## 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

quickfur wrote:Congratz, student5, you've found your first CRF crown jewel.

Well here's a render centered on the o3x3o:

This is basically identical to CJ4.8. But the other side, centered on o3x3x, is different:

Just a question to stella users: are those pairs of trigonal cupolae corealmic? I.e. form cuboctahedra? Or have those a smaller dihedral angle at their hexagonal join?

--- rk
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### Re: Johnsonian Polytopes

Ive written http://hddb.teamikaria.com/forum/viewtopic.php?f=25&t=1881 some noises on Wythoff polytopes that might provide some help to this project. It also gives in some obscure way, the secret behind the spreadsheets. (ie the dynkin-graph is actually a vector, and as student91 correctly observes, wendy really does understand what negative numbers in the CD diagram really mean).

As a bonus, it gives the magic for writing most of these stott-matrices out without having to go through the dynkin matrix inverse. This is passingly handy for those who don't want to do matrix divisions all the time. (hint - wendy uses it all the time).

Here is some ideas i've been tossing around in the direction of names. The idea behind names here is that if something belongs to X, then one can apply magic that is used of X-like things. So for example, 'wythoff polytopes' are things you can feed into my spreadsheet, and non-wythoff polytopes don't work without some retrofit. You can feed a bi-diminished icosahedron in, by pretending it's an icosahedron, and then do the bi-diminished afterwards (it does not affect heights!).

A 'cupola' is a bistrate layer where all the base angles are less than a right angle. These can be 'elongated' by way of a prism-layer.

I was playing with 'keppi' for anything that is bistrate, but not a cupola, but the word is used of the cylinderical form of the france-man's police-hat. The idea is the sort of peaked thing you more often see on police in other parts of the world, such as U.S.o.A. It's better to have a word here than not to.

A 'rotunda' is a polystrate layer say, in the shape of a half-sphere. But i suppose we can use fuller's name somewhere, or prehaps 'dome'.

[I was thinking of layers of vertices, rather than layers between vertices. Richard K picked me up on that: thanks Richard]

Ursulates are turning quite complex, and have now exploded into the hundreds, though nothing fancy here either. The process is currently understood, but a notation hasn't been cooked up enough to allow one to write these things. In essence, you are getting things like

ooxxoo3xfoooo3ooxxoo3oooofx3ooxxooBoooooo&#xt.

I see how hard it is to write this style, cf Richard's version, but we press on.

This is a double-decker expanded ursulated rectified 2_21, but the bases are orientated in different directions.
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### Re: Johnsonian Polytopes

Klitzing wrote:Great finds lately!

Just an idea: are those hexagons of that "equatorial" skew neerly-icosidodecahedron kind of flat, i.e. are either all their vertices at the same circumradius, or otherwise are those those faces "parallel" in some sense, i.e. do they all have a symmetrical slope? Then it could be possible to join either one of these J92 containing halfs as well. Thus providing a third relative. Or am I wrong in this?

--- rk

I think you're right!! look at my previous post:
student91 wrote:
quickfur wrote:[...]I think I'll assign this CRF as CJ4.8.2, since it appears to be related to CJ4.8 (which maybe should be renamed CJ4.8.1?). I wonder if there are any other CRFs that can be joined together in this way via that unusual skew polyhedron?

EDIT: Created a wiki page for it: CJ4.8.2 (you can find the Stella4D model on that page)

Again, nice renders.
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 )
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### Re: Johnsonian Polytopes

Keiji wrote:I have a slight suggestion. As quickfur has said, all the polytopes mentioned lately have been analogs of J91/J92. Could we perhaps just call these set of figures the bilbiro/thawro polychora, and give them BT numbers instead of CJ numbers?

With how many we are finding, they shouldn't really be called crown jewels. It would certainly be nice to enumerate all possible BT polychora, though!

Well, the whole point of the CJ numbers is to provide a unique ID to these CRFs until we have collected enough data to be able to meaningfully classify them. Besides, one has to admit that their constructions were not obvious until we stumbled upon the two general operations that generate them. So in that sense they are crown jewels, though perhaps all belonging to the same general class, like the ursachora, not separate constructions as previously thought.

So I would propose to continue using the CJ numbers, since it's a good way to generate unique IDs for new CRFs of unknown classification, but also to give BT numbers to the CRFs we discovered this month, now that their classifications are clear.

In fact, now that we recognize the BT CRFs as a class of their own, it immediately suggests related directions of research: so far we've found BT CRFs with icosahedral and tetrahedral symmetries. What about octahedral symmetry? Perhaps novel BT CRFs await discovery in that direction!

