## 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:
student91 wrote:And this one can be found by thawro-ing the o5o3x3o too. (the bilbiro-pseudopyramid can be found by bilbiro-ing the o5o3x3o).
a part of the o5o3x3o looks like xoxFof.....3oxoofo.....5ooxoox.....&#xt. Now a "thawro-ing" is done by deleting the o3f5o-layer of vertices, and scaling down the f-hexagons that can be found in the f3o5x-layer (three f's, together with three diagonals of o5x make an f-hexagon). This is done by changing f3o5x to x3x3o. This gives us oxFx3xoox5oxox&#xt.

I think you have a typo there, it should be oxFx3xoox5oxoo&#xt. What you wrote is non-CRF because the bottom x3x5x is too big, and requires either non-CRF lacing edges or self-intersection with the other layers.

I liked this idea. But sadly both of your descriptions look wrong in that bottom figure.

You start with f3o5x. Then you select 8 of those f3o . triangles (in fact those coplanar to a circumscribing large oct). These are connected to f . x rectangles, respectively . o5x Pentagons. Therefore these triangles indeed can be replaced by regular f3f hexagons.

But then, what would be the remainder of this still unscaled figure? YOu'll have 6 rectangles being left (in fact those coplanar to a circumscribing large cube). And the small edges of those will be connected to acute golden triangles x-f-f, while the large ones would be connected to regular triangles f-f-f (the remaining 12, not being replaced by hexagons).

Thus the figure thus described would be a variation of the rectification of the truncated octahedron. I.e. having 8 hexagons, 6 tetragons, and 24 triangles. But this figure supposedly cannot be made CRF. For sure not uniform. And esp. neither x3x3o, nor x3x5x, nor x3x5o does describe this figure!

--- rk
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### Re: Johnsonian Polytopes

Klitzing wrote:
quickfur wrote:
student91 wrote:And this one can be found by thawro-ing the o5o3x3o too. (the bilbiro-pseudopyramid can be found by bilbiro-ing the o5o3x3o).
a part of the o5o3x3o looks like xoxFof.....3oxoofo.....5ooxoox.....&#xt. Now a "thawro-ing" is done by deleting the o3f5o-layer of vertices, and scaling down the f-hexagons that can be found in the f3o5x-layer (three f's, together with three diagonals of o5x make an f-hexagon). This is done by changing f3o5x to x3x3o. This gives us oxFx3xoox5oxox&#xt.

I think you have a typo there, it should be oxFx3xoox5oxoo&#xt. What you wrote is non-CRF because the bottom x3x5x is too big, and requires either non-CRF lacing edges or self-intersection with the other layers.

I liked this idea. But sadly both of your descriptions look wrong in that bottom figure.

You start with f3o5x. Then you select 8 of those f3o . triangles (in fact those coplanar to a circumscribing large oct). These are connected to f . x rectangles, respectively . o5x Pentagons. Therefore these triangles indeed can be replaced by regular f3f hexagons.

But then, what would be the remainder of this still unscaled figure? YOu'll have 6 rectangles being left (in fact those coplanar to a circumscribing large cube). And the small edges of those will be connected to acute golden triangles x-f-f, while the large ones would be connected to regular triangles f-f-f (the remaining 12, not being replaced by hexagons).

Thus the figure thus described would be a variation of the rectification of the truncated octahedron. I.e. having 8 hexagons, 6 tetragons, and 24 triangles. But this figure supposedly cannot be made CRF. For sure not uniform. And esp. neither x3x3o, nor x3x5x, nor x3x5o does describe this figure!

--- rk

Hmm. When I was constructing the vertices for this model, I found that the o5x3x at the bottom of the tower was wrongly-aligned with the rest of it, so there were lots of non-CRF edges, faces, and cells. However, by a simple change in orientation (swap a pair of coordinates in each vertex), the tower became CRF, and thus admits the octa-diminishing described by student91 which produces 8 J92's. I'm not sure if this was just a mistake on my part (messed up order of coordinates at first), or there's an actual orientation flip that produces a CRF figure, in which case I'm not sure what CD symbol could be used to describe it.

