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

Postby Klitzing » Tue Jan 08, 2013 8:38 pm

quickfur wrote:
Klitzing wrote:[...]
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)   |               |
-----------+-------------+---------------+---------------+
...

These ones (24-dim and bi-24-dim of xoox) looks interesting! Maybe I'll make some renders of them! (If I can figure out the coordinates :P) Is the bi-24-dim-xoo5x chiral too?
...

Bidsid pixhi is. In fact it has exactly the same symmetry! It places the tedrids (tri-diminished rhombicosidodecahedra) in the same orientations as the teddies (tri-diminished icosahedra) were placed within bidex. Just that the other cells will be introduced inbetween as well.

For details about the fundamental domain of construction for that fellow you'd read that post. For its application onto bidsid pixhi, you'd read that one. - For what is all this stuff about fundamental domains you should refer to this post, where the concept was explained and examplified with respect to sadi.

Would like to see my baby! :)
The coordinates should be a subset of those of sidpixhi (x3o3o5x) btw.

--- rk
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Re: Johnsonian Polytopes

Postby Klitzing » Tue Jan 08, 2013 8:55 pm

quickfur wrote:
Klitzing wrote:[...]
Code: Select all
-----------+-------------+---------------+---------------+
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)   |               |
-----------+-------------+---------------+---------------+

These ones look interesting too. But the bi-24-dims are non-CRF?
...


Yes. The 24-dims contain the tricu. For the bi-24-dims those would get cut further. Thus there would be half-hexagons, i.e. trapezia of side lengths 1:1:1:2.

...
Code: Select all
-----------+-------------+---------------+---------------+
xxoo       | tex         | -             | -             |
-----------+-------------+---------------+---------------+
xxxo       | grix        | -             | -             |
-----------+-------------+---------------+---------------+
xxox       | prahi       | -             | -             |
-----------+-------------+---------------+---------------+
xxxx       | gidpixhi    | -             | -             |
-----------+-------------+---------------+---------------+


So these don't produce CRF diminishings. Hmm.
...

The same problem occurs here already for the 24-dims: the hexagons of the 0-dims would get halved. :(

--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Tue Jan 08, 2013 9:07 pm

Klitzing wrote:
quickfur wrote:
Klitzing wrote:[...]
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)   |               |
-----------+-------------+---------------+---------------+
...

These ones (24-dim and bi-24-dim of xoox) looks interesting! Maybe I'll make some renders of them! (If I can figure out the coordinates :P) Is the bi-24-dim-xoo5x chiral too?
...

Bidsid pixhi is. In fact it has exactly the same symmetry! It places the tedrids (tri-diminished rhombicosidodecahedra) in the same orientations as the teddies (tri-diminished icosahedra) were placed within bidex. Just that the other cells will be introduced inbetween as well.

For details about the fundamental domain of construction for that fellow you'd read that post. For its application onto bidsid pixhi, you'd read that one. - For what is all this stuff about fundamental domains you should refer to this post, where the concept was explained and examplified with respect to sadi.

Would like to see my baby! :)
The coordinates should be a subset of those of sidpixhi (x3o3o5x) btw.

--- rk

Ah, I see what you were doing now. My current tools don't work directly with symmetry groups and fundamental domains, unfortunately, so they are not of much help to me. However, I am able to work with geometric constructions -- if I understand it correctly, you are basically deleting the vertices of 48 dodecahedra from x3o3o5x, correct? And these are the ones whose hyperplane normals are parallel to the vertices of two 24-cells? I can work with that. I'll see if my tools already can do something similar. If not I'll just have to program them a bit more. :)
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Re: Johnsonian Polytopes

Postby Klitzing » Tue Jan 08, 2013 9:38 pm

quickfur wrote:
Klitzing wrote:[...]
Have to think about the remainder of your post for a while. :)
(Btw., the greek words para and ortho seem not to apply here correctly. If I get your procedure correctly, you rather are investigating an 8-gyration respectively a 4-gyration, right?)

OK, I guess I'm really just borrowing the terminology from the 24-cell diminishings, because of the correspondence between the vertices of the 24-cell and the square faces of the tesseract. The diminishings of the 24-cell correspond with subsets of non-adjacent vertices of the 24-cell. Now since x4o3x3o can be gyrated as long as the rotated 8-prism||square segments are non-adjacent, and these segments correspond with the 2-faces of the tesseract (and therefore the vertices of the 24-cell), I thought it was convenient to borrow the terminology from the 24-cell diminishings to indicate which subset of 8-prism||square segments are rotated.
[...]

That's exactly the relation I described in that recent post. (In fact, there are even more analogues described.)

[...]So I'm using para here in the sense of being antipodes (i.e., opposite vertices of the 24-cell, or antipodal 2-faces of the tesseract, and therefore the 8-prism||square segments corresponding to antipodal 2-faces of the tesseract). There is only one possible combination of non-adjacent vertices of the 24-cell that consists of two pairs of antipodal vertices -- that is, they correspond with the vertices of an inscribed 16-cell, and so "paratetra" uniquely identifies this combination of vertices on the 24-cell, and by extension, the particular combination of two pairs of antipodal 8-prism||square segments on the x4o3x3o.

With ortho, it means a pair of vertices of the 24-cell such that their respective vectors to the center of the 24-cell are 90° to each other. And I think I made a mistake in my usage in my post; a tetraortho- would mean 4 vertices of a tetrahedron in a 16-cell inscribed in the 24-cell, whereas I meant a cyclotetragyration (i.e., the 4 non-adjacent 8-prism||square segments that lie on a great circle, corresponding with the 4 vertices of 16-cell inscribed in a 24-cell that lie on a single great circle).
[...]


Well, that we have 4 axes in 4D and if we use opposite pairs, it is the same to speak of 8 points or of para-4. In any case you mean the vertices of an ico-inscribed hex as to be applied positions. - The ortho means 90°, yes. Cyclo-4 looks much better here. - So I'd propagate using simply 8- resp. cyclo-4- (instead of using the overloaded para), both with respect to ico itself, as to srit (x4o3x3o).

[...]With meta, on the 24-cell it means two vertices A and B such that B is adjacent to A's antipode. The various metadiminishings of the 24-cell happen to be precisely those that do not correspond with any augmentation of the tesseract. It also happens that there are only two maximal diminishings of the 24-cell besides the tesseract itself (maximal means no more diminishing of non-adjacent vertices is possible), and both are metadiminishings. So they represent a kind of interesting configuration of vertices on the 24-cell where there are no remaining vertices that are non-adjacent to the rest. By extension, they also represent maximal subsets of non-adjacent 8-prism||square on the x4o3x3o that do not correspond with augmentations of an 8,8-duoprism, so they are a special kind of diminishing of the x4o3x3o.


Oh, now I see, were you derive from. You're trying to use para- and meta- in the way Johnson did in naming his solids. But note that there para- does not divide the count! From srid (x3o5x) he derives the para-bi-diminished and the meta-bi-diminished one (resp. -gyrated ones). So you might rather specify your ones as para-8- resp. cyclo-4-.

So far as to the positions of actions. Now to what you are gyrating. You multiply mention some 8-prism||square. So far can't visualize those in srit. Would you come in?
In the mentioned 8 positions there are sirco||tic for cupolae. But those would intersect, if applied at 90° positions. Don't believe that this works in gyration, this most probably would produce some non-CRF cell sections. Isn't it?

--- rk
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Re: Johnsonian Polytopes

Postby Klitzing » Tue Jan 08, 2013 9:42 pm

quickfur wrote:Ah, I see what you were doing now. My current tools don't work directly with symmetry groups and fundamental domains, unfortunately, so they are not of much help to me. However, I am able to work with geometric constructions -- if I understand it correctly, you are basically deleting the vertices of 48 dodecahedra from x3o3o5x, correct? And these are the ones whose hyperplane normals are parallel to the vertices of two 24-cells? I can work with that. I'll see if my tools already can do something similar. If not I'll just have to program them a bit more. :)

Completely correct.
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Re: Johnsonian Polytopes

Postby quickfur » Tue Jan 08, 2013 10:43 pm

Klitzing wrote:[...]
Oh, now I see, were you derive from. You're trying to use para- and meta- in the way Johnson did in naming his solids. But note that there para- does not divide the count! From srid (x3o5x) he derives the para-bi-diminished and the meta-bi-diminished one (resp. -gyrated ones). So you might rather specify your ones as para-8- resp. cyclo-4-.

Well, actually I derived my prefixes by studying the enumeration of all 24-cell diminishings. Maybe using the para-, ortho-, meta- prefixes wasn't a good idea after all; but I wanted some kind of nomenclature that is easy to remember. So I use ortho- for the diminishings that have lots of vertices at 90° from each other, and meta- for the cases where some vertices are ortho to another vertex's antipode, and para for when there are many antipodal points.

Clearly, there is a lot of overlap, and a few ambiguous cases, so I devised a system of numerical prefixes to indicate exactly which vertices are meant. The basic idea was that in any diminishing except the null diminishing (which is uninteresting), there must be at least one vertex, so the numbering of everything else will be relative to this vertex. The second vertex then can only be in 3 distinct positions relative to this vertex, so this stratifies the 24-cell into these cross sections: point -- cube -- octahedron (scaled up by 2) -- cube -- point. By convention, the first vertex is always assigned to the first point, which means that the first cube cannot have any vertices (they are all adjacent to the first point).

Using this stratification, any vertex that lies on the middle octahedron cross-section will be ortho to the first vertex, and any vertex on the second cube will be meta, and the vertex on the last point is para (antipodal). Then to reduce the number of duplicate configurations, I adopt the convention that the canonical prefix is the one that gives the smallest numbers (similar to IUPAC naming for chemical compounds), where the octahedral and cube cross sections are numbered like so:

Code: Select all
            1
            | 3'
            |/
        2---+---2'
           /|
          3 |
            1'

          1-----2
         /|    /|
        3-----4 |
        | 5---|-6
        |/    |/
        7-----8

Where 1', 2', 3', are considered to be greater than 1, 2, 3 (I didn't use 4, 5, 6 because I wanted to be able to quickly tell which axis of the octahedron the vertex lies on -- but this scheme can be renamed if it yields a better notation).

