wrote:[...]
Well, you can create the "luna" by cutting two decagons into x5o3x to get a 2-triangles + 1-square surface. It's not a Johnson solid by itself because it's bordered by partial decagons, but these partial decagons are the same which appear through similar construction on o5x3o, so they can be fused and, eventually, closed into bilunabirotunda.

Ah, that's what you mean. That's very interesting indeed, It might give some more CRF's as well. I'll probably check some possiblilties in the (near) future.
The way I looked at the bilunabirotunda is as two o3x5o-lunae placed atop each other at the edge at the end of the pentagon. now because the luna are epic, a distance of 1 occurs, and closes the shape. This way extarpolates difficultly to 4D, so I think your way will be the favored one.
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student91
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### Re: Johnsonian Polytopes

student91 wrote:
quickfur wrote:[...]
In general, a good number of diminishable 600-cell family uniforms can be diminished in the same way as the grand antiprism. And I think the icositetradiminished x5o3o3x may have already been found by Klitzing, as part of a general search for this pattern of diminishing in the 600-cell family. I'll let him confirm that.

What you just said here gave me an interesting idea, although I find it hard to phrase. I'll try anyway:
There is a great similarity between every cuts in the 600-cell family. In fact, the diminishings I've seen so far can all be placed in a number of groups.(e.g. the diminishings of x5o3o3x correspond to the diminishings of the 600-cell, the mesodiminishing of the x3x3o5o corresponds to the "special cuts" of the 600-cell, and so on). Now if we determine the diminishing for one of these, we can say it's applyable to all the others. (e.g. we could state there are the same amound of "special cuts" of the 600-cell as there are "normal" diminishings of the x3x3o5o.)

Well, since all the uniform polychora from the 600-cell family share the same symmetry group, they all have corresponding elements (e.g., 600-cell vertices = 120-cell cells = o5oxo icosahedra = ... etc.). Strictly speaking, any subset of these elements may be deleted in a diminishing. Of course, our interest is in CRF results, so the trick here is identifying for each uniform which symmetry elements admit CRF diminishings, and identifying which two uniforms have corresponding CRF-diminishable elements.

Which uniforms are CRF-diminishable, however, it pretty much a matter of coincidence: the 3D uniforms exhibit a more limited number of diminishings, so in 4D we are constrained by which 3D uniforms have CRF diminishings -- for example, if the dodecahedron had been CRF-diminishable, then the 120-cell and x5oox would also be CRF-diminishable in all positions where x5ox and o5xo cells could be diminished. So really, it's the intersection of the CRF requirement with the general geometric operation (which is universally applicable) that produces the "random" occurrence of CRF-diminishings. The geometric operation itself is completely general and can be applied across the board, and will produce consistent results (truncated element X in one uniform will have the corresponding result to the truncation of element Y in another uniform where X and Y correspond with the same symmetry element). It's just that many of the results will not be CRF.

But you might ask, why do these correspond. I'm still not sure about this, but I think it has something to do with the fundamental domains layout.

Actually, the reason is that all elements of the underlying 600-cell symmetry group have specific hyperplane orientations associated with them. This is easiest to see if you consider the runcinated 120-cell (x5oox), where all symmetry elements are expressed in the cells. The hyperplanes in which the pentagonal prisms sit, for example, are parallel to the hyperplanes of the octagonal prisms in x5xxo and x5xxx. So in a sense, all of these uniforms are "the same", since each symmetry element has a fixed orientation, and therefore dichoral angles among these uniforms are the same few combinations.

Furthermore, because of this same relationship among the 3D uniforms -- the rhombicosidodecahedron x5ox, for example, is just the expanded icosahedron o5ox, so cutting off a pentagonal cupola from x5ox is equivalent to cutting off a pentagonal pyramid from an icosahedron, because pentagonal cupola = Stott-expanded pentagonal pyramid. Similarly, given some CRF diminishing of the 600-cell, if some other 600-cell family uniform has similarly-diminishable elements in the same positions, then it can also be CRF-diminished in the same way. In fact, one could argue that the CRF-diminishing of the latter is just the Stott-expanded version of the 600-cell diminishing.

[...]
One interesting diminishing that I intend to construct at some point is the "swirl-diminished" o5o3x3o: it's possible to delete the top/bottom vertices of each icosahedron in a great circle of icosahedra, to form a ring of alternating pentagonal prisms and antiprisms. Since the vertices of the 600-cell can be partitioned into 12 great circles of 10 vertices each, we can perform this diminishing on 12 great circles on icosahedra in the o5o3x3o. This produces a CRF with swirlprism symmetry, consisting of 12 rings of alternating pentagonal prisms/antiprisms that swirl around each other (according to a subset of the Hopf fibration of the 3-sphere), along with a bunch of square pyramids filling in the gaps between them. This should make a good Polytope of the Month, if I find the time to resume that program.

this one indeed is interesting, do more polytopes have a dept 1 o.o5o-oriented possible diminishing? they too could be "swirl-diminished"
student91

Certainly! Any 600-cell family uniform that has CRF-diminishable elements corresponding with 600-cell edges (resp. 120-cell pentagons) will admit a CRF swirldiminishing. The 600-cell itself admits it, but is a degenerate case (you end up deleting all vertices ); o5oxo admits it as I indicated, so does x5oxo.

In any case, I have just finished constructing the swirldiminished o5oxo: as expected, it consists of 12 rings of alternating pentagonal prisms and antiprisms. Each ring has 10 of each prism, so there are a total of 120 pentagonal prisms and 120 pentagonal antiprisms. The original octahedra get bisected into square pyramids, so that's 600 square pyramids + 120 pentagonal prisms + 120 pentagonal antiprisms = 840 cells.

What's interesting, however, is that not only there are 12 rings of alternating prisms/antiprisms, the original octahedra of the o5oxo share half their faces with each other, and after the diminishing, the square pyramids also share their faces with each other. Furthermore, these square pyramids are isolated by the alternating prisms/antiprisms into individual rings of pyramids that also run around great circles of the 3-sphere. I didn't count them, but I believe there should be 20 such rings, corresponding with the dual swirlprism symmetry to the 12 rings of prisms! So here, in a single polychoron, you have the combination of both the dodecahedral Hopf fibration and its dual icosahedral Hopf fibration, represented by two sets of rings of cells. (One could say this corresponds with an icosidodecahedral Hopf fibration, or an icosidodecahedral uniform polytwister, in which the two types of twisters correspond with the the alternating prism rings and the square pyramid rings).

