## Bilbirothawroids (D4.3 to D4.9)

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

Well, one possibility is to have the inner-most rings that are not on a single torus. A simple case would be a ring of cupolas joined by their n-gons and 2n-gons (two basic forms are available, with ortho- and gyro- joins between the cupolae). After all, isn't the first J92 polychoron we found, which contains 4 J92's in a ring, something similar? Then you also get an additional degree of freedom: by having two different "joins" in the ring, the dichoral angle can be different for each.

BTW - is there a name for the shape you cut off (in two copies) from a 600-cell to get a gap?
Marek14
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### Re: Johnsonian Polytopes

quickfur wrote:
Klitzing wrote:I think you got the right feeling, quickfur, both with the relation of the 2 specific finds (according to my second idea) to sidpith and to stawros. But both weren't exact.
[...]
The important point here is for all that stuff to work: that the wedge angle of 2n-gon||n-p and that of n-gon||2n-p is the same!
[...]

Interesting! So I got 2n-gon||n-prism mixed up with n-gon||2n-prism. It's interesting that they have exactly the same wedge angle. It makes sense in retrospect, but it's not something I would've expected just from being told their structure.

The point here is that in the lace city you have a triangle ABB. And the base side BB is unity (as the base represents a prism) and the lacing element represents a cupolae - in either of both directions. Thus AB would have the same length in both cases. - That's all.
--- rk
Klitzing
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### Re: Johnsonian Polytopes

quickfur wrote:More thoughts about CRFs with duoprismic symmetry: suppose you have a polychoron with a ring of n-prisms and an orthogonal ring of m-prisms, with some lacing edges between them. I'm thinking that if the polychoron is CRF, then it must correspond with some augmentation of some underlying m/d,n/e-duoprism...

Not in that generality, but in my today discussed 2 cases I already ponted that out:
You take some tower of prisms, glue some shapes onto its sides, bend that into 4D, and if you're lucky, the outer shape again displays only squares. Thus this torus of squares would correspond to the supporting duoprism (having been provided explicitely in the discussed cases). And those prisms of this very duoprism, which are parallel to the starting ones of that original tower, then will be the augmented ones.

But, btw., a thing looks surprising so:
When I made a lace city display of some polytope from 4D and above so far, then always the representations of the perp space components at some para space position would have themselves a smaller circumradius, when laying farer out. But here this is not the case, so. Instead rather to the contrary, cf. e.g.:
Klitzing wrote:
Code: Select all
`    x4x   x4x                     x4x x4o   x4o x4x                                  x4x x4o   x4o x4x                     x4x   x4x    `

The perp space x4x clearly has a larger circumradius than the perp space x4o. But the para space position of those x4x here is farer out than that of the x4o. - Crude things happen!

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

Klitzing wrote:
quickfur wrote:More thoughts about CRFs with duoprismic symmetry: suppose you have a polychoron with a ring of n-prisms and an orthogonal ring of m-prisms, with some lacing edges between them. I'm thinking that if the polychoron is CRF, then it must correspond with some augmentation of some underlying m/d,n/e-duoprism...

Not in that generality, but in my today discussed 2 cases I already ponted that out:
You take some tower of prisms, glue some shapes onto its sides, bend that into 4D, and if you're lucky, the outer shape again displays only squares. Thus this torus of squares would correspond to the supporting duoprism (having been provided explicitely in the discussed cases). And those prisms of this very duoprism, which are parallel to the starting ones of that original tower, then will be the augmented ones.

But, btw., a thing looks surprising so:
When I made a lace city display of some polytope from 4D and above so far, then always the representations of the perp space components at some para space position would have themselves a smaller circumradius, when laying farer out. But here this is not the case, so. Instead rather to the contrary, cf. e.g.:
Klitzing wrote:
Code: Select all
`    x4x   x4x                     x4x x4o   x4o x4x                                  x4x x4o   x4o x4x                     x4x   x4x    `

The perp space x4x clearly has a larger circumradius than the perp space x4o. But the para space position of those x4x here is farer out than that of the x4o. - Crude things happen!

--- rk

Actually, this makes me wonder if this particular polychoron is non-convex. I failed to account for this possibility in my analysis.
quickfur
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### Re: Johnsonian Polytopes

Marek14 wrote:Well, one possibility is to have the inner-most rings that are not on a single torus. A simple case would be a ring of cupolas joined by their n-gons and 2n-gons (two basic forms are available, with ortho- and gyro- joins between the cupolae). After all, isn't the first J92 polychoron we found, which contains 4 J92's in a ring, something similar? Then you also get an additional degree of freedom: by having two different "joins" in the ring, the dichoral angle can be different for each.