The notation then, is made by writing the prefix in the form 1,(1,2,3),(1,2,3),1, where the first number corresponds with the first vertex, the first parentheses list the vertices that lie on the octahedral cross section, the second parentheses list the vertices on the second cube, and the last number is for the antipode (omitted if the antipode is not included in the set). Parentheses are omitted if there's only one vertex from the corresponding cross section. If either of the middle two parentheses are empty, then they are written as 0 instead.

Using this naming system, I enumerated all 24-cell diminishings:
Code: Select all
- 1-diminished 24-cell ((mono)diminished)
- 1,1-bidiminished 24-cell (orthobidiminished)
- 1,0,1-bidiminished 24-cell (metabidiminished)
- 1,0,0,1-bidiminished 24-cell (parabidiminished)
- 1,(1,2)-tridiminished 24-cell (orthotridiminished)
- 1,(1,1')-tridiminished 24-cell (paratridiminished)
- 1,1,5-tridiminished 24-cell (metatridiminished)
- 1,(1,2,3)-tetradiminished 24-cell (orthotetradiminished)
- 1,(1,2,1')-tetradiminished 24-cell (paratetradiminished)
- 1,(1,2),6-tetradiminished 24-cell (orthometatetradiminished)
- 1,(1,1'),0,1-tetradiminished 24-cell (cyclotetradiminished)
- 1,1,(5,8)-tetradiminished 24-cell (metametatetradiminished)
- 1,(1,2,3,1')-pentadiminished 24-cell (orthopentadiminished)
- 1,(1,2,3),6-pentadiminished 24-cell (metapentadiminished)
- 1,(1,2,1',2')-pentadiminished 24-cell (parapentadiminished)
- 1,(1,2,3,1',2')-hexadiminished 24-cell (orthohexadiminished)
- 1,(1,2,1',2'),0,1-hexadiminished 24-cell (parahexadiminished)
- Heptadiminished 24-cell (augmented tesseract)
- Octadiminished 24-cell (tesseract)

It's arguable whether the names in the parentheses are the best; if you have a better idea, I'd love to hear it. In any case, the simplest improvement of this notation is to elide the first 1, because it's always 1. After that, there may be a way of simplifying the notation to cover all the cases without needing to spell out all the numbers. But the current system is unambiguous, so that's what I've been using up till now.

So far as to the positions of actions. Now to what you are gyrating. You multiply mention some 8-prism||square. So far can't visualize those in srit. Would you come in?
In the mentioned 8 positions there are sirco||tic for cupolae. But those would intersect, if applied at 90° positions. Don't believe that this works in gyration, this most probably would produce some non-CRF cell sections. Isn't it?

Hmm. Did I mix up which polychoron I was applying the gyration to? I was referring to x4o3x3o (or o3x3o4x in your usual way of writing it). It can be decomposed as an 8,8-duoprism augmented with eight 8prism||square's, with 4 in each ring of 8 prisms, each pair separated by an 8-prism. Eight of the 8prisms from the 8,8-duoprism merge with the square cupolas in the augment to form 8 rhombicuboctahedra, and the square pyramids of the augments merge into 16 octahedra.

The result has higher symmetry than the 8,8-duoprism, because of the coincidence of 4pyr + 4pyr = oct, and there are 3 ways to do this, and also 4cup + 8prism + 4cup = rhombicuboctahedron, which has higher symmetry than 4cup and 8prism. So from the x4o3x3o, we are allowed to cut off two 8prism||square as long as the cutting does not cut the same octahedron from two different planes (it will result in a non-CRF). It's OK to cut the same octahedron from both sides if the cutting plane is the same; in this case the entire octahedron is removed from the result, otherwise a square pyramid remains.

Interestingly, not only you can cut off the 8prism||square, but you can also glue it back with the square rotated 45°. The result is that the 4pyrs don't line up, so they don't form octahedra, but remain as 4pyrs. This is what I mean by gyrating. I've posted about the paratetragyrate before, in this post. It has this image:

Image

If you look at the center part, where the yellow cell and the green cell meet, you can see their common square face, and if you trace the edges outwards by 1 edge length, you can see the outline of an 8-prism. (The triangular prisms + 4pyr are not coplanar so they are not augmented 3prisms or gyrobifastigiums.) This 8-prism is the base of an 8prism||square. If you cut it off, you will get an 8-prism cell here connecting the top part of the yellow cell to the bottom part of the green cell (they will be shortened into elongated square cupola). If you then follow in the same direction along the great circle from the top of the yellow cell to the bottom of the green cell, you can find more 8prism||square augments; you can cut them all off and you will end up with a ring of eight 8prisms.
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Re: Johnsonian Polytopes

Postby Klitzing » Wed Jan 09, 2013 12:39 am

Ah, your system of namings for the diminishings of ico looks not too bad after all. Good work in numerating those all! Now its becoming perfectly clear what you meant with para-4-dim., ortho-4-dim., cyclo-4-dim. From the mere number of possible cases it might serve to divide in meta- and non-meta cases. Esp. for the very reason that the non-meta cases could be sufficently coded by the vertices of an inscribed hex!

The relation of srit (small rhombitesseract = x4o3x3o) to the odip (octagon-duoprism) so far was not apparent to me, I've to admit. :oops:
Your picture then makes the 4g||8p diminishings/gyrations pretty clear. Thank you for that advice. Likewise for the link to the original post. Still have to get my mind used to this inscribed odip, so.

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Re: Johnsonian Polytopes

Postby quickfur » Wed Jan 09, 2013 1:16 am

Klitzing wrote:Ah, your system of namings for the diminishings of ico looks not too bad after all. Good work in numerating those all!

Well actually, I am far from being the first to enumerate them. Wintersolstice on this forum did it a long time before I did, and then Marek after him (albeit indirectly -- he was enumerating augmentations, but it amounts to the same thing since the 24-cell is self-dual). I only did my enumeration because I wanted to explore it for myself -- I was looking for maximal diminishings, which required that I understand exactly what each diminishing represents. It turned out that the maximal diminishings are exactly the tesseract (octadiminished) and the two meta diminishings 1,1,(5,8)-tetradiminished and 1,(1,2,3),6-pentadiminished.

Now its becoming perfectly clear what you meant with para-4-dim., ortho-4-dim., cyclo-4-dim. From the mere number of possible cases it might serve to divide in meta- and non-meta cases. Esp. for the very reason that the non-meta cases could be sufficently coded by the vertices of an inscribed hex!

You're right! That's a useful way to categorize them. And also, the non-meta cases are all tesseract augmentations, so we could approach it from that angle as well.

The relation of srit (small rhombitesseract = x4o3x3o) to the odip (octagon-duoprism) so far was not apparent to me, I've to admit. :oops:

It was only apparent for me after I starting building actual models of x4o3x3o diminishings and making projections with my viewer. It was an unexpected discovery for me too.

It also implies some CRFs that are not directly derivable from diminishing x4oxo: for example, if we denote the x4oxo as augmentations on the two rings of the odip, we may write it as +-+-+-|+-+-+- where | divides the two rings, and + means an augment in that position of the ring, and - means no augment. You cannot have two adjacent +'s, because it will be non-convex; and each sequence of +-+ produces a rhombicuboctahedron in the result. (Non-meta) diminishings of the x4oxo then correspond with replacing +'s with -'s; but it does not cover all the possibilities, for example, you can have +--+-+--|+-+-+-, where the subsequences +-- and --+ produce elongated square cupolae (or partial rhombicuboctahedra, if you will). The subsequence +--+ represents two elongated square cupolae (J19) joined at their bases with a 45° dichoral angle. The list of cells in the first ring are therefore J19, J19, x4o3x, J19, J19 -- a 5-membered ring. Quite an interesting arrangement!

These variations are, of course, all part of the set of odip augmentations. So the set of odip augmentations and the set of x4oxo diminishings intersect, but are not subsets of each other.

Your picture then makes the 4g||8p diminishings/gyrations pretty clear. Thank you for that advice. Likewise for the link to the original post. Still have to get my mind used to this inscribed odip, so. [...]

This makes me wonder, how many ways can you inscribe an odip into a x4oxo?

Off the top of my head, it seems to be equivalent to the number of ways you can decompose a tesseract into two rings of 4 cubes each. Now, if we mark out a single cube on the tesseract, then given any set of two rings it must lie in one of them. Furthermore, the direction that the great circle inscribing the ring intersects this cube uniquely determines the ring, which in turn uniquely determines the second ring. So the number of distinct sets must be equal to the number of opposite pairs of faces on the cube (which corresponds with the direction of the great circle passing through it), which is 3. So there are exactly 3 ways to inscribe an odip into a x4oxo, I think?

(Not 100% sure about my analysis... I've a reputation for always getting combinatorics wrong.)
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Re: Johnsonian Polytopes

Postby Klitzing » Wed Jan 09, 2013 9:34 am

quickfur wrote:[...]
Klitzing wrote:Now its becoming perfectly clear what you meant with para-4-dim., ortho-4-dim., cyclo-4-dim. From the mere number of possible cases it might serve to divide in meta- and non-meta cases. Esp. for the very reason that the non-meta cases could be sufficently coded by the vertices of an inscribed hex!

You're right! That's a useful way to categorize them. And also, the non-meta cases are all tesseract augmentations, so we could approach it from that angle as well.[...]

:)
[...]
The relation of srit (small rhombitesseract = x4o3x3o) to the odip (octagon-duoprism) so far was not apparent to me, I've to admit. :oops:

It was only apparent for me after I starting building actual models of x4o3x3o diminishings and making projections with my viewer. It was an unexpected discovery for me too.[...]

So you were the first to note?
[...]It also implies some CRFs that are not directly derivable from diminishing x4oxo: for example, if we denote the x4oxo as augmentations on the two rings of the odip, we may write it as +-+-+-|+-+-+- where | divides the two rings, and + means an augment in that position of the ring, and - means no augment. You cannot have two adjacent +'s, because it will be non-convex; and each sequence of +-+ produces a rhombicuboctahedron in the result. (Non-meta) diminishings of the x4oxo then correspond with replacing +'s with -'s; but it does not cover all the possibilities, for example, you can have +--+-+--|+-+-+-, where the subsequences +-- and --+ produce elongated square cupolae (or partial rhombicuboctahedra, if you will). The subsequence +--+ represents two elongated square cupolae (J19) joined at their bases with a 45° dichoral angle. [...]