This polychoron is, of course, chiral because the underlying swirlprism symmetry is chiral. Moreover, I'm almost certain that it's vertex-transitive, which is ultracool, because that would make it scaliform.

Anyway, here's a preliminary render of this cute little baby:

I haven't tweaked the colors yet, so it looks a bit ugly, but you can clearly see the swirling rings of alternating prisms/antiprisms here. Or rather, half-rings, since I have visibility clipping on. The red vertical column in the center is actually a half-ring seen from a 90° in the 4D viewpoint, so it appears like a vertical column, but it's actually curved in a half-circle. Its other half, of course, lies on the far side and isn't shown here. It's kinda hard to see in this poorly-colored image, but there are two layers of rings around this central column. There are 5 rings in the inner layer and 5 rings in the outer layer. You can sorta see the slanted green column swirling around the central column, behind the yellow and magenta rings, and the blue ring to the right of it. These are 2 of the 5 inner rings. The yellow and magenta columns, of course, are 2 of the 6 outer rings. There is a ring that runs orthogonal to the red column, but it is not shown here because it lies just behind the limb of the polytope, so it got vis-clipped. So we see the structure of the rings is 1+5+5+1; the outer ring of the central column is the inner ring of the orthogonal ring, and vice versa. All of the rings are equivalent to each other under the underlying swirlprism symmetry group.

Now if you look carefully at the interface between the magenta and yellow rings, you can sorta discern the partial outlines (I omitted some edges to make the image less cluttered) of a wavy trail of triangles and squares: this is where one of the rings of square pyramids are. I didn't render the square pyramids in full because it would make the image too obscured. As you can see, the pyramids interface the pentagonal prisms of one ring to the pentagonal antiprisms of the other ring, in a skewed, alternating fashion. Really fascinating.

So anyway, here you have it, the swirl-diminished rectified 600-cell.

EDIT: Here's the same projection with a slightly better coloring scheme:

Red seems to work poorly when embedded deep inside the projection, so I changed it to white. Other than that, other colors remain the same but with transparencies tweaked.
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### Re: Johnsonian Polytopes

quickfur wrote:The only CRFs I know related to the snub 24-cell is the "runcinated" snub 24-cell, a scaliform that I think either Olshevsky or Johnson himself discovered (or maybe it was Bowers?), made by (pseudo-)Stott expansion of the icosahedral cells and inserting pentagonal prisms between them, and modifying the other cells appropriately.

Ah, you are refering to prissi.
A bit of its history is also provided on my webpage, cf. e.g. here.
(Esp. any of your guesses was wrong!)

In fact the mix of node symbols, as being used in prissi = s3s4o3x, never before has been considered, nor was understood what to mean.
It had to stay undiscovered until the formal setup of alternated facetings.

--- rk
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### Re: Johnsonian Polytopes

Yet another render of the swirl-diminished o5oxo:

Here, I show 5 of the rings of square pyramids. These 5 are the ones that wrap around the central ring of alternating prisms from the previous images. As you can see, these rings are not fully separated from each other; they do not share any faces but they do touch each other at their edges. They also have the interesting property that they not only swirl around each other and around the prism/antiprism rings, but they also have a 3-fold twist to them.

This particular image is also helpful in determining the vertex configuration: each vertex touches 2 pentagonal prisms and 2 pentagonal antiprisms in a chiral formation, flanked by 4 square pyramids. AFAICT all the vertex configurations are the same, so this polychoron is scaliform.
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### Re: Johnsonian Polytopes

student91 wrote:did you get my point on the last line of my previous post?:
student91 wrote:[...]
Just realized, by looking at x5o3o3x, that this one can be diminished in a similar way the grand antiprism is made by diminishig a 600-cell. again, this can also be seen as grand antiprism -> bigyrodecadiminished x5o3o3x. It might even be possible that a diminishing similar to 600-cell => bi-icositetradiminished 600-cell is possible for the x5o3o3x.

(I have to say again, your renders are very helpful) I might have been unclear, so I'll try to explain it again.
when comparing x5o3o3x with o5o3o3x, you see the vertices of o5o3o3x are related to the dodecahedra in x5o3o3x. If we delete a dodecahedron in x5o3o3x, you get a x5o3x-gap. Now compare this diminishing with the (vertex)-diminishings of the 600-cell. It's pretty similar

If we diminish two dodecahedra that are next to each other, they make two gaps that look like diminished x5o3x's. This is similar to diminishing two vertices next to each other in the 600-cell,resulting in two gyroelongated pentagonal pyramid gaps.
The same way, you can delete 20 dodecahedra corresponding to the deleted vertices of the grand antiprism, making a nice grand antiprism-like thing.
If you want to go another step further, you can delete all dodecahedra corresponding to the deleted vertices of the bi-icositetradiminished 600-cell, making a polytope with a lot of (48?) J83's.

Hmm, then you might be looking for those thingies?
• idsrix = icositetra-diminished ( srix = x3o3x5o )
• idsid pixhi = -- " -- ( sidpixhi = x3o3o5x )
• idprix = -- " -- ( prix = x3o3x5x )
• bidsid pixhi = bi-icositetra-diminished ( sidpixhi )

You might want to dig somewhere in this thread. Then the corresponding original research / find of those should show up.

--- rk
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### Re: Johnsonian Polytopes

student91 wrote:What you just said here gave me an interesting idea, although I find it hard to phrase. I'll try anyway:
There is a great similarity between every cuts in the 600-cell family. In fact, the diminishings I've seen so far can all be placed in a number of groups.(e.g. the diminishings of x5o3o3x correspond to the diminishings of the 600-cell, the mesodiminishing of the x3x3o5o corresponds to the "special cuts" of the 600-cell, and so on). Now if we determine the diminishing for one of these, we can say it's applyable to all the others. (e.g. we could state there are the same amound of "special cuts" of the 600-cell as there are "normal" diminishings of the x3x3o5o.)
But you might ask, why do these correspond. I'm still not sure about this, but I think it has something to do with the fundamental domains layout. The fundamental domains in which the "base" of the cut lies, must align a bit like the fundamentall domains of the icosahedral symmetry (seen from a .o3o5o-viewpoint). there are probably more places where this is true. this means that if we determine the maximal cuts for some dept, we can say this is applyable to all polytopes that have such a possiblke diminishing. (e.g. the bi-icositetradiminishing, it can be seen as a maximal diminishing of .o3o5o with dept 1 with single overlap allowed. The 600-cell and x5o3o3x both have such a possible diminishing, meaning these can be bi-icositetradiminished. (they can also be diminished any other way with a dept 1 .o3o5o single overlap cut). if we determine all the general cuts, we only have to say to which polytope it is applied. (in the same way both o5o3x and x3o5x have the same kind of possible diminishings, and thus a comparable naming scheme, while o3x5o has a dept 2 diminishing (making a rotunda))
again, I hope i have been understandable.