BTW - is there a name for the shape you cut off (in two copies) from a 600-cell to get a gap?

In terms of coordinates, it's just deleting two great circles of vertices from the 600-cell, but geometrically speaking, it's deleting two rings of alternating pentagonal antiprism pyramids and 5-pyramid pyramids (ie., wedge made from two pentagonal pyramids). Interestingly enough, now that I think about this, here's another example of bridged augments that are CRF! You could think of the 600-cell as a gap augmented with pentagonal antiprism pyramids, which makes the result non-convex, but made convex again by inserting bridging augments in the shape of 5-pyramid wedges.

This suggests that perhaps non-trivial duoprismic CRFs are possible with heterogenous rings...
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### Re: Johnsonian Polytopes

Wow, you guys typed a lot when I was away.

Anyway, let's talk about this:
quickfur wrote:[...]

This suggests that perhaps non-trivial duoprismic CRFs are possible with heterogenous rings...

I like the Idea of heterogenous rings as well. (In fact, I've thought about it earlier). first note that the gap has a lace-city:
Code: Select all
`                        o5x                                                                                       x5o                     x5o                                    f5o                                         o5f           o5f                                                                         o5x                                   o5x                 f5o                     f5o                                                                                                                              o5f                     o5f                 x5o                                   x5o                                                                         f5o           f5o                                         o5f                                    o5x                     o5x                                                                                       x5o                        `

This is just the lace-city of ex, with a x5x and 10 o5o's removed. You can see one ring of pentagonal antiprisms is made by x5o's and o5x's, and the other ring is made by alternating f5o's and o5f's. I would like to search for CRF's that have a lace-city display, because they're easier to discover. (they are the "soft" crown jewels).
Now a heterogenous ring that has both ways of representation in a lace-city (both something like f5o's and something like x5o's) is a ring made of pentagonal prisms and antiprisms, alternated. The f5o-like city would then look like F5x and x5F alternated in the same way as before. (I think you can in fact just leave whole that part of the lace-city alive, and just change the nodes, h( of5fo&#xt ) seems to equal h( Fx5xF&#xt ).)
The o5x-like ring would look like x5o and o5x alternated differently, namely as x5o x5o o5x o5x x5o x5o etc. with a total of 10 o5x's and 10 x5o's. this thing would clearly become bigger that the original ring of o5x's and x5o's this means that between those rings some place will occur for intermediate lacing vertices, and we might be able to close it up.

I just realized that the insertion of pentagonal prisms corresponds to a stott-expansion. As a consequence, the above-described thing can be closed up by using the lace-city of rox:
Code: Select all
`                                    x5o             x5o                                                                                o5f                                                                    o5x                                     o5x                                                        x5x                     x5x                                                                                                                                               f5o                     x5f                     f5o                                o5x                     F5o             F5o                     o5x                                            o5F                     o5F                                                        f5x                                     f5x                                                                                                                               x5x                     f5f             f5f                     x5x                x5o             o5F                     V5o                     o5F             x5o                                                                                                               F5o             o5V                     o5V             F5o                                        f5f                                     f5f                            o5f                                     x5F                                     o5f                x5f                     F5x             F5x                     x5f                                                                                                       x5o                     V5o                                     V5o                     x5o                                x5F                     x5F                                            F5o                                                             F5o                x5x             f5f                                             f5f             x5x                o5F                                                             o5F                                            F5x                     F5x                                o5x                     o5V                                     o5V                     o5x                                                                                                       f5x                     x5F             x5F                     f5x                f5o                                     F5x                                     f5o                            f5f                                     f5f                                        o5F             V5o                     V5o             o5F                                                                                                               o5x             F5o                     o5V                     F5o             o5x                x5x                     f5f             f5f                     x5x                                                                                                                               x5f                                     x5f                                                        F5o                     F5o                                            x5o                     o5F             o5F                     x5o                                o5f                     f5x                     o5f                                                                                                                                               x5x                     x5x                                                        x5o                                     x5o                                                                    f5o                                                                                o5x             o5x                                    `

This CRF is really soft, it's just a diminishing of a uniform. Therefore, it won't be a CJ. Nevertheless I think it's interesting anyways.
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student91
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### Re: Johnsonian Polytopes

quickfur wrote:
student91 wrote:
[...]
I think there are other ways to close it up, just found a new one (I hope): (whatever)|| o3F3o || x3x3f || F3o3x || x3f3x || o3x3x. let's hope it works
(for whatever you can either use o3x3x or f3x3o || x3o3o.)