135° I would think.
[...]The list of cells in the first ring are therefore J19, J19, x4o3x, J19, J19 -- a 5-membered ring. Quite an interesting arrangement![...]

You could represent those in a 2D diagram by their heights. Getting a pentagon with 3 right angles (being denoted ";" in the following sequence) with side lengths 2+q;1+q,1+q;1+q,1+q; (where q=sqrt2). Kind the crossection of a house. :D
[...]These variations are, of course, all part of the set of odip augmentations. So the set of odip augmentations and the set of x4oxo diminishings intersect, but are not subsets of each other.[...]

Right, important to emphasize! Metadiminishings of srit would run out of the set of odip augmentations. - Am I right in supposing that the superset of odip augmentations and srit diminishings is the set of srit diminishings and/or gyrations?
[...]
Your picture then makes the 4g||8p diminishings/gyrations pretty clear. Thank you for that advice. Likewise for the link to the original post. Still have to get my mind used to this inscribed odip, so. [...]

This makes me wonder, how many ways can you inscribe an odip into a x4oxo?

Off the top of my head, it seems to be equivalent to the number of ways you can decompose a tesseract into two rings of 4 cubes each.[...]

Would serve perfectly. But you even could use srit itself as well: any symmetry breaking of cube -> 4-prism corresponds to any selection of an inscribed op within sirco.
[...]Now, if we mark out a single cube on the tesseract, then given any set of two rings it must lie in one of them. Furthermore, the direction that the great circle inscribing the ring intersects this cube uniquely determines the ring, which in turn uniquely determines the second ring. So the number of distinct sets must be equal to the number of opposite pairs of faces on the cube (which corresponds with the direction of the great circle passing through it), which is 3. So there are exactly 3 ways to inscribe an odip into a x4oxo, I think?

(Not 100% sure about my analysis... I've a reputation for always getting combinatorics wrong.)

Yes, looks correct to me. The geometric realization of tes serves as both: the tes-symmetric regular polychoron, and the subsymmetric 4-dip. Subsymmetry occurs then in the selection of a specific axis for the cubes (i.e. the already mentioned symmetry breaking: cube -> 4-prism). And, each cube then will be part of exactly one great circle. But having chosen one of those great circles, the second one (of that duoprism) will be determined thereby automaticly. Thus your restriction to just one (any) cube makes perfectly sense.

This btw. should give rise to a further finding: there should be a compound of 3 odips, which is vertex inscribed into srit.
Further, like there is a compound of 3 tes, vertex inscribed into ico (having 2 such tes incident to any ico vertex), there clearly is also a corresponding compound of 3 srit, vertex inscribed into spic. And by the above there should be then a compound of 3x3=9 odip, vertex inscribable into spic, as well! (Or could you visualize whether some of those odip therein might coincide? - Without going into coordinate calculations: as mentioned above, 2 tes are incident at any ico vertex each. Thus it seems that at most 2 corresponding odip might coincide. But 9 is not dividable by 2. So it looks to me unlikely; i.e. that 9-odip-compound should exist as well.)

--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Wed Jan 09, 2013 7:19 pm

Klitzing wrote:
quickfur wrote:
[...]The relation of srit (small rhombitesseract = x4o3x3o) to the odip (octagon-duoprism) so far was not apparent to me, I've to admit. :oops:

It was only apparent for me after I starting building actual models of x4o3x3o diminishings and making projections with my viewer. It was an unexpected discovery for me too.[...]

So you were the first to note?

That, I can't say. I've been known to rediscover a lot of things that are already known, but I was unaware of. So maybe somebody else has found this before.

[...]It also implies some CRFs that are not directly derivable from diminishing x4oxo: for example, if we denote the x4oxo as augmentations on the two rings of the odip, we may write it as +-+-+-|+-+-+- where | divides the two rings, and + means an augment in that position of the ring, and - means no augment. You cannot have two adjacent +'s, because it will be non-convex; and each sequence of +-+ produces a rhombicuboctahedron in the result. (Non-meta) diminishings of the x4oxo then correspond with replacing +'s with -'s; but it does not cover all the possibilities, for example, you can have +--+-+--|+-+-+-, where the subsequences +-- and --+ produce elongated square cupolae (or partial rhombicuboctahedra, if you will). The subsequence +--+ represents two elongated square cupolae (J19) joined at their bases with a 45° dichoral angle. [...]

135° I would think.

Yes, you're right. :oops:

[...]The list of cells in the first ring are therefore J19, J19, x4o3x, J19, J19 -- a 5-membered ring. Quite an interesting arrangement![...]

You could represent those in a 2D diagram by their heights. Getting a pentagon with 3 right angles (being denoted ";" in the following sequence) with side lengths 2+q;1+q,1+q;1+q,1+q; (where q=sqrt2). Kind the crossection of a house. :D

Yes, exactly. That's what I thought when I found this, too. :) It's the srit-house!

[...]These variations are, of course, all part of the set of odip augmentations. So the set of odip augmentations and the set of x4oxo diminishings intersect, but are not subsets of each other.[...]

Right, important to emphasize! Metadiminishings of srit would run out of the set of odip augmentations. - Am I right in supposing that the superset of odip augmentations and srit diminishings is the set of srit diminishings and/or gyrations?

No, gyrating the 8prism||square segments do not move the location of the segment on the srit, so it cannot produce the srit-house shape, for example. To extend my earlier notation, we can write + for 8prism||square in one orientation, and x for 8prism||square rotated by 45°. Then starting from srit, which is +-+-+-+-|+-+-+-+-, the paratetragyrated srit can be denoted as x-+-x-+-|x-+-x-+-. Each x-+ or +-x produces a J37 (elongated square gyrobicupola), so this shape has 8 J37's. But gyration can only change + to x and vice versa, it cannot interchange + and -. And diminishing can only change + to -, not - to +, so you cannot produce +--+-+--|+-+-+-. You will still need to add augmentations before you can obtain the entire superset.

[...]
Your picture then makes the 4g||8p diminishings/gyrations pretty clear. Thank you for that advice. Likewise for the link to the original post. Still have to get my mind used to this inscribed odip, so. [...]

This makes me wonder, how many ways can you inscribe an odip into a x4oxo?

Off the top of my head, it seems to be equivalent to the number of ways you can decompose a tesseract into two rings of 4 cubes each.[...]

Would serve perfectly. But you even could use srit itself as well: any symmetry breaking of cube -> 4-prism corresponds to any selection of an inscribed op within sirco.

You're right. So it amounts to choosing an op within each x4ox, and there are exactly 3 choices.

[...]Now, if we mark out a single cube on the tesseract, then given any set of two rings it must lie in one of them. Furthermore, the direction that the great circle inscribing the ring intersects this cube uniquely determines the ring, which in turn uniquely determines the second ring. So the number of distinct sets must be equal to the number of opposite pairs of faces on the cube (which corresponds with the direction of the great circle passing through it), which is 3. So there are exactly 3 ways to inscribe an odip into a x4oxo, I think?

(Not 100% sure about my analysis... I've a reputation for always getting combinatorics wrong.)

Yes, looks correct to me. The geometric realization of tes serves as both: the tes-symmetric regular polychoron, and the subsymmetric 4-dip. Subsymmetry occurs then in the selection of a specific axis for the cubes (i.e. the already mentioned symmetry breaking: cube -> 4-prism). And, each cube then will be part of exactly one great circle. But having chosen one of those great circles, the second one (of that duoprism) will be determined thereby automaticly. Thus your restriction to just one (any) cube makes perfectly sense.

This btw. should give rise to a further finding: there should be a compound of 3 odips, which is vertex inscribed into srit.
Further, like there is a compound of 3 tes, vertex inscribed into ico (having 2 such tes incident to any ico vertex), there clearly is also a corresponding compound of 3 srit, vertex inscribed into spic. And by the above there should be then a compound of 3x3=9 odip, vertex inscribable into spic, as well! (Or could you visualize whether some of those odip therein might coincide? - Without going into coordinate calculations: as mentioned above, 2 tes are incident at any ico vertex each. Thus it seems that at most 2 corresponding odip might coincide. But 9 is not dividable by 2. So it looks to me unlikely; i.e. that 9-odip-compound should exist as well.)

Sad to say, my 4D visualization skills is not advanced enough to deal with self-intersecting shapes just yet. ;)

But one thought I have is, the tesseract decomposes into 2 hexes, and also 3 tesseracts span a 24-cell, and 3 hexes also span a 24-cell. So that means in the compound of 3 tesseracts, there must be many coinciding vertices. Each pair of hexes in the compound of 3 hexes would produce a tesseract. But it's still possible that all odips in each tess are non-coincident. Hmm, I'm not sure. I have trouble seeing self-intersecting shapes in 4D. :lol:
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Re: Johnsonian Polytopes

Postby quickfur » Wed Jan 09, 2013 8:01 pm

Actually, thinking about it a bit more, I think the 9-odip compound does exist.

I think I was just confusing myself over the tess = 2 hex issue. For simplicity, let's use hex: the plane of the rings in each odip must lie in the same plane as 4 vertices of a hex, in the 3-hex decomposition of the 24-cell. This plane must be defined by two axes of the hex (and the other two axes define the plane of the second ring, of course).

Now, let's take the decomposition of the 24-cell into 3 hexes: apacs(2,0,0,0), ecs(1,1,1,1) and ocs(1,1,1,1). (Using wendy's notation: apacs = all permutations of coordinates, all changes of sign; ecs = even changes of sign, ocs = odd changes of sign.) A little algebra shows that none of the axes of the 3 hexes coincide. Which means that there are no pairs of axes that coincide, either. So none of the planes of the odip's rings should coincide either, and therefore there cannot be any coincident odips.

The only concern is that some of the pairs of axes may not be linearly independent of each other, so that it may be possible to have the same plane spanned by different pairs. But this is not the case, because each pair corresponds with two opposite faces of a cube cell in the tes, and in the 3-tes compound, none of the square faces from different tes are coplanar. So there cannot be any coincident planes, either.