This, my dear, is a quite easy question to answer. You probably know there are several different symmetry groups, generated by reflection within hyperplanes (mirrors). These symmetry groups then just depend on the relative angles between those mirrors. in fact what elsewhere is being called the kaleidoscope. And it is the intersection of that kaleidoscope by the surface of the hypersphere, which determines the fundamental region of that symmetry. Within 3D e.g. we have the tetrahedral group, the octahedral/cubical one, the icosahedral/dodecahedral, and all the prismatic ones with any number of sides. Any such group is correlated to some (unmarked) Dynkin graph.

Now all polytopes, which are derived from a single such (so far undecorated) Dynkin graph (by applying node decorations arbitrarely to those graphs, that is, which follow therefrom by Wythoff's kaleidoscopical construction) then will have their axes of sub-symmetries always within the same set of mutual angles. - That is, when e.g. (a scaled copy of) the 24-cell is vertex inscribable into the 600-cell, then this feature holds true for all members of the 600-/120-cell family. Only that the latter vertices might become replaced by some cells with icosahedral/doecahedral symmetry.

Code: Select all
`x3o3o5o = 600-cell = ex. o3o5o = its icosahedral sub-symmetry axis - here no markings, thus vertices of exx . o5o = its pentagonal-prismatic subsymmetry axis - here no markings in right part, thus edges of exx3o . o = its trigonal-prismatic subsymmetry axis - here no markings in the right part, thus triangles of exx3o3o . = its tetrahedral subsymmetry axis - one mark at the left, thus the tetrahedra (tet) of exo3x3o5o = rectified 600-cell = rox. x3o5o = its icosahedral subsymmetry - here icosahedrao . o5o = no markings at all, thus the vertices of roxo3x . o = some of the triangles of rox (but . x3o . will be further ones)o3x3o . = the octahedra of roxo3o3x5x = truncated 120-cell = thi. o3x5x = its icosahedral subsymmetry - here truncated dodecahedra (tid)o . x5x = no marks at the left, thus the decagonso3o . x = no marks at the left, thus some of the edges of thi (others would be . . x .)o3o3x . = the tetrahedra of thi`

Then, according to the directions of those axes, you surely can collect all vertices of any such polytope into a finite set of discrete hyperplanes. Some of those section planes then allow for true polytopal cuts (i.e. edges, faces etc. also arrange within these hyperplanes. Often this is not only one, there might be more than one such. Provided those <i>true</i> polytopal cuts would exist, those then surely have to be mutually parallel for any polytope of the same symmetry group.

Finally it thus is just a question of the deepness of those section planes, whether such sections orthogonal to different (mirror) instances of some axis would mutually intersect, or not.

--- rk
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### Re: Johnsonian Polytopes

quickfur wrote:In any case, I have just finished constructing the swirldiminished o5oxo: as expected, it consists of 12 rings of alternating pentagonal prisms and antiprisms. Each ring has 10 of each prism, so there are a total of 120 pentagonal prisms and 120 pentagonal antiprisms. The original octahedra get bisected into square pyramids, so that's 600 square pyramids + 120 pentagonal prisms + 120 pentagonal antiprisms = 840 cells.

What's interesting, however, is that not only there are 12 rings of alternating prisms/antiprisms, the original octahedra of the o5oxo share half their faces with each other, and after the diminishing, the square pyramids also share their faces with each other. Furthermore, these square pyramids are isolated by the alternating prisms/antiprisms into individual rings of pyramids that also run around great circles of the 3-sphere. I didn't count them, but I believe there should be 20 such rings, corresponding with the dual swirlprism symmetry to the 12 rings of prisms! So here, in a single polychoron, you have the combination of both the dodecahedral Hopf fibration and its dual icosahedral Hopf fibration, represented by two sets of rings of cells. (One could say this corresponds with an icosidodecahedral Hopf fibration, or an icosidodecahedral uniform polytwister, in which the two types of twisters correspond with the the alternating prism rings and the square pyramid rings).

This polychoron is, of course, chiral because the underlying swirlprism symmetry is chiral. Moreover, I'm almost certain that it's vertex-transitive, which is ultracool, because that would make it scaliform.

Anyway, here's a preliminary render of this cute little baby:

I haven't tweaked the colors yet, so it looks a bit ugly, but you can clearly see the swirling rings of alternating prisms/antiprisms here. Or rather, half-rings, since I have visibility clipping on. The red vertical column in the center is actually a half-ring seen from a 90° in the 4D viewpoint, so it appears like a vertical column, but it's actually curved in a half-circle. Its other half, of course, lies on the far side and isn't shown here. It's kinda hard to see in this poorly-colored image, but there are two layers of rings around this central column. There are 5 rings in the inner layer and 5 rings in the outer layer. You can sorta see the slanted green column swirling around the central column, behind the yellow and magenta rings, and the blue ring to the right of it. These are 2 of the 5 inner rings. The yellow and magenta columns, of course, are 2 of the 6 outer rings. There is a ring that runs orthogonal to the red column, but it is not shown here because it lies just behind the limb of the polytope, so it got vis-clipped. So we see the structure of the rings is 1+5+5+1; the outer ring of the central column is the inner ring of the orthogonal ring, and vice versa. All of the rings are equivalent to each other under the underlying swirlprism symmetry group.

Now if you look carefully at the interface between the magenta and yellow rings, you can sorta discern the partial outlines (I omitted some edges to make the image less cluttered) of a wavy trail of triangles and squares: this is where one of the rings of square pyramids are. I didn't render the square pyramids in full because it would make the image too obscured. As you can see, the pyramids interface the pentagonal prisms of one ring to the pentagonal antiprisms of the other ring, in a skewed, alternating fashion. Really fascinating.

So anyway, here you have it, the swirl-diminished rectified 600-cell.

EDIT: Here's the same projection with a slightly better coloring scheme:

Red seems to work poorly when embedded deep inside the projection, so I changed it to white. Other than that, other colors remain the same but with transparencies tweaked.