On second thoughts, though, perhaps the CJ numbers should be renamed to just CRF numbers (though I can't say I like the idea of renumbering the CJ polytopes, because that defeats the purpose of having them as unique identifiers), because it's really not just limited to "crown jewels", but it can just be a canonical indexing ID that we draw upon to give us placeholder names for new discoveries, until we have enough data to properly classify them and name them.
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### Re: Johnsonian Polytopes

Perhaps we should generalise even further and just use some kind of "discovery numbers" for any yet-to-be-classified interesting new finding?

I'd really like to see some BT CRFs with octahedral symmetry. I'm curious as to whether any exist though, since J91 and J92 have pentagons, and octahedral symmetry figures do not. Are there any skew polyhedra with pentagons and octagons, similar to the one with pentagons and hexagons that shows up in CJ4.8.2?

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### Re: Johnsonian Polytopes

Looking at the "Near misses" in Stella, one interesting category of polyhedra are "mixed prisms". They look like antiprisms, with two polygonal bases joined by lacing of triangles, but one base has more sides, with one square inserted in the lacings. I wonder whether those could be realized as skew CRF polyhedra...
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### Re: Johnsonian Polytopes

Keiji wrote:Perhaps we should generalise even further and just use some kind of "discovery numbers" for any yet-to-be-classified interesting new finding?

I agree!! Maybe we should reassign the current CJ4.x numbers to DN4.x. The existing CJ assignments should remain as-is, for the purposes of referential consistency (e.g. when somebody reads this forum in the distant future). But no more CJ numbers will be assigned; all new CRFs will be assigned a DN number instead.

I'd really like to see some BT CRFs with octahedral symmetry. I'm curious as to whether any exist though, since J91 and J92 have pentagons, and octahedral symmetry figures do not.

You forgot the octahedral ursachoron. It has octahedral symmetry and also sports nice regular pentagons.

Are there any skew polyhedra with pentagons and octagons, similar to the one with pentagons and hexagons that shows up in CJ4.8.2?

AFAIK, the skew polyhedron in CJ4.8 and CJ4.8.2 is a new polyhedron; I certainly haven't seen it anywhere else before. It appears to be the rectified version of Near Miss #22 on orchidpalms.com.

My feeling is that we might discover some similar skew polyhedron in the course of constructing CRFs with J91's/J92's in octahedral symmetry.
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### Re: Johnsonian Polytopes

Klitzing wrote:Great finds lately!

Just an idea: are those hexagons of that "equatorial" skew neerly-icosidodecahedron kind of flat, i.e. are either all their vertices at the same circumradius, or otherwise are those those faces "parallel" in some sense, i.e. do they all have a symmetrical slope? Then it could be possible to join either one of these J92 containing halfs as well. Thus providing a third relative. Or am I wrong in this?

--- rk

The hexagons of the skew polyhedron lie on the same 2-sphere, since they are just the hexagonal faces of the J92's surrounding the tetrahedron in perfect tetrahedral symmetry.

Basically, the skew polyhedron is the convex hull (in 3D) of o3F3o, x3x3f, and F3o3x. Note that because the first and last CD symbols do not commute, the surface patch that fits on one side will not fit on the other side, so some modification will be necessary for the pieces to fit together. So what we have is the interesting situation where this skew polyhedron defines an uneven, directed "joining surface" in 4D, where whatever piece that fits on one side of it, will produce a CRF with whatever piece fits on the other side of it, so the total set of CRFs would be the cartesian product of the set of possible top pieces and the set of bottom pieces. Since the polyhedron is not symmetric about its average hyperplane, the set of top pieces is disjoint from the set of bottom pieces.

So far, there are two known "top pieces" - CJ4.8's tetrahedron + 4 J92's, and CJ4.8.2's truncated tetrahedron + 4 (inverted) J92's. There is one known "bottom piece": octahedron + 6 J91's. Student91 suggested one or two other possibilities, but I've yet to verify that they will produce CRFs.
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### Re: Johnsonian Polytopes

Klitzing wrote:
quickfur wrote:Congratz, student5, you've found your first CRF crown jewel.

Well here's a render centered on the o3x3o:

This is basically identical to CJ4.8. But the other side, centered on o3x3x, is different:

Just a question to stella users: are those pairs of trigonal cupolae corealmic? I.e. form cuboctahedra? Or have those a smaller dihedral angle at their hexagonal join?

--- rk

They are definitely not corealmic, otherwise they would show up as cuboctahedra, not trigonal cupolae. My renders are generated from the convex hull of the vertices, so the polytope viewer would not render only half a cell if indeed all vertices are corealmic.
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### Re: Johnsonian Polytopes

quickfur wrote:
Keiji wrote:Perhaps we should generalise even further and just use some kind of "discovery numbers" for any yet-to-be-classified interesting new finding?

I agree!! Maybe we should reassign the current CJ4.x numbers to DN4.x. The existing CJ assignments should remain as-is, for the purposes of referential consistency (e.g. when somebody reads this forum in the distant future). But no more CJ numbers will be assigned; all new CRFs will be assigned a DN number instead.