In any case, whatever it is, the model I have on file is definitely CRF, and surely has 8 J92 cells. I'm just not sure if student91's lace tower accurately describes it.

EDIT: Actually, on second thoughts, it most definitely is o5x3x at the bottom of the lace tower. The orientation flip was my own mistake in my first attempt to construct the model. If you look at the image I posted, you can see that the pentagons of the icosidodecahedron and the pentagons of the outer truncated icosahedron are aligned. So there is no magic here, it's definitely o5x3x.

EDIT 2: Here's a projection of the non-diminished lace tower:

The center yellow icosidodecahedron is o5x3o; the tops of the blue octahedra is x5o3x, the tops of the truncated icosahedra (gyroelongated pentagonal pyramids) that protrude from the red outlined truncated icosahedron is o5o3F, and the red outlined truncated icosahedron itself is o5x3x. Note that the o5o3F overhangs the o5x3x a little, but it lies between x5o3x and o5x3x in the 4th direction.

Hope this helps to clear things up.
Last edited by quickfur on Mon Feb 24, 2014 9:31 pm, edited 1 time in total.
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### Re: Johnsonian Polytopes

It Looks so that the truncated icosahedron will fit here. But you too see at one layer underneath the non-flat Arrangement that I was pointing out not to be possible with regular hexagons. Probably you were adding a further layer to finally close that thing?

For sure there cannot be a CD description of this figure. This is just because pyritohedral symmetry cannot be described as a Coxeter group. And then there will neither be an axial group, having that as surround symmetry.

Quite a fascinating figure, indeed!

--- rk
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### Re: Johnsonian Polytopes

Btw, quickfur, I'd like to see into that figure a bit deeper. Even so your coloring is very appealing, it kind of hides the structure a bit. Perhaps you could set up a full sequence of added layers, one by one, just as for the figures at your homepage?

--- rk
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### Re: Johnsonian Polytopes

Suppose that the naming issue is around.

What about a Thawro Pyrite?
(Or just J92 Pyrite?)

--- rk
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### Re: Johnsonian Polytopes

quickfur wrote:EDIT 2: Here's a projection of the non-diminished lace tower:

That one looks even prettier, is it too CRF?

(and can you check your PMs? )

Klitzing wrote:Suppose that the naming issue is around.

What about a Thawro Pyrite?
(Or just J92 Pyrite?)

--- rk

Hmm, that made me want to call it thawro pyrochoron, but pyrochoron is already taken So it may be rather confusing to call it pyrite...

Keiji

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### Re: Johnsonian Polytopes

Klitzing wrote:
quickfur wrote:
student91 wrote:And this one can be found by thawro-ing the o5o3x3o too. (the bilbiro-pseudopyramid can be found by bilbiro-ing the o5o3x3o).
a part of the o5o3x3o looks like xoxFof.....3oxoofo.....5ooxoox.....&#xt. Now a "thawro-ing" is done by deleting the o3f5o-layer of vertices, and scaling down the f-hexagons that can be found in the f3o5x-layer (three f's, together with three diagonals of o5x make an f-hexagon). This is done by changing f3o5x to x3x3o. This gives us oxFx3xoox5oxox&#xt.

I think you have a typo there, it should be oxFx3xoox5oxoo&#xt. What you wrote is non-CRF because the bottom x3x5x is too big, and requires either non-CRF lacing edges or self-intersection with the other layers.

I liked this idea. But sadly both of your descriptions look wrong in that bottom figure.

You start with f3o5x. Then you select 8 of those f3o . triangles (in fact those coplanar to a circumscribing large oct). These are connected to f . x rectangles, respectively . o5x Pentagons. Therefore these triangles indeed can be replaced by regular f3f hexagons.