That should be the third thing you're talking about (in fact there are two, because you can insert two "whatevers". the one with o3x3x should be the most interesting, as it has two sets of thawro's )

Unfortunately, this tower is non-CRF. The x3f3x layer overhangs the F3o3x, so there are non-CRF lacing edges from x3x3f to x3f3x (edge length 1/phi). (Unless I made a mistake again.... it's rather late and I'm not thinking very clearly... but I did check the difference in coordinates between F3o3x and x3f3x, they are all unit edges, but x3x3f to x3f3x has 1/phi edges. )

oops, that's due to an error of mine, the height between F3o3x and x3f3x should've been sqrt(1/8)/phi instead of sqrt(1/8)/(phi^2). This is fixed by using x3f3o instead of x3f3x. This gives (whatever) || o3F3o || x3x3f || F3o3x || x3f3o || o3x3o. This time it should work.
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

hello everybody!

I have been thinking about quickfur's polytope which led to mine, and have found something quite interesting:
look at the lace tower of CJ4.8.2; oxFxoo3xooxFx3ofxfox&#xt the first tower consists of octahedra and triangular cupolae (o3x||x3o, x3o||x3x and x3x||o3x with faces touching each other). the interesting thing about these is, that they have their 4- and 5-fold symmetry equivalents being antiprisms and cupolae, thus the same lace tower could work in another symetry group, then being oxFxoo4xooxFx3ofxfox&#xt or oxFxoo5xooxFx3ofxfox&#xt

oxFxoo4xooxFx3ofxfox&#xt doesn't work, because the lacing height of -x4o3-v, the distance for the cupolae touching the truncated oct is unreal, oxFxox4xooxFx3ofxfox&#xt doesn't work aswell, (x4-f3f)+(o4f-x) is not (-x4o3-v)

oxFxoo5xooxFx3ofxfox&#xt or oxFxox4xooxFx3ofxfox&#xt might still work however, it seems more promising because pentagons occur more often in id-symetry, so...
oxFxoo5xooxFx3ofxfox&#xt isn't because (o5f3-x) is unreal and oxFxox4xooxFx3ofxfox&#xt is partly D4.5.3, but doesn't work either because f5o3-v is unreal

I haven't found anything, guess February is over or something, but I did find out how incredibly lucky CJ4.8.2 was

I'm sorry to have posted this rather useless information, but now you know where not to search
student5
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### Re: Johnsonian Polytopes

I've found something else which turned out to be nothing (please tell me if it is useless to share my failed attempts)

the lace tower oxox5xofx3xFxo&#xt looked quite promising, all it's lacings were real, but the dichoral angles were different.

has it occured that there is no (small) TB centered on a truncated icosahedron, it might just be possible another way...
student5
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### Re: Johnsonian Polytopes

student91 wrote:
quickfur wrote:
student91 wrote:
[...]
I think there are other ways to close it up, just found a new one (I hope): (whatever)|| o3F3o || x3x3f || F3o3x || x3f3x || o3x3x. let's hope it works
(for whatever you can either use o3x3x or f3x3o || x3o3o.)

That should be the third thing you're talking about (in fact there are two, because you can insert two "whatevers". the one with o3x3x should be the most interesting, as it has two sets of thawro's )

Unfortunately, this tower is non-CRF. The x3f3x layer overhangs the F3o3x, so there are non-CRF lacing edges from x3x3f to x3f3x (edge length 1/phi). (Unless I made a mistake again.... it's rather late and I'm not thinking very clearly... but I did check the difference in coordinates between F3o3x and x3f3x, they are all unit edges, but x3x3f to x3f3x has 1/phi edges. )

oops, that's due to an error of mine, the height between F3o3x and x3f3x should've been sqrt(1/8)/phi instead of sqrt(1/8)/(phi^2). This is fixed by using x3f3o instead of x3f3x. This gives (whatever) || o3F3o || x3x3f || F3o3x || x3f3o || o3x3o. This time it should work.

Hooray! Your corrected lace tower is indeed CRF! Here's a render of it, centered on the o3x3o:

It shows a rather interesting formation of pentagonal pyramids straddled by a pair of square pyramids between the square faces of the J92's. There are, of course, 4 J92's around the central octahedron, and so if we put x3o3o || f3x3o on the other side, we get a tetrahedral-symmetry CRF containing 8 J92's. In that case, the pairs of pentagonal pyramids on this side of the CRF will join with corresponding pairs of pentagonal pyramids on the other side, forming an interesting configuration of two pairs of pentagonal pyramids straddling a pair of square pyramids. Most fascinating!