In that case I'll move all the CJ pages to D pages (the N is redundant, since there's a number after it).

quickfur wrote:
Keiji wrote:I'd really like to see some BT CRFs with octahedral symmetry. I'm curious as to whether any exist though, since J91 and J92 have pentagons, and octahedral symmetry figures do not.

You forgot the octahedral ursachoron. It has octahedral symmetry and also sports nice regular pentagons.

Hmm, I wonder, might there be a way to BT an ursatope?

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### Re: Johnsonian Polytopes

Keiji wrote:
quickfur wrote:
Keiji wrote:Perhaps we should generalise even further and just use some kind of "discovery numbers" for any yet-to-be-classified interesting new finding?

I agree!! Maybe we should reassign the current CJ4.x numbers to DN4.x. The existing CJ assignments should remain as-is, for the purposes of referential consistency (e.g. when somebody reads this forum in the distant future). But no more CJ numbers will be assigned; all new CRFs will be assigned a DN number instead.

In that case I'll move all the CJ pages to D pages (the N is redundant, since there's a number after it).

Sounds good. There should be an index page for it as well, explaining what it's about.

quickfur wrote:
Keiji wrote:I'd really like to see some BT CRFs with octahedral symmetry. I'm curious as to whether any exist though, since J91 and J92 have pentagons, and octahedral symmetry figures do not.

You forgot the octahedral ursachoron. It has octahedral symmetry and also sports nice regular pentagons.

Hmm, I wonder, might there be a way to BT an ursatope?

Possibly. The lace tower constructions are very similar.

On a related note, I have a partially-constructed polychoron on file that begins with a tetrahedron, and sticks 4 J63's around it, with J62's touching the edges of the tetrahedron. Sorta like a "fat ursachoron" (or "fat teddy", so to speak). The analogy here is that the metabidiminished icosahedron may be thought of as a digonal fat teddy, that is, whereas the usual ursatope construction would just have a phi-scaled digon in the next layer, the fat teddy has f2x, then o2F to close the pentagons, then another layer of vertices to close the polyhedron. J92 itself then is the trigonal fat teddy: you start with a triangle x3o, then f3x (where the "skinny" teddy J63 would have just f3o), then o3F to close the pentagons, then a final layer of vertices to close the polyhedron.

In the case of my partial polychoron, it turns out that you get 4 tetrahedra around the initial tetrahedron, with J62's around the edges of the initial tetrahedron and J63's around its faces. This fragment currently appears to be a part of some kind of diminished 600-cell, so I'm not sure yet if there are "interesting" ways to close it up (instead of just producing a 600-cell diminishing). The thing is, it doesn't have to be J62's around the edge; the first two layers of J62's vertices are the same as the first two layers of J91's vertices, so conceivably, you could extend the polychoron a little farther out to make J91 cells, and find some way of closing up the rest of it in a CRF way. Similar modifications could be applied to ursatopes to introduce J91's to the initial fragment. Not sure how the result would close up, though.
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### Re: Johnsonian Polytopes

wendy wrote:...
A 'cupola' is a bistrate layer where all the base angles are less than a right angle.

Sorry, Wendy, definitely not!

A cupola has 2 vertex layers, yes. But it is monostratic. (I.e. it has just 1 region between vertex layers.)
Similarily, a bistratic tower would have 3 vertex layers. Etc.

Stratos was taken from metrology, meaning cloudy, puffy stuff, which floats on top of something, but by its own weight is bounded in height as well. Thus it could be described as being (more or less) between two separated (hyper)"planes". - In the transscription to polytopes those hyperplanes are represented by vertex layers. That is, whenever you have a polytope with N vertex layers, then it will have N-1 strati. (No more, no less.) - At least with respect to the orientation under consideration.

--- rk
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### Re: Johnsonian Polytopes

Klitzing wrote:
wendy wrote:...
A 'cupola' is a bistrate layer where all the base angles are less than a right angle.

Sorry, Wendy, definitely not!

A cupola has 2 vertex layers, yes. But it is monostratic. (I.e. it has just 1 region between vertex layers.)
Similarily, a bistratic tower would have 3 vertex layers. Etc.

Stratos was taken from metrology, meaning cloudy, puffy stuff, which floats on top of something, but by its own weight is bounded in height as well. Thus it could be described as being (more or less) between two separated (hyper)"planes". - In the transscription to polytopes those hyperplanes are represented by vertex layers. That is, whenever you have a polytope with N vertex layers, then it will have N-1 strati. (No more, no less.) - At least with respect to the orientation under consideration.

--- rk

"Stratum" is also used in geology to indicate layers corresponding to different geological eras, so it's a good name in that sense as well.