But then, what would be the remainder of this still unscaled figure? YOu'll have 6 rectangles being left (in fact those coplanar to a circumscribing large cube). And the small edges of those will be connected to acute golden triangles x-f-f, while the large ones would be connected to regular triangles f-f-f (the remaining 12, not being replaced by hexagons).

Thus the figure thus described would be a variation of the rectification of the truncated octahedron. I.e. having 8 hexagons, 6 tetragons, and 24 triangles. But this figure supposedly cannot be made CRF. For sure not uniform. And esp. neither x3x3o, nor x3x5x, nor x3x5o does describe this figure!

--- rk

Indeed a typo, I'll correct that in my previous post. Your renders are very beautifull, they nicely show the pyritohedral symmetry, and make me able to have a real visual to it

First I derived a thing that would give a thawro if diminished, and afterwards I could get a thawro by a simple diminishing. The deriviation of the figure that can be diminished into someting with thawros is called thawro-ing, so I'm not including the diminishing to the thawroing.
The way I derived the thawro'd o5o3x3o was by looking at the f3o5x as a f3f5A, with A=-1. I'm not sure thats the way negative nodes are used, but when I used it this way it worked. I ment with the negative number that the edge had a length 1 the other way around. Now you have a f3f.. that can be shrinked to x3x. The A has to become 0, so that gives me x3x5o. I think I cheated a bit with the negative numbers, but at least it worked.
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### Re: Johnsonian Polytopes

Klitzing wrote:It Looks so that the truncated icosahedron will fit here. But you too see at one layer underneath the non-flat Arrangement that I was pointing out not to be possible with regular hexagons. Probably you were adding a further layer to finally close that thing?

No, the o5x3x is the bottom cell of the polytope. The o5o3F vertices actually lie above this cell (though in projection they protrude beyond the image of the o5x3x -- they are only overhanging, not underneath).

For sure there cannot be a CD description of this figure. This is just because pyritohedral symmetry cannot be described as a Coxeter group. And then there will neither be an axial group, having that as surround symmetry.

Well, student91's initial lace tower is certainly accurately described as o5x3o || x5o3x || o5o3F || o5x3x. However, this is only the initial figure; what follows is a deletion of 8 triangles from x5o3x that produces the J92's. So after this deletion, we cannot describe the second layer of vertices as x5o3x anymore, but it's some kind of non-CRF diminishing of x5o3x (even though the resulting figure itself is CRF due to the J92's thus introduced).

Klitzing wrote:Btw, quickfur, I'd like to see into that figure a bit deeper. Even so your coloring is very appealing, it kind of hides the structure a bit. Perhaps you could set up a full sequence of added layers, one by one, just as for the figures at your homepage?

--- rk

Hmm, that will take some time. But I'll see if I can throw something together.

In my model source coordinates, I actually construct it exactly as student91 describes: first as the lace tower with icosahedral symmetry, then delete the vertices of 8 triangles from it as prescribed. So the second layer vertices are a subset of x5o3x, not the entire x5o3x.
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### Re: Johnsonian Polytopes

quickfur wrote:EDIT 2: Here's a projection of the non-diminished lace tower:

The center yellow icosidodecahedron is o5x3o; the tops of the blue octahedra is x5o3x, the tops of the truncated icosahedra (gyroelongated pentagonal pyramids) that protrude from the red outlined truncated icosahedron is o5o3F, and the red outlined truncated icosahedron itself is o5x3x. Note that the o5o3F overhangs the o5x3x a little, but it lies between x5o3x and o5x3x in the 4th direction.

Hope this helps to clear things up.

=   oxFx3xoox5oxoo&#xt
=   id || pseudo srid || pseudo F-ike || ti

Sure it does!

--- rk
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### Re: Johnsonian Polytopes

Keiji wrote:
quickfur wrote:EDIT 2: Here's a projection of the non-diminished lace tower:

That one looks even prettier, is it too CRF?