(I hope I didn't make a mistake in my reasoning.)
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Re: Johnsonian Polytopes

Postby Klitzing » Wed Jan 09, 2013 10:29 pm

Found it to be much easier! There is a compound of 3 hex inscribed into an ico. Just splitting the vertex count. Now take the dual tes instead of each individual hex. This generates an compound of 3 tes. Moreover this compound of 3 tes again is vertex inscribed into ico. From the vertex counts it becomes obvious that this compound of 3 tes does visit each ico vertex twice. None the less no 2 of those tes are completely coincident.

Next consider the subsymmetrical interpretation of tes as a 4-dip (by reducing the cubes to 4-prisms). We already saw that this could be done in 3 different ways. In fact this applies to the first cube, but this first choice already determines the rest.

Thus we could consider the ico inscribed compound of 3x3 4-dips. Here it is evident that the 3 4-dips within the tes are completely coincident, and therefore no 4-dip within one tes would conflict with a 4-dip of an other tes within ico.

Now transfer this observation to the higher Wythoffian analoges of original interest and we are done. No odip of one srit would conflict with an odip of an other srit within the common spic. But to the contrary, the 3 odips within srit no longer fall together any longer (as the 4-dips did within tes).

--- rk
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Re: Johnsonian Polytopes

Postby Klitzing » Wed Jan 09, 2013 11:07 pm

Got some special interest into your cyclotetra-... So I investigated that one further. Here are 3 specially interesting CRFs:

Name: cyclotetradiminished small rhombitesseract
OBSA could be: cyted srit :?: (@hedrondude)
In fact it can be seen likewise as cyclotetraugmented octagonal duoprism

Pattern
(parts of total size: 8x8 squares)

Code: Select all
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |
1   B   1       1   B   1       1
| / : \ |       | / : \ |       |
A---3---A---4---A---3---A---4---A-...
|   :   |       |   :   |       |
2   :   2       2   :   2       2
|   :   |       |   :   |       |
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |
1   B   1       1   B   1       1
| / : \ |       | / : \ |       |
A---3---A---4---A---3---A---4---A-...
|   :   |       |   :   |       |
2   :   2       2   :   2       2
|   :   |       |   :   |       |
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |


Incidence matrix
Code: Select all
64  * |  1  1  1  1  1  0 | 1  1  1  1  1  1  1  1 0 | 1 1  1  1 1 odip vertices (A)
 * 16 |  0  0  0  0  4  2 | 0  0  0  0  0  2  2  4 1 | 0 0  1  2 2 vertices of B-squares
------+-------------------+--------------------------+------------
 2  0 | 32  *  *  *  *  * | 0  1  1  0  0  1  0  0 0 | 1 0  1  0 1 (1)
 2  0 |  * 32  *  *  *  * | 0  0  0  1  1  0  0  1 0 | 0 1  0  1 1 (2)
 2  0 |  *  * 32  *  *  * | 1  1  0  1  0  0  1  0 0 | 1 1  1  1 0 (3)
 2  0 |  *  *  * 32  *  * | 1  0  1  0  1  0  0  0 0 | 1 1  0  0 1 (4)
 1  1 |  *  *  *  * 64  * | 0  0  0  0  0  1  1  1 0 | 0 0  1  1 1 (/,\)
 0  2 |  *  *  *  *  * 16 | 0  0  0  0  0  0  0  2 1 | 0 0  0  1 2 (:)
------+-------------------+--------------------------+------------
 8  0 |  0  0  4  4  0  0 | 8  *  *  *  *  *  *  * * | 1 1  0  0 0
 4  0 |  2  0  2  0  0  0 | * 16  *  *  *  *  *  * * | 1 0  1  0 0
 4  0 |  2  0  0  2  0  0 | *  * 16  *  *  *  *  * * | 1 0  0  0 1
 4  0 |  0  2  2  0  0  0 | *  *  * 16  *  *  *  * * | 0 1  0  1 0
 4  0 |  0  2  0  2  0  0 | *  *  *  * 16  *  *  * * | 0 1  0  0 1
 2  1 |  1  0  0  0  2  0 | *  *  *  *  * 32  *  * * | 0 0  1  0 1
 2  1 |  0  0  1  0  2  0 | *  *  *  *  *  * 32  * * | 0 0  1  1 0
 2  2 |  0  1  0  0  2  1 | *  *  *  *  *  *  * 32 * | 0 0  0  1 1
 0  4 |  0  0  0  0  0  4 | *  *  *  *  *  *  *  * 4 | 0 0  0  0 2 B-squares
------+-------------------+--------------------------+------------
16  0 |  8  0  8  8  0  0 | 2  4  4  0  0  0  0  0 0 | 4 *  *  * * op
16  0 |  0  8  8  8  0  0 | 2  0  0  4  4  0  0  0 0 | * 4  *  * * op
 4  1 |  2  0  2  0  4  0 | 0  1  0  0  0  2  2  0 0 | * * 16  * * squippy
 4  2 |  0  2  2  0  4  1 | 0  0  0  1  0  0  2  2 0 | * *  * 16 * trip
16  8 |  8  8  0  8 16  8 | 0  0  4  0  4  8  0  8 2 | * *  *  * 4 sirco




Name: cyclotetragyrated small rhombitesseract
OBSA could be: cyte gysrit :?: (@hedrondude)
In fact is also some type of bicyclotetraugmented octagonal duoprism

Pattern
(parts of total size: 8x8 squares)

Code: Select all
A---3---A---4---A---3---A---4---A-...
| \ : / | \   / | \ : / | \   / |
1===B===1===C===1===B===1===C===1=
| / : \ | /   \ | / : \ | /   \ |
A---3---A---4---A---3---A---4---A-...
|   :   |       |   :   |       |
2   :   2       2   :   2       2
|   :   |       |   :   |       |
A---3---A---4---A---3---A---4---A-...
| \ : / | \   / | \ : / | \   / |
1===B===1===C===1===B===1===C===1=
| / : \ | /   \ | / : \ | /   \ |
A---3---A---4---A---3---A---4---A-...
|   :   |       |   :   |       |
2   :   2       2   :   2       2
|   :   |       |   :   |       |
A---3---A---4---A---3---A---4---A-...
| \ : / | \   / | \ : / | \   / |


Incidence matrix
Code: Select all
64  *  * |  1  1  1  1  1  1  0  0 |  1  1  1  1  1  1  1  1  1  1 0 0 |  1  1  1  1 1 1 odip vertices (A)
 * 16  * |  0  0  0  0  4  0  2  0 |  0  0  0  0  2  2  0  0  4  0 1 0 |  1  0  2  0 2 0 vertices of B-squares
 *  * 16 |  0  0  0  0  0  4  0  2 |  0  0  0  0  0  0  2  2  0  4 0 1 |  0  1  0  2 0 2 vertices of C-squares
---------+-------------------------+-----------------------------------+----------------
 2  0  0 | 32  *  *  *  *  *  *  * |  1  1  0  0  1  0  1  0  0  0 0 0 |  1  1  0  1 1 0 (1)
 2  0  0 |  * 32  *  *  *  *  *  * |  0  0  1  1  0  0  0  0  1  0 0 0 |  0  0  1  0 1 1 (2)
 2  0  0 |  *  * 32  *  *  *  *  * |  1  0  1  0  0  1  0  0  0  1 0 0 |  1  0  1  1 0 1 (3)
 2  0  0 |  *  *  * 32  *  *  *  * |  0  1  0  1  0  0  0  1  0  0 0 0 |  0  1  0  0 1 1 (4)
 1  1  0 |  *  *  *  * 64  *  *  * |  0  0  0  0  1  1  0  0  1  0 0 0 |  1  0  1  0 1 0
 1  0  1 |  *  *  *  *  * 64  *  * |  0  0  0  0  0  0  1  1  0  1 0 0 |  0  1  0  1 0 1
 0  2  0 |  *  *  *  *  *  * 16  * |  0  0  0  0  0  0  0  0  2  0 1 0 |  0  0  1  0 2 0 (:)
 0  0  2 |  *  *  *  *  *  *  * 16 |  0  0  0  0  0  0  0  0  0  2 0 1 |  0  0  0  1 0 2 (=)
---------+-------------------------+-----------------------------------+----------------
 4  0  0 |  2  0  2  0  0  0  0  0 | 16  *  *  *  *  *  *  *  *  * * * |  1  0  0  1 0 0
 4  0  0 |  2  0  0  2  0  0  0  0 |  * 16  *  *  *  *  *  *  *  * * * |  0  1  0  0 1 0
 4  0  0 |  0  2  2  0  0  0  0  0 |  *  * 16  *  *  *  *  *  *  * * * |  0  0  1  0 0 1
 4  0  0 |  0  2  0  2  0  0  0  0 |  *  *  * 16  *  *  *  *  *  * * * |  0  0  0  0 1 1
 2  1  0 |  1  0  0  0  2  0  0  0 |  *  *  *  * 32  *  *  *  *  * * * |  1  0  0  0 1 0
 2  1  0 |  0  0  1  0  2  0  0  0 |  *  *  *  *  * 32  *  *  *  * * * |  1  0  1  0 0 0
 2  0  1 |  1  0  0  0  0  2  0  0 |  *  *  *  *  *  * 32  *  *  * * * |  0  1  0  1 0 0
 2  0  1 |  0  0  0  1  0  2  0  0 |  *  *  *  *  *  *  * 32  *  * * * |  0  1  0  0 0 1
 2  2  0 |  0  1  0  0  2  0  1  0 |  *  *  *  *  *  *  *  * 32  * * * |  0  0  1  0 1 0
 2  0  2 |  0  0  1  0  0  2  0  1 |  *  *  *  *  *  *  *  *  * 32 * * |  0  0  0  1 0 1
 0  4  0 |  0  0  0  0  0  0  4  0 |  *  *  *  *  *  *  *  *  *  * 4 * |  0  0  0  0 2 0 B-squares
 0  0  4 |  0  0  0  0  0  0  0  4 |  *  *  *  *  *  *  *  *  *  * * 4 |  0  0  0  0 0 2 C-squares
---------+-------------------------+-----------------------------------+----------------
 4  1  0 |  2  0  2  0  4  0  0  0 |  1  0  0  0  2  2  0  0  0  0 0 0 | 16  *  *  * * * B-squippy
 4  0  1 |  2  0  0  2  0  4  0  0 |  0  1  0  0  0  0  2  2  0  0 0 0 |  * 16  *  * * * C-squippy
 4  2  0 |  0  2  2  0  4  0  1  0 |  0  0  1  0  0  2  0  0  2  0 0 0 |  *  * 16  * * * B-trip
 4  0  2 |  2  0  2  0  0  4  0  1 |  1  0  0  0  0  0  2  0  0  2 0 0 |  *  *  * 16 * * C-trip
16  8  0 |  8  8  0  8 16  0  8  0 |  0  4  0  4  8  0  0  0  8  0 2 0 |  *  *  *  * 4 * B-sirco
16  0  8 |  0  8  8  8  0 16  0  8 |  0  0  4  4  0  0  0  8  0  8 0 2 |  *  *  *  * * 4 C-sirco