EDIT: assigned D4.8.3 to this CRF. The software model files can be found on that page.

EDIT 2: this means that we now have four CRFs in the D4.8 class, because this particular lace tower can be attached to two different other halves, and vice versa. For simplicity of description, let's define A1 = tetrahedron + 4 J92's, found in the original D4.8, and B1 = octahedron + 4 J91's, found on the other side of the original D4.8. Let A2 = truncated tetrahedron + 4 (inverted) J92's found in D4.8.2, as discovered by student5, and B2 = octahedron + 4 J92's, as described by student91 here. The A's and B's have matching boundaries in the shape of that skew polyhedron that I discovered (whose projection is the rectified near-miss #22), so either of the A's can be glued to either of the B's. So we have 4 combinations:

A1 + B1 = D4.8 (maybe this should be renamed to D4.8.1?)
A2 + B1 = D4.8.2
A1 + B2 = D4.8.3
A2 + B2 = (currently unassigned: maybe assigning this to D4.8.4 would not be amiss?)

What I find most fascinating about this, is that the joining boundary of these pieces is non-planar (non-corealmar), yet there exist multiple cell configurations that exhibit the same boundary, thus leading to these combinatorial possibilities!

EDIT 3: here's a slightly better render, with the tridiminished icosahedra rendered in green so that the J92's are easier to pick out visually:

quickfur
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### Re: Johnsonian Polytopes

quickfur: Yes, reminds me of an egg broken in two irregular patches of shell which nevertheless fit perfectly together.
Marek14
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### Re: Johnsonian Polytopes

quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)

Done, and while we're renumbering discovery indices, can we shift the D4.9's up by one, so D4.9.0 becomes D4.9.1, D4.9.1 becomes D4.9.2 and D4.9.2 becomes D4.9.3? Not really a fan of having that .0 there. Hopefully it won't have too much impact given that they don't have wiki pages yet.

Keiji
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### Re: Johnsonian Polytopes

I'd suggest modifying the definition of "augmented" a bit -- wiki says that augmentation is done with pyramids, but even in 3D you have Johnson solids which are augmented with cupolas (augmented truncated cube, for example).

EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.
Marek14
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### Re: Johnsonian Polytopes

Keiji wrote:
quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)

Done, and while we're renumbering discovery indices, can we shift the D4.9's up by one, so D4.9.0 becomes D4.9.1, D4.9.1 becomes D4.9.2 and D4.9.2 becomes D4.9.3? Not really a fan of having that .0 there. Hopefully it won't have too much impact given that they don't have wiki pages yet.

It was numbered .0 because it does not actually have any bilbiro or thawro cells, so it doesn't fit very well with the current definition of BT polytopes. Arguably, the definition of BT polytopes is a bit arbitrary... we should be basing the classification on the operations used to derive these things, rather than the occurrence of bilbiro or thawro cells, which is only incidental, not fundamental. Case in point, D4.4 arguably doesn't really belong in this category, because it isn't obvious how it is derived from any existing uniform polychora by the "bilbiro-ing" and "thawro-ing" operations, so it seems to stand apart from the others it's currently lumped with.

I'm not against renumbering, though, but if we're going to do that, we'll need to fix the references to those numbers in previous posts in this thread, otherwise we will lose referential consistency for when we come back to read this thread in the future.
quickfur
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### Re: Johnsonian Polytopes

Marek14 wrote:I'd suggest modifying the definition of "augmented" a bit -- wiki says that augmentation is done with pyramids, but even in 3D you have Johnson solids which are augmented with cupolas (augmented truncated cube, for example).

Speaking of which, I think it's high time we looked at the augmentations of 2m,2n-duoprisms with 2m-prism||m-gon's and 2n-prism||n-gon's. We should write a script for enumerating these things and see if we get the same results?

EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.

What, really? Where do they occur?
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:[...]
EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.

What, really? Where do they occur?

After thinking a bit, I concluded marek was right, the pentagonla cupola's and the half id's are at an 180 degrees dichoral angle, making them fuse together, resulting in orthocupolarotunda's!! that's indeed really cool.

the pentagonal cupola's that are inserted with the pentagonal prisms, are part of an x3x3o5x. this means those cupola's are comparable with the hemiated id's, and thus the rotunda and the cupola are comparably aligned, making them fuse together .
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student91
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### Re: Johnsonian Polytopes

Let's have a look at this polychoron with Stella data.

It has 10 shapes of cells, further subdivided into 19 distinct types.