Definitely, yes. It's a kind of modified tristratic cap cut from a rectified 600-cell. Basically, you cut off an icosahedron||icosidodecahedron from the rectified 600-cell, then you go down 4 layers of vertices below that, and cut off the rest of the rectified 600-cell. The bottom of the remaining tristratic polychoron will not be CRF, because the bottom layer of vertices has non-unit edges. So what you do, is to replace that bottom layer with a CRF truncated icosahedron, and all edges become unit length again, making the lace tower CRF. Quite an ingenious construction, I must say!

(and can you check your PMs? )

Whoa, one thing at a time! I post a couple o' *cough* awesome images and suddenly you guys are flooding me with messages!

Klitzing wrote:Suppose that the naming issue is around.

What about a Thawro Pyrite?
(Or just J92 Pyrite?)

--- rk

Hmm, that made me want to call it thawro pyrochoron, but pyrochoron is already taken So it may be rather confusing to call it pyrite...

I already created an entry for it on the crown jewels page as CJ4.7, so once you guys duke it out over the naming, you can just rename the page.

Personally, though, I would name it "octa-diminished <something>", where <something> is the name of student91's ingenious lace tower. But I have no idea what to call that.
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### Re: Johnsonian Polytopes

You even could ignore those F-ike vertices.
Then it becomes a simple stack of 2 segmentochora:

id || srid (= ox3xo5ox&#x) with cells being: 1 id + 20 oct (=3ap) + 30 squippy + 12 pap + 1 srid

srid || ti (= xx3ox5xo&#x) with cells being: 1 srid + 20 tricu + 30 trip (=2cu) + 12 pap + 1 ti

Note that the former one has a circumradius of sqrt[5+2 sqrt(5)] = 3.077684 (and does belong to a true section of rox), while the second one has a radius of sqrt[(106+41 sqrt(5))/32] = 2.485450 only.

If you further would consider the latter paps each together with the therein deleted F-ike vertex, then you'd get a local circumradius of (1+sqrt(5))/2 = 1.618034 there...

--- rk
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### Re: Johnsonian Polytopes

quickfur wrote:
Keiji wrote:
quickfur wrote:EDIT 2: Here's a projection of the non-diminished lace tower:

That one looks even prettier, is it too CRF?

Definitely, yes. It's a kind of modified tristratic cap cut from a rectified 600-cell. Basically, you cut off an icosahedron||icosidodecahedron from the rectified 600-cell, then you go down 4 layers of vertices below that, and cut off the rest of the rectified 600-cell. The bottom of the remaining tristratic polychoron will not be CRF, because the bottom layer of vertices has non-unit edges. So what you do, is to replace that bottom layer with a CRF truncated icosahedron, and all edges become unit length again, making the lace tower CRF. Quite an ingenious construction, I must say!
well, it's basically the same as what we did to thawro the o5x3o3o, with the only difference that in this thing the thawro's overlap. I think a lot more such constructions are possible, but they are not that interesting, as they don't allow thawro-diminishings. So basically it's the same operation as thawroing the o5x3o3o. Now maybe, because that thawroing had another version (compare CJ4.5.2 with 4.5.3) this one has another version as well.
(and can you check your PMs? )

Whoa, one thing at a time! I post a couple o' *cough* awesome images and suddenly you guys are flooding me with messages!

Klitzing wrote:Suppose that the naming issue is around.

What about a Thawro Pyrite?
(Or just J92 Pyrite?)

--- rk

Hmm, that made me want to call it thawro pyrochoron, but pyrochoron is already taken So it may be rather confusing to call it pyrite...

I already created an entry for it on the crown jewels page as CJ4.7, so once you guys duke it out over the naming, you can just rename the page.
Personally, though, I would name it "octa-diminished <something>", where <something> is the name of student91's ingenious lace tower. But I have no idea what to call that.