Name: bicyclotetragyrated small rhombitesseract
OBSA could be: bicyte gysrit :?: (@hedrondude)
In fact it could be given also as some type of bicyclotetraugmented octagonal duoprism

Pattern
(parts of total size: 8x8 squares)

Code: Select all
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |
1   B   1       1   B   1       1
| / : \ |       | / : \ |       |
A---3---A---4---A---3---A---4---A-...
|   :   | \   / |   :   | \   / |
2===:===2===C===2===:===2===C===2=
|   :   | /   \ |   :   | /   \ |
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |
1   B   1       1   B   1       1
| / : \ |       | / : \ |       |
A---3---A---4---A---3---A---4---A-...
|   :   | \   / |   :   | \   / |
2===:===2===C===2===:===2===C===2=
|   :   | /   \ |   :   | /   \ |
A---3---A---4---A---3---A---4---A-...
| \ : / |       | \ : / |       |

(pattern-symmetry here equates 1=4, 2=3, B=C!)

Incidence matrix
Code: Select all
64  * |  2  2   2  0 |  1  1  1  2  2  2 0 |  2  2 2 odip vertices (A)
 * 32 |  0  0   4  2 |  0  0  0  2  2  4 1 |  1  2 2 vertices of squares (B,C)
------+--------------+---------------------+--------
 2  0 | 64  *   *  * |  1  1  0  1  0  0 0 |  1  0 2 (1,4)
 2  0 |  * 64   *  * |  0  1  1  0  1  1 0 |  1  2 1 (2,3)
 1  1 |  *  * 128  * |  0  0  0  1  1  1 0 |  1  1 1
 0  2 |  *  *   * 32 |  0  0  0  0  0  2 1 |  0  1 2 (:,=)
------+--------------+---------------------+--------
 4  0 |  4  0   0  0 | 16  *  *  *  *  * * |  0  0 2
 4  0 |  2  2   0  0 |  * 32  *  *  *  * * |  1  0 1
 4  0 |  0  4   0  0 |  *  * 16  *  *  * * |  0  2 0 (*)
 2  1 |  1  0   2  0 |  *  *  * 64  *  * * |  1  0 1
 2  1 |  0  1   2  0 |  *  *  *  * 64  * * |  1  1 0
 2  2 |  0  1   2  1 |  *  *  *  *  * 64 * |  0  1 1
 0  4 |  0  0   0  4 |  *  *  *  *  *  * 8 |  0  0 2
------+--------------+---------------------+--------
 4  1 |  2  2   4  0 |  0  1  0  2  2  0 0 | 32  * * squippy
 4  2 |  0  4   4  1 |  0  0  1  0  2  2 0 |  * 32 * trip
16  8 | 16  8  16  8 |  4  4  0  8  0  8 2 |  *  * 8 sirco

(*: do those squares really exist, or do those 2 trips lie in the same realm, i.e. becoming gyrobifastigia?)


Btw. pattern here always relate to the usual display of the toroid surface as a square (which has to be connected right to left (tube) and bottom to top (ring)). With respect to duoprisms this is just the coat of squares. The prisms then are to be seen before as horizontal stack respectively behind as vertical stack. Here the mere square pattern (vertices A) becomes a bit more complex as to display also those vertices to be introduced into some of the prisms (B resp. C vertices). But I think this now should be understandable even so.

--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Thu Jan 10, 2013 1:03 am

Klitzing wrote:Got some special interest into your cyclotetra-... So I investigated that one further. Here are 3 specially interesting CRFs:

Name: cyclotetradiminished small rhombitesseract
OBSA could be: cyted srit :?: (@hedrondude)
In fact it can be seen likewise as cyclotetraugmented octagonal duoprism

Yeah, I think it's easiest to understand it as a cyclotetraaugmented 8,8-duoprism. In one ring, it would have 4 rhombicuboctahedra, 16 triangular prisms, and 16 square pyramids. In the other ring, 8 octagonal prisms.

[...]
Name: cyclotetragyrated small rhombitesseract
OBSA could be: cyte gysrit :?: (@hedrondude)
In fact is also some type of bicyclotetraugmented octagonal duoprism

This one is interesting. It is equivalent to a srit in which you cut off one ring from the other (in the process cut the octahedra into square pyramids), then rotate the entire ring 45°, and then glue the two rings back together. It's sortof like a "misaligned" srit. :) It has 8 rhombicuboctahedra, 32 triangular prisms, and 32 square pyramids (did I miscount?).

[...]Name: bicyclotetragyrated small rhombitesseract
OBSA could be: bicyte gysrit :?: (@hedrondude)
In fact it could be given also as some type of bicyclotetraugmented octagonal duoprism

Hmm. Are you sure this isn't the same thing as a rotated srit?

P.S. Actually, now that I think of it carefully, it should be distinct, because the (mono)cyclotetragyration splits up the octahedra into 4pyramids and rotates them so that they lie between two 4pyramids on the other ring. The second gyration does not rotate them back; the rotation is on a different plane and moves the 4pyramids out of the original rotation plane, so that they now lie between the vertices of four 4pyramids in the other ring. So they should form a kind of checkerboard pattern over the surface of the polychoron.

Interesting!! I should make a model of this and render it!

[...]
(*: do those squares really exist, or do those 2 trips lie in the same realm, i.e. becoming gyrobifastigia?)[...]

I'm quite sure none of the triangular prisms will be co-realmar, so they will not form gyrobifastigia.

It should be simple to check, because the dichoral angle of the prisms within a ring in the 8,8-duoprism is 135°, and the dichoral angle of prisms between two different rings is always 90°. There cannot be adjacent augments within the same ring, because it will be non-convex, so the only possibility for potential cell merging is between augments on different rings. So for the triangular prisms to form gyrobifastigia, the dichoral angle of the triangular prism with the octagonal prism in 8prism||square must be exactly 45°. But they are not; only the square pyramids have 45° angle with the 8prism, so when you glue the augments on the 8,8-duoprism, only adjacent square pyramids will merge into octahedra. The square pyramids will not merge with the 3prisms, and the 3prisms also will not merge among themselves. (The orientation in which the augment is glued on is irrelevant, because rotating in the plane of the octagonal faces does not change the dichoral angle of the lateral cells with the base.)
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Re: Johnsonian Polytopes

Postby Klitzing » Thu Jan 10, 2013 10:39 am

quickfur wrote:
Klitzing wrote:Got some special interest into your cyclotetra-... So I investigated that one further. Here are 3 specially interesting CRFs:

Name: cyclotetradiminished small rhombitesseract
OBSA could be: cyted srit :?: (@hedrondude)
In fact it can be seen likewise as cyclotetraugmented octagonal duoprism

Yeah, I think it's easiest to understand it as a cyclotetraaugmented 8,8-duoprism. In one ring, it would have 4 rhombicuboctahedra, 16 triangular prisms, and 16 square pyramids. In the other ring, 8 octagonal prisms.
[..]
In that it resembles somehow to ordinary 4D prisms, having 3D prisms between the bases. But here the orientation of the 3D prisms is different, they align in a great circle. And there aren't 2 bases either. Those combine here to a single further great circle.
[...]
[...]Name: cyclotetragyrated small rhombitesseract
OBSA could be: cyte gysrit :?: (@hedrondude)
In fact is also some type of bicyclotetraugmented octagonal duoprism

This one is interesting. It is equivalent to a srit in which you cut off one ring from the other (in the process cut the octahedra into square pyramids), then rotate the entire ring 45°, and then glue the two rings back together. It's sortof like a "misaligned" srit. :) It has 8 rhombicuboctahedra, 32 triangular prisms, and 32 square pyramids (did I miscount?).

Very easy to cross-check your numbers, just read my given incidence matrix! :P (Yes your counts are correct.)
Yes, that makes it kind of similar to the triangular orthobicupola: cut into 2, one half gyrated, glued back. Just that here we don't use an equatorial, i.e. spherical cut, but a toroidal one!
[...]
[...]Name: bicyclotetragyrated small rhombitesseract
OBSA could be: bicyte gysrit :?: (@hedrondude)
In fact it could be given also as some type of bicyclotetraugmented octagonal duoprism

Hmm. Are you sure this isn't the same thing as a rotated srit?
[...]
No! Just confer the provided pattern: for a srit the B and the C vertices would fall into the same squares. Only then the 2 squippies could connect to octahedra.
[...]
P.S. Actually, now that I think of it carefully, it should be distinct, because the (mono)cyclotetragyration splits up the octahedra into 4pyramids and rotates them so that they lie between two 4pyramids on the other ring. The second gyration does not rotate them back; the rotation is on a different plane and moves the 4pyramids out of the original rotation plane, so that they now lie between the vertices of four 4pyramids in the other ring. So they should form a kind of checkerboard pattern over the surface of the polychoron.

Interesting!! I should make a model of this and render it!
[...]
Always interested in your pics. Esp. when you render my findings. :D
[...]
[...]
(*: do those squares really exist, or do those 2 trips lie in the same realm, i.e. becoming gyrobifastigia?)[...]

I'm quite sure none of the triangular prisms will be co-realmar, so they will not form gyrobifastigia.