On that note, I like the word "dome" to refer to cup-like polytopes. So perhaps we could adopt this terminology:
- pyramid - for when the top layer is a point
- pseudo-pyramid - for when the top layer is larger than a point, but still subdimensional (e.g. digon||J91, trigon||J92)
- cupola - for when the upper layer is full-dimensional but smaller than the lower layer (thus base angles are less than 90°) - this includes Stott-expanded pyramids but encompasses other variations as well
- prism - for when the upper and lower layers are identical
- antiprism - for when the upper and lower layers are dual to each other
- dome (or cap / cup?) - for possibly multistratics that are dome-shaped (i.e., bottom layers are progressively larger than top layers).
- rotunda - for hemispherical shapes.
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### Re: Johnsonian Polytopes

I got stratum wrong. I was thinking in terms of layers of vertices rather than layers of flesh.

But quickfur got the idea right. If you stick names to things, it's easier to revisit them and say, let's try x, y, z to this.
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### Re: Johnsonian Polytopes

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. )
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### Re: Johnsonian Polytopes

I was wondering about making a batch of *.off files for all possible 24-cell diminishings, but the pvtest on site is still rather clunky (creating a convex hull requires way too many clicking and tab-switching). Since all the coordinate files are quite easy to make (just removing some vertices from 24-cell), would it be possible to just do a batch conversion?
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### Re: Johnsonian Polytopes

Done.

You can already batch upload many files, so do that, and make sure you enter a box number when you upload, preferably an empty one - let's say box number 7 for your 24-cell diminishings.

Then, go to the pvtest page and enter box number 7 where you're supposed to enter the file hash, and click submit. It'll batch process everything in that box.

You should see something like this: http://hddb.teamikaria.com/wiki/:pvtest?coords_hash=6

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### Re: Johnsonian Polytopes

I was thinking about grand antiprism and whether other rings of antiprisms could be connected in similar way, maybe with more layers of vertices in-between. Maybe a good place to start would be simply to double the antiprism? I.e., start with two rings of 20 decagonal antiprisms. Could a layer of vertices fit between them to make a CRF figure?
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### Re: Johnsonian Polytopes

Marek14 wrote:I was thinking about grand antiprism and whether other rings of antiprisms could be connected in similar way, maybe with more layers of vertices in-between. Maybe a good place to start would be simply to double the antiprism? I.e., start with two rings of 20 decagonal antiprisms. Could a layer of vertices fit between them to make a CRF figure?

I've often wondered about that too. But I haven't able to answer that yet. Maybe the thing to do is to try it out and see what happens.
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### Re: Johnsonian Polytopes

More polydimensional ursulate stuff.

I see that someone (prolly Kr litzing, sounds like something he would do), has done a stott expansion on the ursulates. xfo3oox4xxx would have both pentagons and octagons. A truly interesting mix. And this occurs in every dimension!

The icosahedral expanded-ursulate is given as tentitive. It actually exists, the face consist is an ID, 50 pentagonal prisms, 20 pentagonal cupolae, and a truncated dodecahedron. It's a crf as these things go.

In any case, it got me expanding the ursulates in higher dimensions too. This pretty much means that the list i gave earlier can be stott-expanded, as long as the two or three nodes used by the ursulation are not also expanded.

So a middle section of x3o3x3o3xBo can arise from the ursulation of o3x3o3o3xBo, or x3o3o3x3oBo, and because these ursulates are base-convex (ie cupola-like, in that perpendiculars to the base are all in figure), then these can be joined together in any way, with or without a medial prism. The last node can be 'x' ed for an other full set.

The interesting thing is that some of these ursulates are very flat, and so one can summount these on top of the faces of the cross-polytope or even the simplex, which would have similar faces.

So it's prolly better if we shunt the ursulates and the extended ursulates off into their own group, because even though they give small numbers in 3 and 4 dim, they do give lots more in the higher ones. I just have to weasel a notation to find them all.
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### Re: Johnsonian Polytopes

quickfur wrote:
Marek14 wrote:I was thinking about grand antiprism and whether other rings of antiprisms could be connected in similar way, maybe with more layers of vertices in-between. Maybe a good place to start would be simply to double the antiprism? I.e., start with two rings of 20 decagonal antiprisms. Could a layer of vertices fit between them to make a CRF figure?

I've often wondered about that too. But I haven't able to answer that yet. Maybe the thing to do is to try it out and see what happens.

Hmm, first step should probably be coming up with an analogue of lace tower notation for these figures (lace toruses?).

Each layer in lace tower is composed of vertices with identical 4th coordinate. In fact, the layers are usually spherical, so each vertex (x,y,z,w) satisfies conditions (x^2 + y^2 + z^2 = a, w^2 = b) for a and b constant for each layer. In lace torus, each vertex layer would be composed of vertices (x,y,z,w) with (x^2 + y^2 = a, z^2 + w^2 = b), and a and b would be once again constant for each layer.