1. Pentagonal prisms A (mid-layer). 12. 5-fold dihedral symmetry.
Pentagons joined to pentagonal orthocupolarotundas.
Squares joined to hexagonal prisms.

2. Pentagonal prisms B. 120. Reflection symmetry.
One pentagon joined to a bilbro.
Other pentagon joined to a icosidodecahedron.
Two adjacent squares joined to cuboctahedra A.
Two squares adjacent to them joined to triangular cupolas B.
Fifth square adjacent to a triangular cupola A.

3. Pentagonal prisms C. 120. Reflection symmetry.
One pentagon joined to a icosidodecahedron.
Other pentagon joined to a pentagonal rotunda.
Two adjacent squares joined to cuboctahedra B.
Two squares adjacent to them joined to cuboctahedra C.
Fifth square adjacent to a triangular cupola C.

4. Pentagonal prisms D. 24. 5-fold pyramidal symmetry.
One pentagon joined to a pentagonal rotunda.
Other pentagon joined to a pentagonal orthocupolarotunda.
Squares joined to cuboctahedra B.

5. Pentagonal prisms E. 120. Reflection symmetry.
One pentagon joined to a icosidodecahedron.
Other pentagon joined to a pentagonal orthocupolarotunda.
Two adjacent squares joined to cuboctahedra B.
Two squares adjacent to them joined to cuboctahedra A.
Fifth square joined to a triangular cupola B.

6. Pentagonal prisms F. 60. 2-fold pyramidal symmetry.
Pentagons joined to icosidodecahedra.
Two adjacent squares joined to cuboctahedra A.
Two squares adjacent to them joined to cuboctahedra B.
Fifth square joined to a cuboctahedron C.

7. Triangular cupola A. 40. 3-fold pyramidal symmetry.
Hexagon joined to another triangular cupola A.
Squares joined to pentagonal prisms B.
Lateral triangles joined to triangular cupolas B.
Top triangle joined to an icosidodecahedron.

8. Triangular cupola B. 120. Reflection symmetry.
Hexagon joined to a truncated tetrahedron.
Two squares joined to pentagonal prisms B.
Third square joined to a pentagonal prism E.
One lateral triangle between two "pentagonal prism B" squares joined to a triangular cupola A.
Two remaining lateral triangles joined to cuboctahedra A.
Top triangle joined to an icosidodecahedron.

9. Triangular cupola C. 40. 3-fold pyramidal symmetry.
Hexagon joined to a great icosidodecahedron.
Squares joined to pentagonal prisms C.
Lateral triangles joined to cuboctahedra C.
Top triangle joined to an icosidodecahedron.

10. Truncated tetrahedron. 60. 2-fold pyramidal symmetry.
Two hexagons joined to hexagonal prisms.
Other two hexagons joined to triangular cupolas B.
Triangles adjacent to two "hexagonal prism" hexagons joined to pentagonal orthocupolarotundas.
Triangles adjacent to two "triangular cupola B" hexagons joined to bilbros.

11. Hexagonal prism. 60. 2-fold pyramidal symmetry.
Hexagons joined to truncated tetrahedra.
One square joined to a pentagonal prism A.
Two squares adjacent to it joined to pentagonal orthocupolarotundas.
Two squares adjacent to those joined to cuboctahedra A.
Sixth square joined to a bilbro.

12. Cuboctahedron A. 120. Reflection symmetry.
Two adjacent squares joined to pentagonal prisms B.
Two squares that form a ring with them joined to pentagonal prisms E.
Fifth square joined to a pentagonal prism F.
Sixth square cross from it joined to a hexagonal prism.
Two triangles adjacent to "pentagonal prism B", "pentagonal prism E" and "pentagonal prism F" squares joined to an icosidodecahedron.
Two triangles adjacent to "pentagonal prism B", "pentagonal prism E" and "hexagonal prism" squares joined to a triangular cupola B.
A triangle adjacent to two "pentagonal prism B" squares and the "pentagonal prism F" square joined to another cuboctahedron A.
A triangle adjacent to two "pentagonal prism B" squares and the "hexagonal prism" square joined to a bilbro.
A triangle adjacent to two "pentagonal prism E" squares and the "pentagonal prism F" square joined to a cuboctahedron B.
A triangle adjacent to two "pentagonal prism E" squares and the "hexagonal prism" square joined to a pentagonal orthocupolarotunda.