It should be simple to check, because the dichoral angle of the prisms within a ring in the 8,8-duoprism is 135°, and the dichoral angle of prisms between two different rings is always 90°. There cannot be adjacent augments within the same ring, because it will be non-convex, so the only possibility for potential cell merging is between augments on different rings. So for the triangular prisms to form gyrobifastigia, the dichoral angle of the triangular prism with the octagonal prism in 8prism||square must be exactly 45°. But they are not; only the square pyramids have 45° angle with the 8prism, so when you glue the augments on the 8,8-duoprism, only adjacent square pyramids will merge into octahedra. The square pyramids will not merge with the 3prisms, and the 3prisms also will not merge among themselves. (The orientation in which the augment is glued on is irrelevant, because rotating in the plane of the octagonal faces does not change the dichoral angle of the lateral cells with the base.)

Yes, you are right. The relevant base ops are at 90°. So the angle between the trip and the op in 4g||8p counts. That one I did just calculate to be atan(1/sqrt(2))=35.264°. And the angle between squippy and op is 45°. Therefore we have 35.264°+90°+35.264°=160.528°<180°, but 45°+90°+45°=180°, i.e. those squippies become corealmic and combine into octahedra, whereas those trips fall back to become corealmic. (Btw. that trip-trip angle well could have become greater than 180°, I feared, and then the polychoron wouldn't be convex anymore. But now that angle is proven to be still convex, uff!)

--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Thu Jan 10, 2013 7:38 pm

Klitzing wrote:
quickfur wrote:[...]
P.S. Actually, now that I think of it carefully, it should be distinct, because the (mono)cyclotetragyration splits up the octahedra into 4pyramids and rotates them so that they lie between two 4pyramids on the other ring. The second gyration does not rotate them back; the rotation is on a different plane and moves the 4pyramids out of the original rotation plane, so that they now lie between the vertices of four 4pyramids in the other ring. So they should form a kind of checkerboard pattern over the surface of the polychoron.

Interesting!! I should make a model of this and render it!
[...]
Always interested in your pics. Esp. when you render my findings. :D

I really should work on that soon, both this one, and the bi-24-dim-x5oox that you discovered. I've been meaning to since 2 days ago, but got busy with other things.

[...](Btw. that trip-trip angle well could have become greater than 180°, I feared, and then the polychoron wouldn't be convex anymore. But now that angle is proven to be still convex, uff!)[...]

I'm not too worried about that, actually, because I've actually made a model of the paratetragyrated x4oxo, so if the angle was >180°, the convex hull algo would have produced something completely different from the images I posted. The fact that the rendered image was what I expected is proof enough for me that the construction is valid. :) (But of course, you'd need actual mathematical proof in a research paper.)
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Re: Johnsonian Polytopes

Postby Klitzing » Thu Jan 10, 2013 8:35 pm

quickfur wrote:[...]
I'm not too worried about that, actually, because I've actually made a model of the paratetragyrated x4oxo, so if the angle was >180°, the convex hull algo would have produced something completely different from the images I posted. The fact that the rendered image was what I expected is proof enough for me that the construction is valid. :) (But of course, you'd need actual mathematical proof in a research paper.)

:lol: great fun, that pragmatix!
Well, even then it would be a valid and interesting polychoron, just not a CRF.
(But now we know it is convex, though.)
--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Thu Jan 10, 2013 10:25 pm

Alrighty, some images are in order!

Image

Here's a snapshot of the octagyrated x4o3x3o. I wanted to see the checkerboard pattern of square pyramids, so I used my program to make a graph of how the pyramids are connected, then made a 2-coloring of them. This image has visibility clipping turned on, so some of the pyramids can't be seen, but you can tell where some of them are by the isolated colored polygons in various places.

Next, let's look at the cells that were culled (i.e., all the cells facing away from the 4D viewpoint, and none of the cells facing the 4D viewpoint):

Image

These are all the square pyramid cells on the far side.

Alright. Now let's look at the rhombicuboctahedral cells:

Image

This is the only rhombicuboctahedron visible on the near side, although you can see the outline of some of the other rhombicuboctahedra if you look at the edges on the limb.

Let's look at the rhombicuboctahedra on the far side:

Image

I went all-out in the coloring scheme here. Which hopefully is helpful to see all the rhombicuboctahedral cells distinctly. As you can see, they are all connected to each other in "twisted" orientations. Pretty fascinating!
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Re: Johnsonian Polytopes

Postby Klitzing » Thu Jan 10, 2013 11:49 pm

quickfur wrote:Alrighty, some images are in order![...]
Great!
[...]Here's a snapshot of the octagyrated x4o3x3o.[...]
i.e. ...bi-cyclotetra-gyrated... :nod:
[...]I went all-out in the coloring scheme here. Which hopefully is helpful to see all the rhombicuboctahedral cells distinctly. As you can see, they are all connected to each other in "twisted" orientations. Pretty fascinating!

Not completely. There should be 2 rings of sircoes, each one is aligned perfectly. On the other hand it is that their 4 equatorial non-cubical squares (of each sirco) do cross-connect through that toroidal manifold of odip-squares to same ones of the other, orthogonal ring.

So it might look better, just as you've already colored the squippies of one ring in yellow and those of the other ring in red, that you'd similarily color the sircoes of one ring in one color and the sircoes of the other ring in a different one.
Perhaps you even could complement that by a third pic, featuring the trips; again those of one ring in one color, those of the other in a second one. Those rather will serv as digonal cupolae, cross-conected with their square-base to the ones of the other, orthogonal ring.

In fact, it might serve for guidance of visualization also to have a zeroth pic highlighting only all the 64 odip-squares of that toroidal boundary between those 2 rings. (Or, even better, having those 3 colored: 16, which cross-connect those sircoes (i.e. the 1-4 squares of my pattern), 16, which cross-connect those trips (i.e. the 2-3 squares of my pattern), and 32 which are the various squippy bases (i.e. the 1-3 and the 2-4 ones).)

--- rk
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Re: Johnsonian Polytopes

Postby quickfur » Fri Jan 11, 2013 12:52 am

Klitzing wrote:
quickfur wrote:[...]I went all-out in the coloring scheme here. Which hopefully is helpful to see all the rhombicuboctahedral cells distinctly. As you can see, they are all connected to each other in "twisted" orientations. Pretty fascinating!

Not completely. There should be 2 rings of sircoes, each one is aligned perfectly. On the other hand it is that their 4 equatorial non-cubical squares (of each sirco) do cross-connect through that toroidal manifold of odip-squares to same ones of the other, orthogonal ring.

So it might look better, just as you've already colored the squippies of one ring in yellow and those of the other ring in red, that you'd similarily color the sircoes of one ring in one color and the sircoes of the other ring in a different one.
Perhaps you even could complement that by a third pic, featuring the trips; again those of one ring in one color, those of the other in a second one. Those rather will serv as digonal cupolae, cross-conected with their square-base to the ones of the other, orthogonal ring.

Hmm. The triangular prisms will be problematic, because there are a lot of them, and showing all of them at once will probably clutter the image and make it hard to understand. I'll have to think about that one.

For the two rings of sircoes, I think it's best to separate them out in different images. Here's one cycle, which lies on the far side (so I culled the near side cells, otherwise there will be too many tangled edges):

Image

Here's the other cycle:

Image

Due to the overlapping in the projection images, I had to use different textures for the 4 cells. The nearest to the 4D viewpoint is outlined in blue, and the rest are in shades of red. I'm not very happy with the result though --- when I get home tonight I'll see if I can improve it.

Showing both cycles at the same time doesn't produce a good image, because they share too many faces and there's not enough blank space in the projection volume to see things clearly.

In fact, it might serve for guidance of visualization also to have a zeroth pic highlighting only all the 64 odip-squares of that toroidal boundary between those 2 rings. (Or, even better, having those 3 colored: 16, which cross-connect those sircoes (i.e. the 1-4 squares of my pattern), 16, which cross-connect those trips (i.e. the 2-3 squares of my pattern), and 32 which are the various squippy bases (i.e. the 1-3 and the 2-4 ones).)

--- rk

I have to run now, I'll get to this later.
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Re: Johnsonian Polytopes

Postby Klitzing » Sat Jan 12, 2013 5:17 pm

There is also further quite interesting figure in this gyration series of srit:
Name: bipara(bi)gyrated small rhombitesseract
OBSA: bipgy srit (? @hedrondude)
(again some type of bicyclotetraugmented octagonal duoprism)

Pattern
(parts of total size: 8x8 squares)

Code: Select all
A---7---A---8---H---9---H---8---A-...
| \ : / |       | \ : / |       |
1==B,E==1=======4===F===4=======1=  (EF.. remains)
| / : \ |       | / : \ |       |
A---7---A---8---H---9---H---8---A-...
|   :   |       | \ : / |       |
2   :   2       5   D   5       2
|   :   |       | / : \ |       |
H---a---H---b---I---c---I---b---H-...
| \ : / | \   / |   :   | \   / |
3===C===3===G===6===:===6===G===3=  (GG.. gyrated)
| / : \ | /   \ |   :   | /   \ |
H---a---H---b---I---c---I---b---H-...
|   :   |       | \ : / |       |
2   :   2       5   D   5       2
|   :   |       | / : \ |       |
A---7---A---8---H---9---H---8---A-...
| \ : / |       | \ : / |       |

   (BC..           (DD..            (etc.)
 remains)        gyrated)

(pattern-symmetry equates 1=7, 2=8, 3=9, 4=a, 5=b, 6=c, B=E, C=F, D=G!)