While a basic layer for lace tower has a 3D symmetry group, the symmetry groups for layers of lace torus would be different. For example, any duoprism would be monostratic, as its vertices always lie on a single duocylinder ridge -- this shows that there's unlimited amount of symmetry groups based on square tilings. Even-membered rings of antiprisms would be another basic type, as well as simple polygons (for example, a 16-cell is what Wendy calls, I think, a "square tegum") -- taken as lace torus it has a square in one layer and orthogonal square in second layer.

So the first question is what kinds of symmetries are exactly possible in these layers...

EDIT: Another interesting example: If you combine vertices of (8,4)-duoprism and a (4,8)-duoprism, you'll get runcinated tesseract.

EDIT: I can see the tradeoff: it seems that while there is more symmetry groups, it's generally harder to choose the layers so the lacing would be unit.

I tried combining (3,6)-duoprism with (6-3)-duoprism and got a strange non-CRF polychoron with 9 irregular tetrahedra, 9 cuboids and 24 triangular prisms of 2 different shapes. But I wonder - maybe it would be possible to rotate one of the duoprisms in a way that would make the lacings unit?

Anyway, this suggests that a new method of augmenting duoprisms might exist -- instead of building pyramids and cupolas etc. on their cells, you could remove a whole loop of cells and build something on the square faces in the duoprism ridge.
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### Re: Johnsonian Polytopes

OK, still wondering about the generic duoprism||duoprism lace toruses.

The basic possible forms seem to be:

2m,n||m,n -- not sure if possible

2m,n||m,2n

2m,2n||m,n

(m,n||m,n is impossible because it's still only 4D -- both duoprisms would have to be on the same layer).

In each case, there seem to be 4 sub-cases as there are 2 possibilities of orienting m vs. 2m (with edges of m parallel to edges of 2m or without) and 2 possibilities of orienting n vs. n (ortho or gyro). Well, the second orientation of m vs. 2m is never used in 3D solids, so it might be impossible here as well.

But now I seem to see a bit more. What's the problem with antiprismatic rings? Well, the structure of the layer corresponds to a triangular tiling of the plane. So the symmetry of the tiling isn't the same as symmetry of the layer. I wonder what happens when you go from one layer of grand antiprism vertices to the other. What's the structure of the layer exactly halfway? Well, some of the tetrahedra go from triangle in one layer to point, so they would be triangular, some go from point to triangle, some go from line to line, so they'd be squares...

Wait a sec.

Could the structure of middle layer of grand antiprism (halfway between both rings) be a snub square tiling that's made of triangles and squares? The picture on wikipedia (http://en.wikipedia.org/wiki/Snub_square_tiling) shows it in orientation where pairs of triangles are horizontal and vertical, so it can be easily seen as a transition point between triangular tiling with horizontal lines and triangular tiling with vertical lines.

And this means that if the double-decagonal antiprismatic ring is possible, it could have a structure very much like this as its middle layer of vertices.

EDIT: And this actually gives me an idea how to represent these lace toruses. Their basic form might be imagined as a stack of planar tilings with top and bottom having a unit edge length (even nonuniform ones like various combinations of square and triangle rows). As you go through the stack, the tiling will transform. However, you have four fixed points in the tiling (with rectangle defined by them repeating to tile the whole plane) and you trace the movement of those points as the tiling transforms.

So there is a general transformation on whole plane, and the lace torus is this transformation limited to a certain rectangular subsections of top and bottom tilings. I have a hunch that there will be severe constraints on how this subsection can be chosen (for example, when going between two orientations of triangular tiling, grand antiprism is the only valid choice), but given the freedom in choosing it, it SHOULD be possible to make such a choice in lots of cases.

And tilings that can be turned 90 degrees like square tiling or snub square tiling have special property of being able to serve as an "equator" of sorts.

Maybe next thing to do should be to make a ridge of duocylinder with equal radii, tile it with snub square tiling and try to make a CRF polychoron out of it...

EDIT2: Not quite sure of the duocylinder ridge geometry -- would the planar representation have to use non-unit edges so they could map to unit edges in the resulting polychoron or are distances preserved?
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### Re: Johnsonian Polytopes

wendy wrote:More polydimensional ursulate stuff.

I see that someone (prolly Kr litzing, sounds like something he would do), has done a stott expansion on the ursulates. xfo3oox4xxx would have both pentagons and octagons. A truly interesting mix. And this occurs in every dimension!

The icosahedral expanded-ursulate is given as tentitive. It actually exists, the face consist is an ID, 50 pentagonal prisms, 20 pentagonal cupolae, and a truncated dodecahedron. It's a crf as these things go.