Great renders, as always, quickfur!

Jonathan so seems to have beaten you by far in investigating that figure. He called that one an swirlprismatodiminished rectified hexacosachoron, being abbreviated to spidrox. - I too have the corresponding incidence matrix already online.

--- rk
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### Re: Johnsonian Polytopes

Klitzing wrote:[...]
Hmm, then you might be looking for those thingies?
• idsrix = icositetra-diminished ( srix = x3o3x5o )
• idsid pixhi = -- " -- ( sidpixhi = x3o3o5x )
• idprix = -- " -- ( prix = x3o3x5x )
• bidsid pixhi = bi-icositetra-diminished ( sidpixhi )

You might want to dig somewhere in this thread. Then the corresponding original research / find of those should show up.

--- rk

Wait, so this means most of the icositetradiminishings had been discovered, except those of o5o3x3x and o5o3x3o, i.e. the trunc/rect snub 24-cell?? I assumed, because the icositetradiminishings of o5o3x3o and o5o3x3x hadn't been discovered, those of other .5.3.3.-things wouldn't've been discovered as well.

Klitzing wrote:[...]
Then, according to the directions of those axes, you surely can collect all vertices of any such polytope into a finite set of discrete hyperplanes. Some of those section planes then allow for true polytopal cuts (i.e. edges, faces etc. also arrange within these hyperplanes. Often this is not only one, there might be more than one such. Provided those <i>true</i> polytopal cuts would exist, those then surely have to be mutually parallel for any polytope of the same symmetry group.
[..]

Well, I thought because 1. if two fundamental domains touch each other, you get an edge, and 2. you only get an edge where two fundamental domains touch, that the possible diminishings can be derived from the fundamental domains layout. (If you cut the fundamental domain layout of .5.3.3. at the equator, you get a sphere with triangles that more or less outline a icosidodecahedron, the same icosidodecahedron you get at the equator-cut of the 600-cell). seeing things this way, we will peut-être be able to prove we've found all diminishings. On the other hand, we might want to think twice before we do such a big investigation.
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### Re: Johnsonian Polytopes

quickfur wrote:Yet another render of the swirl-diminished o5oxo:

Here, I show 5 of the rings of square pyramids. These 5 are the ones that wrap around the central ring of alternating prisms from the previous images. As you can see, these rings are not fully separated from each other; they do not share any faces but they do touch each other at their edges. They also have the interesting property that they not only swirl around each other and around the prism/antiprism rings, but they also have a 3-fold twist to them.

This particular image is also helpful in determining the vertex configuration: each vertex touches 2 pentagonal prisms and 2 pentagonal antiprisms in a chiral formation, flanked by 4 square pyramids. AFAICT all the vertex configurations are the same, so this polychoron is scaliform.

Indeed it is!

Can't wait to see that one on your own website. Your elaborations together with the renders always are that useful.
I surely then would link my spidrox page to that one. As soon as it is up.

--- rk
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### Re: Johnsonian Polytopes

Great renders of Spidrox!, you can easily see the swirlprism and polytwister symmetries with these renders. By the way, it was George Olshevsky who originally found it.
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### Re: Johnsonian Polytopes

Finally digged out that old post of mine:

Klitzing wrote:
quickfur wrote:Would someone kindly summarize what's been going on here? I've to confess I haven't been able to keep up with all the detailed posts here.
...

Well, the idea of my researches was as follows:
Both, sadi and bidex are based on ex (x3o3o5o). In fact they are the 24-diminished ex and the bi-24-diminished ex. I wondered if the same diminishings would apply as well to the higher Wythoffian analogues as well (i.e. on tex (x3x3o5o), srix (x3o3x5o), sidpith (x3o3o5x), etc.). - The outcome, in detail described in those posts, is that the neighbouring node can be ringed only (when considering CRFs only) if we don't apply such diminishings (thus remaining within the mere Wythoffians). That the one but next node could be ringed if we want to apply 24-diminishing only onto it. And that the opposite node might be ringed if we want to apply bi-24-diminishing onto it.
This results in:
Code: Select all
`-----------+-------------+---------------+---------------+decoration | non-dim.    | 24-dim.       | bi-24-dim.    |-----------+-------------+---------------+---------------+xooo       | ex          | sadi          | bidex         |           | (600 tets)  | (120 tets,    | (48 teddies)  |           |             |  600 ikes)    |               |-----------+-------------+---------------+---------------+xoox       | sidpixhi    | idsid pixhi   | bidsid pixhi  |           | (600 tets,  | (120 tets,    | (120 trips,   |           | 1200 trips, |  480 trips,   |  216 pips,    |           |  720 pips,  |  432 pips,    |   72 does,    |           |  120 does)  |   96 does,    |   48 tedrids) |           |             |   24 srids)   |               |-----------+-------------+---------------+---------------+xoxo       | srix        | idsrix        | -             |           | (600 coes,  | (120 coes,    |               |           |  720 pips,  |  432 pips,    |               |           |  120 ids)   |   96 ids,     |               |           |             |  480 tricues, |               |           |             |   24 ties)    |               |-----------+-------------+---------------+---------------+xoxx       | prix        | idprix        | -             |           | (600 coes,  | (120 coes,    |               |           | 1200 trips, |  480 trips,   |               |           |  720 dips,  |  432 dips,    |               |           |  120 tids)  |   96 tids,    |               |           |             |  480 tricues, |               |           |             |   24 grids)   |               |-----------+-------------+---------------+---------------+xxoo       | tex         | -             | -             |-----------+-------------+---------------+---------------+xxxo       | grix        | -             | -             |-----------+-------------+---------------+---------------+xxox       | prahi       | -             | -             |-----------+-------------+---------------+---------------+xxxx       | gidpixhi    | -             | -             |-----------+-------------+---------------+---------------+`

And, most recently, I thought about some parallels. Sadi, and all these, were produced by an inscribed compound of 5 (tau scaled) icositetrachora (= ico) into the hexacosachoron (= ex). (In fact, the vertices of any of those icoes gives rise to a further 24-diminishing.) So we could see, what could be derived by the compound of 3 hexadecachora inscribed into an ico. So I investigated the application of 8-diminishings onto ico (x3o4o3o) and its higher Wythoffian relatives (i.e. tico (x3x4o3o), srico (x3o4x3o), spic (x3o4o3x), etc.). - The outcame was given in my last post of this thread.