Incidence matrix
Code: Select all
16  *  * * *  * |  2  2  0  0  0  0  2  0  0  0  0 0 0 |  2  1 0  0 0  2  2  0  0  0  0  0  2  0  0 0 0 | 1 0  0  2 0 0 2 (A)
 * 32  * * *  * |  0  1  1  1  1  0  0  1  1  0  0 0 0 |  1  1 1  1 0  0  0  1  1  1  1  0  1  1  0 0 0 | 0 1  1  1 1 0 2 (H)
 *  * 16 * *  * |  0  0  0  0  2  2  0  0  0  2  0 0 0 |  0  1 0  2 1  0  0  0  0  0  2  2  0  0  2 0 0 | 0 0  2  0 0 2 2 (I)
 *  *  * 8 *  * |  0  0  0  0  0  0  4  0  0  0  2 0 0 |  0  0 0  0 0  2  2  0  0  0  0  0  4  0  0 1 0 | 1 0  0  2 0 0 2 (B,E)
 *  *  * * 8  * |  0  0  0  0  0  0  0  4  0  0  2 0 0 |  0  0 0  0 0  0  0  2  2  0  0  0  4  0  0 1 0 | 0 1  0  2 0 0 2 (C,F)
 *  *  * * * 16 |  0  0  0  0  0  0  0  0  2  2  0 1 1 |  0  0 0  0 0  0  0  0  0  1  2  1  0  2  2 0 1 | 0 0  1  0 1 1 2 (D,G)
----------------+--------------------------------------+------------------------------------------------+----------------
 2  0  0 0 0  0 | 16  *  *  *  *  *  *  *  *  *  * * * |  1  0 0  0 0  1  1  0  0  0  0  0  0  0  0 0 0 | 1 0  0  1 0 0 1 (1,7)
 1  1  0 0 0  0 |  * 32  *  *  *  *  *  *  *  *  * * * |  1  1 0  0 0  0  0  0  0  0  0  0  1  0  0 0 0 | 0 0  0  1 0 0 2 (2,8)
 0  2  0 0 0  0 |  *  * 16  *  *  *  *  *  *  *  * * * |  0  0 1  1 0  0  0  1  0  1  0  0  0  0  0 0 0 | 0 1  1  0 1 0 1 (3,9)
 0  2  0 0 0  0 |  *  *  * 16  *  *  *  *  *  *  * * * |  1  0 1  0 0  0  0  0  1  0  0  0  0  1  0 0 0 | 0 1  0  1 1 0 1 (4,a)
 0  1  1 0 0  0 |  *  *  *  * 32  *  *  *  *  *  * * * |  0  1 0  1 0  0  0  0  0  0  1  0  0  0  0 0 0 | 0 0  1  0 0 0 2 (5,b)
 0  0  2 0 0  0 |  *  *  *  *  * 16  *  *  *  *  * * * |  0  0 0  1 1  0  0  0  0  0  0  1  0  0  1 0 0 | 0 0  1  0 0 2 1 (6,c)
 1  0  0 1 0  0 |  *  *  *  *  *  * 32  *  *  *  * * * |  0  0 0  0 0  1  1  0  0  0  0  0  1  0  0 0 0 | 1 0  0  1 0 0 1
 0  1  0 0 1  0 |  *  *  *  *  *  *  * 32  *  *  * * * |  0  0 0  0 0  0  0  1  1  0  0  0  1  0  0 0 0 | 0 1  0  1 0 0 1
 0  1  0 0 0  1 |  *  *  *  *  *  *  *  * 32  *  * * * |  0  0 0  0 0  0  0  0  0  1  1  0  0  1  0 0 0 | 0 0  1  0 1 0 1
 0  0  1 0 0  1 |  *  *  *  *  *  *  *  *  * 32  * * * |  0  0 0  0 0  0  0  0  0  0  1  1  0  0  1 0 0 | 0 0  1  0 0 1 1
 0  0  0 1 1  0 |  *  *  *  *  *  *  *  *  *  * 16 * * |  0  0 0  0 0  0  0  0  0  0  0  0  2  0  0 1 0 | 0 0  0  1 0 0 2
 0  0  0 0 0  2 |  *  *  *  *  *  *  *  *  *  *  * 8 * |  0  0 0  0 0  0  0  0  0  0  0  0  0  2  0 0 1 | 0 0  0  0 1 0 2
 0  0  0 0 0  2 |  *  *  *  *  *  *  *  *  *  *  * * 8 |  0  0 0  0 0  0  0  0  0  0  0  0  0  0  2 0 1 | 0 0  0  0 0 1 2
----------------+--------------------------------------+------------------------------------------------+----------------
 2  2  0 0 0  0 |  1  2  0  1  0  0  0  0  0  0  0 0 0 | 16  * *  * *  *  *  *  *  *  *  *  *  *  * * * | 0 0  0  1 0 0 1
 1  2  1 0 0  0 |  0  2  0  0  2  0  0  0  0  0  0 0 0 |  * 16 *  * *  *  *  *  *  *  *  *  *  *  * * * | 0 0  0  0 0 0 2
 0  4  0 0 0  0 |  0  0  2  2  0  0  0  0  0  0  0 0 0 |  *  * 8  * *  *  *  *  *  *  *  *  *  *  * * * | 0 1  0  0 1 0 0
 0  2  2 0 0  0 |  0  0  1  0  2  1  0  0  0  0  0 0 0 |  *  * * 16 *  *  *  *  *  *  *  *  *  *  * * * | 0 0  1  0 0 0 1
 0  0  4 0 0  0 |  0  0  0  0  0  4  0  0  0  0  0 0 0 |  *  * *  * 4  *  *  *  *  *  *  *  *  *  * * * | 0 0  0  0 0 2 0
 2  0  0 1 0  0 |  1  0  0  0  0  0  2  0  0  0  0 0 0 |  *  * *  * * 16  *  *  *  *  *  *  *  *  * * * | 1 0  0  1 0 0 0
 2  0  0 1 0  0 |  1  0  0  0  0  0  2  0  0  0  0 0 0 |  *  * *  * *  * 16  *  *  *  *  *  *  *  * * * | 1 0  0  0 0 0 1
 0  2  0 0 1  0 |  0  0  1  0  0  0  0  2  0  0  0 0 0 |  *  * *  * *  *  * 16  *  *  *  *  *  *  * * * | 0 1  0  0 0 0 1
 0  2  0 0 1  0 |  0  0  0  1  0  0  0  2  0  0  0 0 0 |  *  * *  * *  *  *  * 16  *  *  *  *  *  * * * | 0 1  0  1 0 0 0
 0  2  0 0 0  1 |  0  0  1  0  0  0  0  0  2  0  0 0 0 |  *  * *  * *  *  *  *  * 16  *  *  *  *  * * * | 0 0  1  0 1 0 0
 0  1  1 0 0  1 |  0  0  0  0  1  0  0  0  1  1  0 0 0 |  *  * *  * *  *  *  *  *  * 32  *  *  *  * * * | 0 0  1  0 0 0 1
 0  0  2 0 0  1 |  0  0  0  0  0  1  0  0  0  2  0 0 0 |  *  * *  * *  *  *  *  *  *  * 16  *  *  * * * | 0 0  1  0 0 1 0
 1  1  0 1 1  0 |  0  1  0  0  0  0  1  1  0  0  1 0 0 |  *  * *  * *  *  *  *  *  *  *  * 32  *  * * * | 0 0  0  1 0 0 1
 0  2  0 0 0  2 |  0  0  0  1  0  0  0  0  2  0  0 1 0 |  *  * *  * *  *  *  *  *  *  *  *  * 16  * * * | 0 0  0  0 1 0 1
 0  0  2 0 0  2 |  0  0  0  0  0  1  0  0  0  2  0 0 1 |  *  * *  * *  *  *  *  *  *  *  *  *  * 16 * * | 0 0  0  0 0 1 1
 0  0  0 2 2  0 |  0  0  0  0  0  0  0  0  0  0  4 0 0 |  *  * *  * *  *  *  *  *  *  *  *  *  *  * 4 * | 0 0  0  0 0 0 2
 0  0  0 0 0  4 |  0  0  0  0  0  0  0  0  0  0  0 2 2 |  *  * *  * *  *  *  *  *  *  *  *  *  *  * * 4 | 0 0  0  0 0 0 2
----------------+--------------------------------------+------------------------------------------------+----------------
 4  0  0 2 0  0 |  4  0  0  0  0  0  8  0  0  0  0 0 0 |  0  0 0  0 0  4  4  0  0  0  0  0  0  0  0 0 0 | 4 *  *  * * * * oct
 0  4  0 0 1  0 |  0  0  2  2  0  0  0  4  0  0  0 0 0 |  0  0 1  0 0  0  0  2  2  0  0  0  0  0  0 0 0 | * 8  *  * * * * squippy (J1)
 0  2  2 0 0  1 |  0  0  1  0  2  1  0  0  2  2  0 0 0 |  0  0 0  1 0  0  0  0  0  1  2  1  0  0  0 0 0 | * * 16  * * * * squippy (J1)
 2  2  0 1 1  0 |  1  2  0  1  0  0  2  2  0  0  1 0 0 |  1  0 0  0 0  1  0  0  1  0  0  0  2  0  0 0 0 | * *  * 16 * * * trip
 0  4  0 0 0  2 |  0  0  2  2  0  0  0  0  4  0  0 1 0 |  0  0 1  0 0  0  0  0  0  2  0  0  0  2  0 0 0 | * *  *  * 8 * * trip
 0  0  4 0 0  2 |  0  0  0  0  0  4  0  0  0  4  0 0 1 |  0  0 0  0 1  0  0  0  0  0  0  2  0  0  2 0 0 | * *  *  * * 8 * trip
 4  8  4 2 2  4 |  2  8  2  2  8  2  4  4  4  4  4 2 2 |  2  4 0  2 0  0  2  2  0  0  4  0  4  2  2 1 1 | * *  *  * * * 8 esquigybcu (J37)


There is just a single of this one, as can be seen from the pattern. Gyrating just alternate ones of the odip augmentations result in all possible combinations at once.
So we get for cells in here octs as well as squippies (J1), trips, and esp. esquigybcu (i.e. J37)!

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Re: Johnsonian Polytopes

Postby quickfur » Sat Jan 12, 2013 5:42 pm

Klitzing wrote:There is also further quite interesting figure in this gyration series of srit:
Name: bipara(bi)gyrated small rhombitesseract
OBSA: bipgy srit (? @hedrondude)
(again some type of bicyclotetraugmented octagonal duoprism) [...]

Isn't this the same as the paratetragyrated srit that I posted about? It has eight J37 cells, 4 octahedra, 24 square pyramids, and 32 triangular prisms.