Don't know anymore who brought up the expansion stuff applied to the ursachora.
Those sure are all existent, as soon as the non-expanded ones are. And that in turn was showed by Wendy.
So this wiki here definitely is way behind my page, where all 7 ones (yes, indeed, not just 6!) have been described for long.

--- rk
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### Re: Johnsonian Polytopes

The current number of ursulachora is 15, but i have not looked too hard yet.
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the dream we dream together is reality.

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### Re: Johnsonian Polytopes

Marek14 wrote:[...]
Could the structure of middle layer of grand antiprism (halfway between both rings) be a snub square tiling that's made of triangles and squares? The picture on wikipedia (http://en.wikipedia.org/wiki/Snub_square_tiling) shows it in orientation where pairs of triangles are horizontal and vertical, so it can be easily seen as a transition point between triangular tiling with horizontal lines and triangular tiling with vertical lines.

I have studied this before. Well, it started off as an attempt to generalize antiprisms to 4D using a different approach from the usual X || dual_X. I considered instead analysing an n-gonal antiprism as two n-gons surrounded by (almost) perpendicular triangles, with a sawtooth edge, joining together to form an antiprism. So I considered the question of how sawtooth-style cell arrangements would fit together in 4D. I approached this from a purely topological perspective.

First, suppose that there is some kind of 2-manifold on which pyramids would be erected (to serve as the "teeth" in the sawtooth configuration). Suppose we have some kind of n-gonal tiling of this 2-manifold, such that we can erect n-gonal pyramids on it. Suppose further, that the other side of the prospective "antiprism" has an m-gonal tiling of another 2-manifold. What are the conditions where these two sets of "teeth" would fit together? If the tips of the n-gonal pyramids have to lace to the other 2-manifold, and correspondingly the tips of the m-gonal pyramids from the other 2-manifold has to lace to the first 2-manifold, then topologically speaking the requirement is that there must be m n-gonal pyramids surrounding each m-gonal pyramid, and there must be n m-gonal pyramids surrounding each n-gonal pyramid. Now, if the 2-manifolds are compact tilings of the n-gons and m-gons of the respective pyramids, then this implies that the tilings on the two 2-manifolds must be dual to each other. When this requirement is fulfilled, then you can insert tetrahedra between the lacing edges, and this would close up the two sets of "teeth" to form a band of n-gonal pyramids, m-gonal pyramids, and tetrahedra that lace the two 2-manifolds.

Now, this nice topological conclusion has one problem: it doesn't explain the grand antiprism's structure. However, upon closer inspection, it turns out that the tetrahedra that lace the two rings of antiprisms actually correspond to the square tiling of the respective toroidal 2-manifolds that form the boundary of the rings of antiprisms -- except that the square pyramid "teeth" corresponding with such a tiling have been split into a pair of conjoined tetrahedra. So the tetrahedra in the grand antiprism are basically of two types: (1) pairs of tetrahedra between two antiprism cells in the same ring, whose tips touch the other ring, and (2) tetrahedra that fill in the gaps between the "split teeth" of the square tiling of the ring boundaries.

It just so happens, that these "split teeth" of the two rings line up just right, in order to have unit lacing edges. The structure is quite dependent on the exact ring angles in the grand antiprism, and trying to generalize the same pattern in a CRF way seems, at present, rather difficult.
quickfur
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### Re: Johnsonian Polytopes

quickfur wrote:
Marek14 wrote:[...]
Could the structure of middle layer of grand antiprism (halfway between both rings) be a snub square tiling that's made of triangles and squares? The picture on wikipedia (http://en.wikipedia.org/wiki/Snub_square_tiling) shows it in orientation where pairs of triangles are horizontal and vertical, so it can be easily seen as a transition point between triangular tiling with horizontal lines and triangular tiling with vertical lines.

I have studied this before. Well, it started off as an attempt to generalize antiprisms to 4D using a different approach from the usual X || dual_X. I considered instead analysing an n-gonal antiprism as two n-gons surrounded by (almost) perpendicular triangles, with a sawtooth edge, joining together to form an antiprism. So I considered the question of how sawtooth-style cell arrangements would fit together in 4D. I approached this from a purely topological perspective.

First, suppose that there is some kind of 2-manifold on which pyramids would be erected (to serve as the "teeth" in the sawtooth configuration). Suppose we have some kind of n-gonal tiling of this 2-manifold, such that we can erect n-gonal pyramids on it. Suppose further, that the other side of the prospective "antiprism" has an m-gonal tiling of another 2-manifold. What are the conditions where these two sets of "teeth" would fit together? If the tips of the n-gonal pyramids have to lace to the other 2-manifold, and correspondingly the tips of the m-gonal pyramids from the other 2-manifold has to lace to the first 2-manifold, then topologically speaking the requirement is that there must be m n-gonal pyramids surrounding each m-gonal pyramid, and there must be n m-gonal pyramids surrounding each n-gonal pyramid. Now, if the 2-manifolds are compact tilings of the n-gons and m-gons of the respective pyramids, then this implies that the tilings on the two 2-manifolds must be dual to each other. When this requirement is fulfilled, then you can insert tetrahedra between the lacing edges, and this would close up the two sets of "teeth" to form a band of n-gonal pyramids, m-gonal pyramids, and tetrahedra that lace the two 2-manifolds.