You might ask, what about that there being mentioned impossibility of the ico-symmetric diminishing of tex, which you now have been shown to exist. - This is quite easy to answer. All those mentioned diminishings, respectively their possibilities or impossibilities, were given with respect to monostratic diminishings only. Whereas your recently found diminishing of tex was in contrast a bistratic one! (A monostratic diminishing of tex would ask to diddect the truncated tetrahedra at the vertex layer between the top hexagon and the bottom triangle, thereby producing larger triangles (their section) of 2 units sides. So the result cannot be CRF any more.)

That is,
Code: Select all
`*  sadi          =  ex       - 24x ( pt || ike )    =  x3o3o5o - 24x ( ox3oo5oo&#x )*  idsrix        =  srix     - 24x ( id || ti )     =  x3o3x5o - 24x ( ox3xx5oo&#x )*  idsid pixhi   =  sidpixhi - 24x ( doe || srid )  =  x3o3o5x - 24x ( ox3oo5xx&#x )*  idprix        =  prix     - 24x ( tid || grid )  =  x3o3x5x - 24x ( ox3xx5xx&#x )*  bidex         =  ex       - 48x ( pt || ike )    =  x3o3o5o - 48x ( ox3oo5oo&#x )*  bidsid pixhi  =  sidpixhi - 48x ( doe || srid )  =  x3o3o5x - 48x ( ox3oo5xx&#x )`

where "24x" always means: within icositetrachoral arrangement, resp. "48x" means: within arrangement of 2 inscribed icositetrachora. (In fact, there could be up to 5 inscribed icositetrachora. But any higher number would lead to more intricate mutual dissections of the icosahedra, resulting in no longer just unit edged cells...)

Never considered then deeper diminishings... E.g. your truncation of sadi would rather follow:
Code: Select all
`*  tisadi        =  tex      - 24x ( ike || pseudo u-ike || ti )  =  x3x3o5o - 24x ( xux3oox5ooo&#xt )`

where "u" denotes an edge length of 2 units.

And it now occurs to me, as you also considered the rectified version of sadi, that this one again would follow the same sheme as the above mentioned ones! In fact:
Code: Select all
`*  risadi        =  rox      - 24x ( ike || id )    =  o3x3o5o - 24x ( xo3ox5oo&#x )*  ...`

i.e. you started to diminish (in that case again monostratically) some figures of o3o3o5o symmetry, which leave the first node unringed. - Such cases neither have been considered in those times. I.e. those still have to be investigated...

--- rk
Last edited by Klitzing on Sun Feb 02, 2014 10:50 am, edited 2 times in total.
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### Re: Johnsonian Polytopes

Klitzing wrote:Jonathan so seems to have beaten you by far in investigating that figure. He called that one an swirlprismatodiminished rectified hexacosachoron, being abbreviated to spidrox. - I too have the corresponding incidence matrix already online.

Polyhedron Dude wrote:By the way, it was George Olshevsky who originally found it.

Indeed. I just wanted to point out, that it can be found on Jonathan's webpage (scroll down here) already. But yes, it was found by DinoGeorge way back in 2000.

--- rk
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### Re: Johnsonian Polytopes

In January of 2012 we applied the sadi type diminishings (i.e. according to icositetrachoral subsymmetry) by chopping off according segmentochora to different Wythoffian polychora of the hecatonicosachoral (= hyic) symmetry group. Then it turned out that there are only 3 such figures - as long as the first node of the Dynkin symbol of that to be diminished polychoron is ringed. Those are idsrix (the icositetra-diminished x3o3x5o (srix)), idsid pixhi (the 24-diminished x3o3o5x (sidpixhi)), and idprix (24-diminished x3o3x5x (prix)).

Recently student91 found 2 closely related figures: tisadi (the truncation of sadi) and risadi (the rectification of sadi). Tisadi then turned out to be a bistratic 24-diminishing of x3x3o5o (tex) - which because of not being a monostratic one clearly does not contradict the former result. - Risadi OTOH turned out to be a monostratic 24-diminishing of o3x3o5o (rox). - The latter one thus extends the former result, applying it to cases with the first node being unringed!

A short investigation thus results in the following: disregarding the ringing restriction of the first node, we would result in exactly 5 different monostratic 24-diminishings: the 3 of 2012, then risadi, and finally 1 more, which could be called (within the sense of those 3 others) an "idsrahi".

idsrahi = srahi (o3x3o5x) - 24x (srid || tid = xo3ox5xx&#x)

Here comes a quick investigation about its (total) element content:
Code: Select all
`            | srahi | 1-dim      | idsrahi------------+-------+------------+--------vertices    |  3600 |  -60       |    2160------------+-------+------------+--------edges (sum) | 10800 | -240       |    5040------------+-------+------------+--------{3} (sum)   |  4800 | -140       |    1440{4}         |  3600 |  -90       |    1440{5}         |   720 |  -12       |     432------------+-------+------------+--------oct         |   600 |  -20       |     120trip        |  1200 |  -30       |     480srid        |   120 |   -1 -12/3 |       0tid         |     0 |   +1       |      24tedrid      |     0 |      +12/3 |      96`

--- rk
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### Re: Johnsonian Polytopes

Klitzing wrote:In January of 2012 we applied the sadi type diminishings (i.e. according to icositetrachoral subsymmetry) by chopping off according segmentochora to different Wythoffian polychora of the hecatonicosachoral (= hyic) symmetry group. Then it turned out that there are only 3 such figures - as long as the first node of the Dynkin symbol of that to be diminished polychoron is ringed. Those are idsrix (the icositetra-diminished x3o3x5o (srix)), idsid pixhi (the 24-diminished x3o3o5x (sidpixhi)), and idprix (24-diminished x3o3x5x (prix)).