13. Cuboctahedron B. 120. Reflection symmetry.
Two adjacent squares joined to pentagonal prisms C.
Two squares that form a ring with them joined to pentagonal prisms E.
Fifth square joined to a pentagonal prism D.
Sixth square cross from it joined to a pentagonal prism F.
Two triangles adjacent to "pentagonal prism C", "pentagonal prism E" and "pentagonal prism D" squares joined to other cuboctahedra B.
Two triangles adjacent to "pentagonal prism C", "pentagonal prism E" and "pentagonal prism F" squares joined to icosidodecahedra.
A triangle adjacent to two "pentagonal prism C" squares and the "pentagonal prism D" square joined to a pentagonal rotunda.
A triangle adjacent to two "pentagonal prism C" squares and the "pentagonal prism F" square joined to a cuboctahedron C.
A triangle adjacent to two "pentagonal prism E" squares and the "pentagonal prism D" square joined to a pentagonal orthocupolarotunda.
A triangle adjacent to two "pentagonal prism E" squares and the "pentagonal prism F" square joined to a cuboctahedron A.

14. Cuboctahedron C. 60. 2-fold pyramidal symmetry.
A ring of four squares joined to pentagonal prisms C.
Fifth square joined to a pentagonal prism F.
Sixth square across from it joined to a great icosidodecahedron.
Two triangles adjacent to opposite sides of the "pentagonal prism F" square joined to icosidodecahedra.
Two remaining triangles adjacent to the "pentagonal prism F" square joined to cuboctahedra B.
Two triangles adjacent to the "great icosidodecahedron" square, opposite the "icosidodecahedron" ones joined to triangular cupolas C.
Two triangles adjacent to the "great icosidodecahedron" square, opposite the "cuboctahedron B" ones joined to pentagonal rotundas.

15. Bilbro. 30. 2-fold dihedral symmetry.
Pentagons joined to pentagonal prisms B.
Squares joined to hexagonal prisms.
Rotunda triangles joined to cuboctahedra A.
Luna triangles joined to truncated tetrahedra.

16. Pentagonal rotunda. 24. 5-fold pyramidal symmetry.
Decagon joined to great icosidodecahedron.
Lateral pentagons joined to pentagonal prisms C.
Top pentagon joined to a pentagonal prism D.
Bottom triangles joined to cuboctahedra C.
Top triangles joined to cuboctahedra B.

17. Pentagonal orthocupolatorunda. 24. 5-fold pyramidal symmetry.
Cupola pentagon joined to a pentagonal prism A.
Rotunda lateral pentagons joined to pentagonal prisms E.
Rotunda top pentagon joined to a pentagonal prism D.
Squares joined to hexagonal prisms.
Cupola triangles joined to truncated tetrahedra.
Bottom rotunda triangles joined to cuboctahedra A.
Top rotunda triangles joined to cuboctahedra B.

18. Icosidodecahedron. 40. 3-fold pyramidal symmetry.
Three pentagons around "top" triangle joined to pentagonal prisms C.
Three pentagons in next layer joined to pentagonal prisms F.
Three pentagons in next layer joined to pentagonal prisms E.
Three pentagons around "bottom" triangle joined to pentagonal prisms B.
Top triangle joined to a triangular cupola C.
Three triangles adjacent to two "C" pentagons and one "F" pentagon joined to cuboctahedra C.
Six triangles adjacent to one "C" pentagon, one "F" pentagon and one "E" pentagon joined to cuboctahedra B.
Six triangles adjacent to one "F" pentagon, one "E" pentagon and one "B" pentagon joined to cuboctahedra A.
Three triangles adjacent to one "E" pentagon and two "B" pentagons joined to triangular cupolas B.
Bottom triangle joined to a triangular cupola A.

19. Great icosidodecahedron. 2. Icosahedral symmetry.
Decagons joined to pentagonal rotundas.
Hexagons joined to triangular cupolas C.
Squares joined to cuboctahedra C.

Important cycles:
Great icosidodecahedron -10- pentagonal rotunda -5- pentagonal prism D -5- pentagonal orthocupolarotunda -5- pentagonal prism A (quarter circle, mirror to get whole)
Great icosidodecahedron -6- triangular cupola C -3- icosidodecahedron -3- triangular cupola A -6- (quarter circle)
Great icosidodecahedron -4- cuboctahedron C -4- pentagonal prism F -edge- triangle between two cuboctahedra A -point- bilbro (quarter circle)
Marek14
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### Re: Johnsonian Polytopes

student91 wrote:
quickfur wrote:[...]
EDIT: D4.6's page doesn't mention perhaps the most interesting cell it has -- 24 pentagonal orthocupolarotundas.

What, really? Where do they occur?