Now, this nice topological conclusion has one problem: it doesn't explain the grand antiprism's structure. However, upon closer inspection, it turns out that the tetrahedra that lace the two rings of antiprisms actually correspond to the square tiling of the respective toroidal 2-manifolds that form the boundary of the rings of antiprisms -- except that the square pyramid "teeth" corresponding with such a tiling have been split into a pair of conjoined tetrahedra. So the tetrahedra in the grand antiprism are basically of two types: (1) pairs of tetrahedra between two antiprism cells in the same ring, whose tips touch the other ring, and (2) tetrahedra that fill in the gaps between the "split teeth" of the square tiling of the ring boundaries.

It just so happens, that these "split teeth" of the two rings line up just right, in order to have unit lacing edges. The structure is quite dependent on the exact ring angles in the grand antiprism, and trying to generalize the same pattern in a CRF way seems, at present, rather difficult.

Well, my idea is to start from the snub square tiling. In one direction, build triangular prisms on its squares, triangular prisms on its "horizontal" triangles and tetrahedra on its "vertical" triangles. In the other direction, exchange triangular prisms and tetrahedra built on triangles and change the orientation of triangular prisms built on the squares.

And don't forget that since the original tiling is planar, the version taken for the polytope can be of any size (in this particular example I'd like to use it as an "equator", so a square patch would be needed) -- this works against the "exact ring angles" objection. Yes, the angles have to be just right, but there's an infinite amount of options to choose from, so there's a chance one of them DOES have the numbers right

And I have a weird hunch that the exact double of grand antiprism (20-membered rings of decagonal antiprisms) might just turn out to fit in this structure. What would it be called? The "grander" antiprism?

To be even more specific, you'd build on the antiprismatic ring by putting triangular prisms on triangles (instead of tetrahedra of grand antiprism). The tetrahedra built on vertices would be left as they are. The tetrahedra that are normally built on edges would be replaced by triangular prisms. This should lead to a second layer with snub square tiling geometry, and then second, orthogonal, ring of antiprisms can be attached in the same way. The only thing to decide is if it works with 20-10 ring, and if not, whether there is a combination that works...
Marek14
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### Re: Johnsonian Polytopes

Marek14 wrote:[...]
Well, my idea is to start from the snub square tiling. In one direction, build triangular prisms on its squares, triangular prisms on its "horizontal" triangles and tetrahedra on its "vertical" triangles. In the other direction, exchange triangular prisms and tetrahedra built on triangles and change the orientation of triangular prisms built on the squares.

And don't forget that since the original tiling is planar, the version taken for the polytope can be of any size (in this particular example I'd like to use it as an "equator", so a square patch would be needed) -- this works against the "exact ring angles" objection. Yes, the angles have to be just right, but there's an infinite amount of options to choose from, so there's a chance one of them DOES have the numbers right

And I have a weird hunch that the exact double of grand antiprism (20-membered rings of decagonal antiprisms) might just turn out to fit in this structure. What would it be called? The "grander" antiprism?

It's certainly an interesting direction to explore. My little research into "sawteeth" basically assumes a direct interfacing of the two rings of antiprisms; if we allow one or more intermediate layers in between, it may be possible to introduce enough degrees of freedom for things to close up in a CRF way.
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### Re: Johnsonian Polytopes

quickfur wrote:
Marek14 wrote:[...]
Well, my idea is to start from the snub square tiling. In one direction, build triangular prisms on its squares, triangular prisms on its "horizontal" triangles and tetrahedra on its "vertical" triangles. In the other direction, exchange triangular prisms and tetrahedra built on triangles and change the orientation of triangular prisms built on the squares.

And don't forget that since the original tiling is planar, the version taken for the polytope can be of any size (in this particular example I'd like to use it as an "equator", so a square patch would be needed) -- this works against the "exact ring angles" objection. Yes, the angles have to be just right, but there's an infinite amount of options to choose from, so there's a chance one of them DOES have the numbers right

And I have a weird hunch that the exact double of grand antiprism (20-membered rings of decagonal antiprisms) might just turn out to fit in this structure. What would it be called? The "grander" antiprism?

It's certainly an interesting direction to explore. My little research into "sawteeth" basically assumes a direct interfacing of the two rings of antiprisms; if we allow one or more intermediate layers in between, it may be possible to introduce enough degrees of freedom for things to close up in a CRF way.

Basically, various planar tilings would map to different types of layers.