[...]
--- rk

That got every misconception out of the way, thanks
It's cool that the 24-dim. of x5o3x3o does exist as well, I wasn't sure about that in my previous post

I think I can conclude my previous posts. my conclusions are:
1. I once more typed quite some bullshit
2. we could add a uniform "dept" to diminishings, defined as follows:
looking at the o5o3o3.-based diminishings. First we take the polytope that has a vertex on every thing that can be diminished, so we take o5o3o3x. the vertices of o5o3o3x come in 9 hyperplanes when we place a single vertex on the first hyperplane. This means, we can define a "general distance" from one vertex to another by looking how many hyperplanes further it lies. These distances would range from 1 to 8.
example: if we place one vertex of o5o3o3x on the first hyperplane, and make sections at all the other hyperplanes, we get a o3o5o, a x3o5o, a o3o5x, a f3o5o, a o3x5o, and back the same way. Now if we take another vertex, and it lies in the o3x5o-section, it has distance 4. Does it lay in the o3o5x-section, it would have distance 2 or 6 (6 if it is on the "south side"). and so on.
now we make a o5o3o3.-cut, let's say a diminishing of x5o3o3x. The "dept" then is defined as the minimal distance two cuts must have in order to not intersect. The x5o3o3x-diminishing would then have dept "2", because if we diminish it at two dodecahedra with distance <2, they would intersect. If we diminish it at two dodecahedra with distance >1, it would not intersect.
this way of looking at diminishings isn't unambigious (both the "normal" and the mesodiminishing of o5o3x3o have dept 2), but it does make it easier to see what diminishings are possible (every diminishing with dept 2 will allow a 24-diminishing, and in the same way every diminishing with dept 4 will allow a 8-diminishing (based on an inscribed 16-cell))
furthermore some diminishings allow overlap, e.g. x5o3o3x can be diminished with distance "1" between two dodecahedra, making diminished x5o3x-cells, and o5o3o3x can have a similar diminishing. we could say the x5o3o3x-diminishing has dept 2 with a CRF-intersection of 1.

Quickfur, as always your renderings are beautyfull and clarifying. I especially like the one with square pyramids , because it's much clearer than the other one (the other one is a bit crowded)

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### Re: Johnsonian Polytopes

Polyhedron Dude wrote:Great renders of Spidrox!, you can easily see the swirlprism and polytwister symmetries with these renders. By the way, it was George Olshevsky who originally found it.

Thanks!

Ironically enough, I've skimmed over your swirlprisms page before, but since most of them were non-convex, I wrongly assumed that they were all non-convex, so for a long time I've been thinking about searching for CRFs (convex) that show swirlprism symmetry. Eventually, I found this one, and it turned out to be one of the convex entries on your page. So what goes around, comes around.

Now, I also realized that the same diminishing can be applied to x5o3x3o to produce something with swirlprism symmetry; it is basically a Stott expansion of Spidrox, and has rings of alternating decagonal prisms and parabidiminished x5o3x's. Not sure if it's scaliform, though; is this one also known? (I'm pretty sure it should be, but just in case. )
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### Re: Johnsonian Polytopes

student91 wrote:[...]
Quickfur, as always your renderings are beautyfull and clarifying. I especially like the one with square pyramids , because it's much clearer than the other one (the other one is a bit crowded)
[...]

Yeah I know. I should've stuck with my original rendering that shows only the first 6 prism/antiprism rings. The one with the square pyramids is only good because it only shows 5 rings. If I added more rings it would quickly become too crowded as well, probably even more crowded than the other one because there are 30 ringsEDIT: it should be 20 rings, not 30 of square pyramids!

But, these are all just preliminary renders. I should do a Polytope of the Month render for this beautiful polychoron sometime with properly-considered renders. Maybe in April, once I finish the uniforms (and yes I have every intention to finish all the uniforms, esp. the grand-daddy of them all, the omnitruncated 120-cell, that I've been looking forward to render for many years!). Speaking of which, February's Polytope of the Month is ready, so I'll be posting it soon, just have to update a few things to link the pages in.
Last edited by quickfur on Mon Feb 03, 2014 9:05 pm, edited 1 time in total.
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### Re: Johnsonian Polytopes

Application of the ideas, ligned out here (and there, resp. in subsequent posts then being applied to idsrix, idsid pixhi, and idprix, which helped to elaborate the respective incidence matrices), now in respect to idsrahi, results in the followings.

Again, instead of looking at the full polychoron idsrahi, we restrict our investigations just to a Voronoi domain of sadi (s3s4o3o), here superimposed to srahi (o3x3o5x). (We want to recapitulate here that this Voronoi domain of sadi is a figure bounded by 6 golden congruent low isoceles triangles and 3 golden kites. In fact it is nothing but a dodecahedron, being stellated by 3 tall pentagonal pyramids.)