After thinking a bit, I concluded marek was right, the pentagonla cupola's and the half id's are at an 180 degrees dichoral angle, making them fuse together, resulting in orthocupolarotunda's!! that's indeed really cool.

the pentagonal cupola's that are inserted with the pentagonal prisms, are part of an x3x3o5x. this means those cupola's are comparable with the hemiated id's, and thus the rotunda and the cupola are comparably aligned, making them fuse together .

Whoa, you are right!! How did I miss this before? Your comments inspired me to go back to take a more careful look at this CRF. And indeed, there are pentagonal orthocupolarotunda cells!! (Which means I have to add the pentagonal orthocupolarotunda to my website too, since I plan to do renders of all of these CRFs some day. )

The blue cells are pentagonal orthocupolarotundae. The red pentagons indicate where they touch the equatorial pentagonal prisms. The green cells are, of course, bilbiro's.

Here's a side-view that shows how these orthocupolarotunda attach north and south of the pentagonal prisms:

This CRF just upped several notches of cool in my book.
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### Re: Johnsonian Polytopes

It kind of like looks like a goldfish bowl in this view
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the dream we dream together is reality.

wendy
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### Re: Johnsonian Polytopes

wendy wrote:It kind of like looks like a goldfish bowl in this view

Not too surprising in view of its construction:
chopping off the poles of a spherical thing, and cutting out some equatorial stratum, with some further local structure change in that region...

Btw., quickfur, your new pics are fascinating as ever! Looks indeed like a gem, quite appealing, by highlighting all that special stuff. - Only that the normal parts inbetween, within my personal view, are too less visible. I'd like to have the edges at least to remain visible, in order to be at least somehow guided what shall be filled inbetween.

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

Klitzing wrote:
wendy wrote:It kind of like looks like a goldfish bowl in this view

Not too surprising in view of its construction:
chopping off the poles of a spherical thing, and cutting out some equatorial stratum, with some further local structure change in that region...

Btw., quickfur, your new pics are fascinating as ever! Looks indeed like a gem, quite appealing, by highlighting all that special stuff. - Only that the normal parts inbetween, within my personal view, are too less visible. I'd like to have the edges at least to remain visible, in order to be at least somehow guided what shall be filled inbetween.

--- rk

The problem with showing all the edges, is that there are too many of them. This is what it looks like:

Originally, when I first started rendering 4D polytopes, my dream was to show everything in one image, but it become obvious pretty soon that it won't work because of the clutter. That's what led me to do layer-by-layer renderings on my website; it's the only way to show the entire structure in a comprehensible way.
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:The problem with showing all the edges, is that there are too many of them. ...

Originally, when I first started rendering 4D polytopes, my dream was to show everything in one image, but it become obvious pretty soon that it won't work because of the clutter. That's what led me to do layer-by-layer renderings on my website; it's the only way to show the entire structure in a comprehensible way.

I agree. It is a clutter.
And yes, your layer-by-layer approach is really educative!

But I got some idea for that point (and you would have to implement it, haha):
So far you display all edge as bold struts, and in that other pic you just omitted several edges - as those would become too dominant for being only guide lines for subordinate structures.
So what would you think about a 3-state logic for edges?
• shown as struts (as in both of your pic versions - as far as those were displayed at all)
• shown as mere thin lines (perhaps additionally with applicable coloring, so that you might can fade those out and in rather easily, just by adapting that color a bit)
• not shown (as in the former pic in most cases)
--- rk
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### Re: Johnsonian Polytopes

Klitzing wrote:[...] But I got some idea for that point (and you would have to implement it, haha):
So far you display all edge as bold struts, and in that other pic you just omitted several edges - as those would become too dominant for being only guide lines for subordinate structures.
So what would you think about a 3-state logic for edges?
• shown as struts (as in both of your pic versions - as far as those were displayed at all)
• shown as mere thin lines (perhaps additionally with applicable coloring, so that you might can fade those out and in rather easily, just by adapting that color a bit)
• not shown (as in the former pic in most cases)
--- rk

The renders are all done as povray 3D models, and the edges are basically thin cylinders (there's no way to only draw thin lines in povray; they must be 3D objects). They can be assigned different textures, though, and I've used that in the past (e.g. to highlight the outline of a cell that overlaps with other elements so that its faces don't obscure the rest of the image). I suppose I could use it more. It's not just a 3-state logic, it is n-state, since you can assign any texture to any edge. The tricky part comes from which texture to assign to which edges to make the resulting image clear, and this is usually not as straightforward as it looks.