A "degenerate" tiling would be a 1D tiling, or a loop of vertices. The decagon in 600-cell you have to remove to get a ring of grand antiprism would be an example.

A triangular tiling corresponds to a ring of antiprisms -- or, more precisely to the tube of their triangular faces. Triangles in a rectangular patch of such tiling can be oriented horizontally or vertically, which corresponds to two orientations of the tube. This is reason why rings of antiprisms cannot be directly blended, unlike rings of prisms.

A square tiling corresponds to a ring of prisms -- any such tilings can be completed in two ways to form a duoprism.
What if we turn it 45 degrees to form a "chainlink fence layer"? Will that lead to anything interesting?

A hexagonal tiling... that's an interesting one, a "carbon nanotube" layer. Not sure what it could connect to.

Snub square tiling -- it seems to be the "middle layer" of grand antiprism (formed by midpoints of its between-layer edges). It has 90-degree rotational symmetry, allowing it to be used as equatorial layer. However, it lacks long straight lines, which is why it probably can't be filled on its own like square tiling has.

Laminate tiling -- by itself, it can be completed, in one direction, to form a ring formed of alternating prisms and antiprisms. Antiprismatic prisms have this structure. There are possibilities in various patterns of square and triangular layers alternating, but it would be necessary to find how to enclose it from the other side.

Haven't though much about other tilings yet...
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### Re: Johnsonian Polytopes

Let's have a back-of-the-envelope calculation:

I have a basic 2m-membered ring of n-gonal antiprisms. What are its coordinates?

The centers of each n-gon would be vertices of regular 2m-gon lying in zw plane, whose edge is equal to height of the n-gonal antiprism.
Each of them would then expand into regular n-gon of unit edge in xy plane, with alternating orientations.

The layer has 2mn vertices

The next layer (our middle one) would be made by:

1. Extruding each triangle face into a triangular prism. The edges joining two antiprisms should become squares joining two of these prisms.
2. Every lacing edge of the antiprisms will become a triangular prism. It's joined to square faces of two triangular prisms built on the triangular faces the edge joins.
3. Every vertex expands into a pair of tetrahedra. Their triangular faces will join with triangular faces of triangular prisms built on the edges.

The total number of vertices in the middle layer should be four times higher than that of the first layer, i.e. 8mn (since each vertex is joined to 4 vertices of middle layer and none of these are shared).

Reversing the operations to get next antiprism layer leads to conclusion that the second layer should be a 2n-membered ring of m-gonal antiprisms.

What we need is an algorithm, that can take the numbers m and n and compute whether the middle layer with unit edges exists, and how it looks like. Theoretically, the polychoron should close if middle layer exists for both 2m/n and 2n/m, or if it exists for 2m/m.

BTW, I took a look in Stella on sections of grand antiprism. The snub square tiling is quite clear if you know what to look for

So... how many equations are needed?

Let's assume we already have the basic antiprism ring. We know that exists.

How many types of vertices are there in the middle layer? It turns out there are just two. There's 4mn vertices A adjacent to a triangular face joining 2 tetrahedra, and 4mn vertices B adjacent to a square face joining 2 triangular prisms built on the triangular faces of the original ring.

So we need 1 equation for antiprism ring vertex - A and 1 for antiprism ring vertex - B.
In order to fix the distances in the middle layer, we need to divide snub square tiling into squares, horizontal triangles and vertical triangles. We have square/horizontal, square/vertical, horizontal/horizontal and vertical/vertical edges. I *think* that we don't need an additional equation to ensure the quadrilateral will be really square -- that should follow from the symmetry of the figure.

All in all, it looks like there are 6 equations for 8 coordinates (4 for vertex A and 4 for vertex B, one of each should be sufficient). On the first look it looks like it might work, though there are probably mistakes in it somewhere.

BTW -- for the interlocking sawtooth approach: there might still be possible to have other shapes in this category than grand antiprism: Grand antiprism has 2*5,5 and 2*5,5 ring, but other 2*m,n and 2*n,m rings might be possible as well. If m and n are not equal, these polychora wouldn't be uniform, so they might not have been found before. And my hunch is that there is connection between both types of figures... that solution for m,n in one might mean a solution for 2m,2n (or another set of numbers closely related to m,n) in the other.
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### Re: Johnsonian Polytopes

Before we get too deep into the details, don't forget that the triangles in the triangular tiling of a grand antiprism's rings are not coplanar, and they are not all equivalent. Consider pairs of triangles that share an edge perpendicular to the direction of the ring. Some pairs form an obtuse angle where 2 tetrahedra may fit, whereas the other pairs form a reflex angle where three tetrahedra may meet. So, the triangles are not all equivalent, and this may change the possible CRF topologies in a profound way.

Edit: hmm, it seems that I was wrong, the pairs are all equivalent. I take that back.
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