The contained former single vertex (. o3o5o) of x3o3o5o (ex) now gets replaced by a full . x3o5x (srid) of o3x3o5x (srahi). But this srid is scaled such within the fundamental domain of consideration, that its pentagons lie in the face planes of its inscribed dodecahedron. According to the overall 3-fold axial symmetry of the domain we could devide those vertices of srahi into ((3(A)+6(B)+2*3(C)+6(D)+2*6(E)+6(F)+2*6(G)+6(H)+6(I)+3(J)+6(K)+3(L))*96/2 = 144(A)+288(B)+288(C)+288(D)+576(E)+288(F)+576(G)+288(H)+288(I)+144(J)+288(K)+144(L). Note the general division by 2, resp. the converse multiplication for some vertex classes. This is because those vertices lie at the domain border, i.e. that srid there conncets to a further srid of srahi, and therefore would else be counted double. These border-vertices of the centrally contained srids of srahi will be mapped into those of tedrids (tri-diminished rhombicosidodecahedra). But the internal vertices (types C, E, and G - thus not being counted double) would be the connections to those srids of srahi, which here will be replaced by tids (o3x5x = truncated dodecahedra). That is those vertices become eaten by those, chopping off the 3 pentagonal cupolae (pecus) from that central srid within the domain. Thus we will be left only with 144(A)+288(B)+288(D)+288(F)+288(H)+288(I)+144(J)+288(K)+144(L) = 2160 vertices in total.
Next are the edges. Both for srahi and for idsrahi it is valid that all edges will be contained in those of the srids, rep. of the tedrids. So we have to count those only. Again we have to distinguish between those, which run completely within the borders of that Voronoi domain, as thos contribute only half, and those which run through their inner, which then contribute full. Here we'll have: (2*3(AA)+6(AB)+2*3(BB)+6(BD)+3(DD)+2*6(DF)+3(FF)+6(FH)+2*6(HI)+6(HJ)+3(II)+6(IK)+2*6(JK)+2*3(KK)+6(KL)+2*3(LL))*96/2 = 288(AA)+288(AB)+288(BB)+288(BD)+144(DD)+576(DF)+144(FF)+288(FH)+576(HI)+288(HJ)+144(II)+288(IK)+576(JK)+288(KK)+288(KL)+288(LL)
With respect to the faces first there are those of the tedrid. The pentagons of which will count half, as those will fall on the domain border. Additionally there are the triangles perpendiculary to the domain edges. - Within this context we will have to consider the connectivity between the Voronoi domains. Obviously the (kite shaped) tetragons can connect to tetragons only. Then 3 such domains will connect around any of the short sides of the kites, providing a 3-fold dimple there. Thus it follows, that the 6 congruent golden triangles, which fall into 2 sets of 3 by their relative arrangement within a single domain, now just would connect in the opposite way, i.e. always mixing these sets! Btw. right from this fact it follows moreover, that the following former mentioned vertex calsses can be identified (due to outer symmetry): (A)=(J), (B)=(H), and (D)=(F). - Those orthogonal triangles thus are, already using this identification: (AAK), (BBI), (DDD), (KKK), (LLL). Obviously all those do contribute by one third to every domain only. Therefore: (2(AAA)+2*3(AABB)+3(ABDDB)+2*3(BBDFHIIHFD)+2*3(DDFF)+3(FFHJH)+2*6(HIKJ)+3(IIKLK)+2*3(JKK)+2*3(KKLL)+2(LLL))96/2 + ((3+6)(AAK)+(3+6)(BBI)+(3+3)(DDD)+3(KKK)+3(LLL)')96/3 = 96(AAA)+288(AABB)+288(AAK)+288((ABDDB)+(FFHJH))+288(BBDFHIIHFD)+288(BBI)+192(DDD)+288(DDFF)+576(HIKJ)+144(IIKLK)+288(JKK)+96(KKK)+288(KKLL)+96(LLL)+96(LLL)'
The cells finally are quite direct. Every Voronoi domain represents 1 tidrid. Thus there are 96 tidrids (tridiminished rhombicosidodecahedra) in total. The squares of the tedrids all attach to trips (trigonal prisms). Those are of types (AABBIK), (DDDDDD), and (KKKLLL). They amount in (3*(1+2)(AABBIK)+3(DDDDDD)+3(KKKLLL))*96/3 = 288(AABBIK)+96(DDDDDD)+96(KKKLLL). Then there are the octahedra. Those are alternatingly attached to the triangles of those tetdrid, respectively of those trips. We have the 2 types (AAAKKK) and (LLLLLL). Everyone contibutes to one fourth to the domain. Thus we get ((1+3)(AAAKKK)+(LLLLLL))*96/4 = 96(AAAKKK)+24(LLLLLL). Last but not least there are the tids (truncated dodecahedra), centered at the acute vertices of the domain. Each then contributes with one twelfth to the domain, thus we get a total of 96*3/12=24 tids.