Anyway, here's an attempt to use two different textures to show all edges:

It's a little better than before, but the sheer number of edges still makes a big mess of it. The blue edges here are assigned a highly transparent texture in order to minimize their visual obstruction of the image. Originally, I tried to just use a solid color, but the result wasn't that much different from the previous image. Making them 90% transparent helped a little, but as you can see, there are simply too many of them, and they're still cluttering up the image. The transparency has also made them look like somebody scrawled all over the image with a blue marker. Worse yet, because they're now so transparent, it's almost impossible to discern individual edges except on the outer boundary of the projection, which isn't very helpful when you're trying to locate individual cells. Reducing the transparency only brings back the original clutter.

Basically, I've tried similar techniques before, but eventually my conclusion was that the only way to show the structure clearly is to show the elements layer-by-layer. When you highlight only a relatively small number of cells, it's possible to show everything together, but once the number of elements reaches a certain threshold, it just becomes an incomprehensible clutter, and no matter what you do, you just can't get around it. Now, if only we had a way to deliver 3D images directly to the brain, then we won't have this problem, but our 2D retinas are a big limitation here.
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:
Keiji wrote:
quickfur wrote: D4.8 (maybe this should be renamed to D4.8.1?)

Done, and while we're renumbering discovery indices, can we shift the D4.9's up by one, so D4.9.0 becomes D4.9.1, D4.9.1 becomes D4.9.2 and D4.9.2 becomes D4.9.3? Not really a fan of having that .0 there. Hopefully it won't have too much impact given that they don't have wiki pages yet.

It was numbered .0 because it does not actually have any bilbiro or thawro cells, so it doesn't fit very well with the current definition of BT polytopes. Arguably, the definition of BT polytopes is a bit arbitrary... we should be basing the classification on the operations used to derive these things, rather than the occurrence of bilbiro or thawro cells, which is only incidental, not fundamental. Case in point, D4.4 arguably doesn't really belong in this category, because it isn't obvious how it is derived from any existing uniform polychora by the "bilbiro-ing" and "thawro-ing" operations, so it seems to stand apart from the others it's currently lumped with.

I'm not against renumbering, though, but if we're going to do that, we'll need to fix the references to those numbers in previous posts in this thread, otherwise we will lose referential consistency for when we come back to read this thread in the future.

I fully agree on D4.4 not being a bilbirothawroid. I think we should make it CJ4.4 again. If it weren't a crown jewel, we wouldn't be consistent:
the wiki wrote:Crown jewels are a catch-all term for unusual CRF polytopes with unique structures that cannot be obtained from the uniform polytopes or other simpler CRFs by simple "cut-and-paste" operations.
D4.4 clearly can't be obtained by simple cut-n-paste operations. For the same reason, I think the D4.3, and the D4.8's may be reconsidered as well.
Furthermore I would pleat to use a naming scheme for the "left-over" bilbirothawriods with a following format:
[optional icosahedral/tetrahedral prefix] [polar/equatorial] [bilbiro'd/thawro'd] [whatever].
This format highly corresponds to your proposition of compression and pseudo-bisection. The difference is, that my format is specific for the cases where bilbiro's and thawro's occur, and yours doesn't have to do so. I like the specific approach, because this makes it more clear what the result is. Furthermore, an ("obvious") bilbiroing resp. thawroing can be done at exactly one place. This means there aren't any not-well-defined numbers needed. (I think making the number-prefixes will either yield big numbers or ambiguity) However, I'm still not happy with the verbs. maybe we can use compression and pseudo-bisection instead of bilbiro-ing resp thawroing? The icosahedral/tetrahedral prefix is optional because so far we haven't found any tetrahedral bilbiroing resp. thawroing that leaves most of the original intact. Therefore, such a prefix won't be needed when we don't find any tetrahedral things with this property. polar/equatorial is only needed when you are thawro-ing. because the ("obvious") thawroing has a well-defined location, it only needs a side. again, the terms "polar" and "equatorial" can be changed. finally the whatever is the thing that is thawro'd/bilbiro'd.

the D4.9.x-things are numbered that way, because I saw D4.9.0 as the "basic" thing. Just as with an update, the "basic" thing doesn't get a number, or a .0 The further derivatives ("patches" when speaking about updates )get a corresponding postfixed number. In my proposed format I would suggest we call D4.9.0 the polar thawro'd o3x3o5o, and then the 4.9.x's are called octahedral diminished polar thawro'd o3x3o5o, pentagonal antiprismatically diminished polar thawro'd o3x3o5o etc. To these proposed names the following applies as well:
quickfur wrote:I know this nomenclature is still a bit rough around the edges, so suggestions for improvements are welcome,

. Point is that I mostly agree with quickfur about his naming convention, with the difference that I would rather like specific operations rather that global ones.
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