I guess my naming of it is not that great, because the gyrated segments are in two different rings, so bipara-bigyrated is a better designation. But this does sound like the same thing we were talking about before.
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Re: Johnsonian Polytopes

Postby Klitzing » Sat Jan 12, 2013 5:58 pm

quickfur wrote:
Klitzing wrote:There is also further quite interesting figure in this gyration series of srit:
Name: bipara(bi)gyrated small rhombitesseract
OBSA: bipgy srit (? @hedrondude)
(again some type of bicyclotetraugmented octagonal duoprism) [...]

Isn't this the same as the paratetragyrated srit that I posted about? It has eight J37 cells, 4 octahedra, 24 square pyramids, and 32 triangular prisms.
Might be, was not aware of.
I guess my naming of it is not that great, because the gyrated segments are in two different rings, so bipara-bigyrated is a better designation. But this does sound like the same thing we were talking about before.

No, we were talking about the cyclo-tetra-diminished srit, the cyclo-tetra-gyrated srit, and esp. (with your pics) about the bi-cyclo-tetra-gyrated srit. But this bi-para-bi-gyrated srit is quite different. None of the former, for example, were using J37 cells! They all were using sircoes instead.

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Re: Johnsonian Polytopes

Postby quickfur » Sat Jan 12, 2013 6:09 pm

Yes, but before that we were talking about this, which appears to be exactly the same thing you're describing now (it has eight J37 cells).
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Re: Johnsonian Polytopes

Postby Klitzing » Sat Jan 12, 2013 11:27 pm

admitted :oops:
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Re: Johnsonian Polytopes

Postby quickfur » Sun Jan 13, 2013 12:55 am

At any rate, the x4o3x3o is definitely a gold mine of CRFs, due to all the ways you can diminish/gyrate it. As well as its relationship with the 8,8-duoprism, which gives a few more CRFs via augmentations that don't coincide with a srit diminishing.

The only thing is, I think we should agree on a consistent naming scheme for these things. I have been using the scheme <modifier> + <count> + <operation> + <base name>, where <count> is just a simple count of the number of <operation>s performed. So I got para- (from 24-cell diminishings) + tetra- (total number of diminishings) + diminished + cantellated tesseract (Olshevsky's nomenclature). But I think in this case, it may be better to indicate how many operations are performed on each ring, such as bi-cyclo-bi-diminished or something along those lines. But we need a consistent scheme to generate all these names, in a way that covers all of the possibilities. Any suggestions? :)

I'd like also for the scheme to be generic, if possible, to be at least extensible to cover the 600-cell diminishings (and, by extension, the various kinds of CRF modifications one may do to the 600-cell family uniform polychora). The number of CRF diminishings of the 600-cell is astounding; I haven't been able to enumerate all of them. I think somebody has enumerated the non-adjacent diminishings already (I remember seeing a website that did it), but the 600-cell also admits many other diminishings that delete some adjacent vertices. The bi-24-diminished 600-cell (the one with 48 teddies) is among these. There's also the (pseudo-)bisected 600-cell, and the whole series of 600-cell wedges that I posted about some time back, all of which are CRFs and additionally admit further CRF diminishings, but I think for the sake of naming it's probably OK to consider these separately from the non-bisected 600-cell diminishings.
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Re: Johnsonian Polytopes

Postby Klitzing » Sun Jan 13, 2013 11:37 am

quickfur wrote:At any rate, the x4o3x3o is definitely a gold mine of CRFs, due to all the ways you can diminish/gyrate it. As well as its relationship with the 8,8-duoprism, which gives a few more CRFs via augmentations that don't coincide with a srit diminishing.[...]

In fact, as srit is contained in turn in spic, you'd have even more possibilities in considering its diminishings/gyrations.
[...]
The only thing is, I think we should agree on a consistent naming scheme for these things. I have been using the scheme <modifier> + <count> + <operation> + <base name>, where <count> is just a simple count of the number of <operation>s performed. So I got para- (from 24-cell diminishings) + tetra- (total number of diminishings) + diminished + cantellated tesseract (Olshevsky's nomenclature). But I think in this case, it may be better to indicate how many operations are performed on each ring, such as bi-cyclo-bi-diminished or something along those lines. But we need a consistent scheme to generate all these names, in a way that covers all of the possibilities. Any suggestions? :) [...]

Well we should not only use the total count, but use the internal symmetry as well. So cyclo tetra is unique. But not so para tetra. Therefore I used bi para bi instead. And, as para itself refers to 2 opposite ones, the second bi here is obsolete.
[...]
I'd like also for the scheme to be generic, if possible, to be at least extensible to cover the 600-cell diminishings (and, by extension, the various kinds of CRF modifications one may do to the 600-cell family uniform polychora). The number of CRF diminishings of the 600-cell is astounding; I haven't been able to enumerate all of them. I think somebody has enumerated the non-adjacent diminishings already (I remember seeing a website that did it), but the 600-cell also admits many other diminishings that delete some adjacent vertices. [...]

Me too remember having seen such. Don't not know out of my head where, so. Remember too that there was a computer based research on those diminishings, amounting in several millions.
[...]
The bi-24-diminished 600-cell (the one with 48 teddies) is among these. There's also the (pseudo-)bisected 600-cell,
:?:
and the whole series of 600-cell wedges that I posted about some time back, all of which are CRFs and additionally admit further CRF diminishings, but I think for the sake of naming it's probably OK to consider these separately from the non-bisected 600-cell diminishings.
:?:

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Re: Johnsonian Polytopes

Postby quickfur » Sun Jan 13, 2013 8:32 pm

Klitzing wrote:
quickfur wrote:At any rate, the x4o3x3o is definitely a gold mine of CRFs, due to all the ways you can diminish/gyrate it. As well as its relationship with the 8,8-duoprism, which gives a few more CRFs via augmentations that don't coincide with a srit diminishing.[...]

In fact, as srit is contained in turn in spic, you'd have even more possibilities in considering its diminishings/gyrations.

What is spic?

[...]
The only thing is, I think we should agree on a consistent naming scheme for these things. I have been using the scheme <modifier> + <count> + <operation> + <base name>, where <count> is just a simple count of the number of <operation>s performed. So I got para- (from 24-cell diminishings) + tetra- (total number of diminishings) + diminished + cantellated tesseract (Olshevsky's nomenclature). But I think in this case, it may be better to indicate how many operations are performed on each ring, such as bi-cyclo-bi-diminished or something along those lines. But we need a consistent scheme to generate all these names, in a way that covers all of the possibilities. Any suggestions? :) [...]

Well we should not only use the total count, but use the internal symmetry as well. So cyclo tetra is unique. But not so para tetra. Therefore I used bi para bi instead. And, as para itself refers to 2 opposite ones, the second bi here is obsolete.

I just realized that the same 24-cell diminishing may give rise to multiple srit diminishings, due to the lower symmetry of the srit. For example, the cyclotetradiminished 24-cell corresponds with both the cyclotetradiminished srit and the biparabidiminished srit, by changing the orientation of the resp. 8prism||square segments.

This means that the maximal metadiminishings of the 24-cell may give rise to distinct CRF srit diminishings! This may be worth further exploration.

[...]
I'd like also for the scheme to be generic, if possible, to be at least extensible to cover the 600-cell diminishings (and, by extension, the various kinds of CRF modifications one may do to the 600-cell family uniform polychora). The number of CRF diminishings of the 600-cell is astounding; I haven't been able to enumerate all of them. I think somebody has enumerated the non-adjacent diminishings already (I remember seeing a website that did it), but the 600-cell also admits many other diminishings that delete some adjacent vertices. [...]

Me too remember having seen such. Don't not know out of my head where, so. Remember too that there was a computer based research on those diminishings, amounting in several millions.

Yes, I think 600-cell family polychora will produce the majority of 4D CRFs. :) Duoprism augmentations produce about 3000+ CRFs, but this is nothing compared to what you can get from the 600-cell family!

[...]
The bi-24-diminished 600-cell (the one with 48 teddies) is among these. There's also the (pseudo-)bisected 600-cell,
:?:
and the whole series of 600-cell wedges that I posted about some time back, all of which are CRFs and additionally admit further CRF diminishings, but I think for the sake of naming it's probably OK to consider these separately from the non-bisected 600-cell diminishings.
:?: [...]

See this post.
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Re: Johnsonian Polytopes

Postby Klitzing » Sun Jan 13, 2013 11:01 pm

As to your question:
spic = smal prismated icositetrachoron = x3o4o3x.

As to your provided cross-link:
Even so hemidemi makes a nice ringing rhyme it seems inapropriate for those ex-wedges. Hemi is appropriate for the first cut. Hemi has greek root. The same word with latin root is demi. Thus hemidemi would be something which is a quarter. But your angle between the 2 equator planes is meant to vary.

Wedges are what those are indeed. But then again wedge alone might be not specific enough (on how those are to become wedges, i.e. where the faceting planes are to be chosen).

So we might try a completely different way. Any such cut results individually in a hemispherical remainder. So you are, more specifically, dealing with differnent possibilities of an intersection of hemispheres. Might this result in a better (more specific) naming advice?

That very building procedure itself is quite intersting! The same could most probably be applied to other figures too. E.g. to ico, which has a co as its equatorial section. Thus you'd get 2 tricues for wedge facets, 2*3 squippies (halved octs) and some remaining octs. (I suppose there should be 3 such wedges, according to neighbouring, orthogonal, and meta positions of normal directions, right?) Or to hex, halving an oct as equatorial section. Thus the single case here would result in the quater hex with 2 squippies for wedge facets and some few remaining tets. (In fact, that latter one is nothing but the pyramid above squippy, i.e. squippypy).

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Re: Johnsonian Polytopes

Postby Klitzing » Sun Jan 13, 2013 11:34 pm

Marek14 wrote:Hmm, what's latin for "watermelon slice"? :)


Looked for a translation of wedge (german: keil) into latin. Got cuneus. Same in spanish: la cuña.

On the other hand, your watermelon prefix clearly does not relate to that special fruit. An orange slice might serve here equally well. But both somehow relate onto the position where the sharp edge is: it runs through the former center. This is not being transfered in wedge, keil, ...

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