Thus:
Code: Select all
`idsrahiA=J         | 288   *   *   *   *   * |   2   2   0   0   0   0   0   0   0   2   0   0   0 |  1   2   2   1   0   0   0   0   2   0   1  0   0  0  0 |  2   2  0  0  1  0  0B=H         |   * 576   *   *   *   * |   0   1   1   1   0   0   1   0   0   0   0   0   0 |  0   1   0   1   2   1   0   0   1   0   0  0   0  0  0 |  2   1  0  0  0  0  1D=F         |   *   * 576   *   *   * |   0   0   0   1   1   2   0   0   0   0   0   0   0 |  0   0   0   1   2   0   1   2   0   0   0  0   0  0  0 |  2   0  1  0  0  0  1I           |   *   *   * 288   *   * |   0   0   0   0   0   0   2   1   1   0   0   0   0 |  0   0   0   0   2   1   0   0   2   1   0  0   0  0  0 |  2   1  0  0  0  0  1K           |   *   *   *   * 288   * |   0   0   0   0   0   0   0   0   1   2   2   1   0 |  0   0   1   0   0   0   0   0   2   1   2  1   2  0  0 |  2   1  0  1  1  0  0L           |   *   *   *   *   * 144 |   0   0   0   0   0   0   0   0   0   0   0   2   4 |  0   0   0   0   0   0   0   0   0   1   0  0   4  2  2 |  2   0  0  2  0  1  0------------+-------------------------+-----------------------------------------------------+---------------------------------------------------------+----------------------AA          |   2   0   0   0   0   0 | 288   *   *   *   *   *   *   *   *   *   *   *   * |  1   1   1   0   0   0   0   0   0   0   0  0   0  0  0 |  1   1  0  0  1  0  0AB=HJ       |   1   1   0   0   0   0 |   * 576   *   *   *   *   *   *   *   *   *   *   * |  0   1   0   1   0   0   0   0   1   0   0  0   0  0  0 |  2   1  0  0  0  0  0BB          |   0   2   0   0   0   0 |   *   * 288   *   *   *   *   *   *   *   *   *   * |  0   1   0   0   1   1   0   0   0   0   0  0   0  0  0 |  1   1  0  0  0  0  1BD=FH       |   0   1   1   0   0   0 |   *   *   * 576   *   *   *   *   *   *   *   *   * |  0   0   0   1   2   0   0   0   0   0   0  0   0  0  0 |  2   0  0  0  0  0  1DD=FF       |   0   0   2   0   0   0 |   *   *   *   * 288   *   *   *   *   *   *   *   * |  0   0   0   1   0   0   0   2   0   0   0  0   0  0  0 |  2   0  1  0  0  0  0DF          |   0   0   2   0   0   0 |   *   *   *   *   * 576   *   *   *   *   *   *   * |  0   0   0   0   1   0   1   1   0   0   0  0   0  0  0 |  1   0  1  0  0  0  1HI          |   0   1   0   1   0   0 |   *   *   *   *   *   * 576   *   *   *   *   *   * |  0   0   0   0   1   1   0   0   1   0   0  0   0  0  0 |  1   1  0  0  0  0  1II          |   0   0   0   2   0   0 |   *   *   *   *   *   *   * 144   *   *   *   *   * |  0   0   0   0   2   0   0   0   0   1   0  0   0  0  0 |  2   0  0  0  0  0  1IK          |   0   0   0   1   1   0 |   *   *   *   *   *   *   *   * 288   *   *   *   * |  0   0   0   0   0   0   0   0   2   1   0  0   0  0  0 |  2   1  0  0  0  0  0JK          |   1   0   0   0   1   0 |   *   *   *   *   *   *   *   *   * 576   *   *   * |  0   0   1   0   0   0   0   0   1   0   1  0   0  0  0 |  1   1  0  0  1  0  0KK          |   0   0   0   0   2   0 |   *   *   *   *   *   *   *   *   *   * 288   *   * |  0   0   0   0   0   0   0   0   0   0   1  1   1  0  0 |  1   0  0  1  1  0  0KL          |   0   0   0   0   1   1 |   *   *   *   *   *   *   *   *   *   *   * 288   * |  0   0   0   0   0   0   0   0   0   1   0  0   2  0  0 |  2   0  0  1  0  0  0LL          |   0   0   0   0   0   2 |   *   *   *   *   *   *   *   *   *   *   *   * 288 |  0   0   0   0   0   0   0   0   0   0   0  0   1  1  1 |  1   0  0  1  0  1  0------------+-------------------------+-----------------------------------------------------+---------------------------------------------------------+----------------------AAA         |   3   0   0   0   0   0 |   3   0   0   0   0   0   0   0   0   0   0   0   0 | 96   *   *   *   *   *   *   *   *   *   *  *   *  *  * |  1   0  0  0  1  0  0AABB        |   2   2   0   0   0   0 |   1   2   1   0   0   0   0   0   0   0   0   0   0 |  * 288   *   *   *   *   *   *   *   *   *  *   *  *  * |  1   1  0  0  0  0  0AAK         |   2   0   0   0   1   0 |   1   0   0   0   0   0   0   0   0   2   0   0   0 |  *   * 288   *   *   *   *   *   *   *   *  *   *  *  * |  0   1  0  0  1  0  0ABDDB=FFHJH |   1   2   2   0   0   0 |   0   2   0   2   1   0   0   0   0   0   0   0   0 |  *   *   * 288   *   *   *   *   *   *   *  *   *  *  * |  2   0  0  0  0  0  0BBDFHIIHFD  |   0   4   4   2   0   0 |   0   0   1   4   0   2   2   1   0   0   0   0   0 |  *   *   *   * 288   *   *   *   *   *   *  *   *  *  * |  1   0  0  0  0  0  1BBI         |   0   2   0   1   0   0 |   0   0   1   0   0   0   2   0   0   0   0   0   0 |  *   *   *   *   * 288   *   *   *   *   *  *   *  *  * |  0   1  0  0  0  0  1DDD         |   0   0   3   0   0   0 |   0   0   0   0   0   3   0   0   0   0   0   0   0 |  *   *   *   *   *   * 192   *   *   *   *  *   *  *  * |  0   0  1  0  0  0  1DDFF        |   0   0   4   0   0   0 |   0   0   0   0   2   2   0   0   0   0   0   0   0 |  *   *   *   *   *   *   * 288   *   *   *  *   *  *  * |  1   0  1  0  0  0  0HIKJ        |   1   1   0   1   1   0 |   0   1   0   0   0   0   1   0   1   1   0   0   0 |  *   *   *   *   *   *   *   * 576   *   *  *   *  *  * |  1   1  0  0  0  0  0IIKLK       |   0   0   0   2   2   1 |   0   0   0   0   0   0   0   1   2   0   0   2   0 |  *   *   *   *   *   *   *   *   * 144   *  *   *  *  * |  2   0  0  0  0  0  0JKK         |   1   0   0   0   2   0 |   0   0   0   0   0   0   0   0   0   2   1   0   0 |  *   *   *   *   *   *   *   *   *   * 288  *   *  *  * |  1   0  0  0  1  0  0KKK         |   0   0   0   0   3   0 |   0   0   0   0   0   0   0   0   0   0   3   0   0 |  *   *   *   *   *   *   *   *   *   *   * 96   *  *  * |  0   0  0  1  1  0  0KKLL        |   0   0   0   0   2   2 |   0   0   0   0   0   0   0   0   0   0   1   2   1 |  *   *   *   *   *   *   *   *   *   *   *  * 288  *  * |  1   0  0  1  0  0  0LLL         |   0   0   0   0   0   3 |   0   0   0   0   0   0   0   0   0   0   0   0   3 |  *   *   *   *   *   *   *   *   *   *   *  *   * 96  * |  1   0  0  0  0  1  0LLL'        |   0   0   0   0   0   3 |   0   0   0   0   0   0   0   0   0   0   0   0   3 |  *   *   *   *   *   *   *   *   *   *   *  *   *  * 96 |  0   0  0  1  0  1  0------------+-------------------------+-----------------------------------------------------+---------------------------------------------------------+----------------------tidrid      |   6  12  12   6   6   3 |   3  12   3  12   6   6   6   3   6   6   3   6   3 |  1   3   0   6   3   0   0   3   6   3   3  0   3  1  0 | 96   *  *  *  *  *  *AABBIK-trip |   2   2   0   1   1   0 |   1   2   1   0   0   0   2   0   1   2   0   0   0 |  0   1   1   0   0   1   0   0   2   0   0  0   0  0  0 |  * 288  *  *  *  *  *DDDDDD-trip |   0   0   6   0   0   0 |   0   0   0   0   3   6   0   0   0   0   0   0   0 |  0   0   0   0   0   0   2   3   0   0   0  0   0  0  0 |  *   * 96  *  *  *  *KKKLLL-trip |   0   0   0   0   3   3 |   0   0   0   0   0   0   0   0   0   0   3   3   3 |  0   0   0   0   0   0   0   0   0   0   0  1   3  0  1 |  *   *  * 96  *  *  *AAAKKK-oct  |   3   0   0   0   3   0 |   3   0   0   0   0   0   0   0   0   6   3   0   0 |  1   0   3   0   0   0   0   0   0   0   3  1   0  0  0 |  *   *  *  * 96  *  *LLLLLL-oct  |   0   0   0   0   0   6 |   0   0   0   0   0   0   0   0   0   0   0   0  12 |  0   0   0   0   0   0   0   0   0   0   0  0   0  4  4 |  *   *  *  *  * 24  *tid         |   0  24  24  12   0   0 |   0   0  12  24   0  24  24   6   0   0   0   0   0 |  0   0   0   0  12  12   8   0   0   0   0  0   0  0  0 |  *   *  *  *  *  * 24`

--- rk
Klitzing
Pentonian

Posts: 1638
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: Johnsonian Polytopes

After reading everything carefully, and looking up what a voroni-cell is, I think I understood your post.
Am I right when I say voroni-cells look a bit like the dual cells of s3s4o3o, connected to the center of s3s4o3o to make them 4D-bodies?

I think this polytope is the coolest of the 24-diminished .5.3.3.'s. I mean, it has J83's!!
How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
-Stern/Multatuli/Eduard Douwes Dekker
student91
Tetronian

Posts: 328
Joined: Tue Dec 10, 2013 3:41 pm

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