I would go for thawro'd o5o3x3o for the undiminished stack. If the other version exists as well, we have to distinguish between a polar thawroing and a (possible ) equatorial thawroing. Maybe Keji can come up with a more appealing verb for thawroing? Anyway, this would also solve the naming problem of CJ4.5.2 and CJ4.5.3. After all, it's just the same procedure applied to a different uniform. (Sidethought: maybe we should make a list of procedures that can be applied to uniforms, rather than uniforms with the procedure applied to it. Aftef all, CJ 4.5.4 is "just" a o5x3o3o with both bilbiroing and thawroing apllied to it. The general procedures bilbiroing and thawroing already imply this one, together with loads of combinations with other cuts. So we would distinguish between procedures and combinations. After all, the procedures are the most elementary, and wheather my stack dows or doesn't have an ike at the end is just considered unimportant.)
Note that a bilbroing of o5o3x3o is also possible, but I haven't explicitely derived its lace construction yet.
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### Re: Johnsonian Polytopes

student91 wrote:[...]
(Sidethought: maybe we should make a list of procedures that can be applied to uniforms, rather than uniforms with the procedure applied to it. Aftef all, CJ 4.5.4 is "just" a o5x3o3o with both bilbiroing and thawroing apllied to it. The general procedures bilbiroing and thawroing already imply this one, together with loads of combinations with other cuts. So we would distinguish between procedures and combinations. After all, the procedures are the most elementary, and wheather my stack dows or doesn't have an ike at the end is just considered unimportant.)

This is a good point. Now that we have found a good number of crown jewels with common trends among them, we should look at the common parts of their construction and investigate where the same operations can be applied. This will help not only with the naming, but also with the categorization of these CRFs.

On a more general note, we might want to look over the current list of known CRFs and identify common operations that apply to each significant subset. I'm not very happy with the organization of the CRF polychora discovery project page right now; I think we can and should take a much more fundamental approach to it by identifying the underlying constructions / operations, rather than just lumping things together based on ad hoc associations.

Note that a bilbroing of o5o3x3o is also possible, but I haven't explicitely derived its lace construction yet.

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### Re: Johnsonian Polytopes

Just going over some more Johnson polyhedra, I'm curious about the following three with threefold symmetry (similar to J63 and J92)...

*triaugmented triangular prism (J51)
*triaugmented hexagonal prism (J57)
*augmented truncated tetrahedron (J65)

In particular, might J51 || triangle, J57 || triangle and J65 || triangular prism exist as CRFs? looking at the vertex positions I don't think so, but just some shots in the dark

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### Re: Johnsonian Polytopes

Keiji wrote:Just going over some more Johnson polyhedra, I'm curious about the following three with threefold symmetry (similar to J63 and J92)...

*triaugmented triangular prism (J51)
*triaugmented hexagonal prism (J57)
*augmented truncated tetrahedron (J65)

In particular, might J51 || triangle, J57 || triangle and J65 || triangular prism exist as CRFs? looking at the vertex positions I don't think so, but just some shots in the dark

I'm also thinking about Marek's J91 || gyro J91 (in some sense of gyro that preserves bilateral symmetry). It may not be possible to make a monostratic CRF from it, but a polystratic might work?
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### Re: Johnsonian Polytopes

quickfur wrote:[...]

Note that a bilbroing of o5o3x3o is also possible, but I haven't explicitely derived its lace construction yet.

I already tried to explain it in my previous posts, but I think everyone missed that:
student91 wrote:
student91 wrote:
quickfur wrote:
It might just be possible that things with id-cuts will allow a relatively simple bilbiro-ing. The only things with id-cuts are as far as I know the o5o3x3o and the o5o3o3x. The latter probably won't be bilbiro-able. This means from now on I'll be looking for a bilbiro-ing of the o5o3x3o. I think that when I've investigated that one, I'll be able to say something about the other ways of bilbiro-ing you proposed.

I'd love to hear about that!

I don't have anything yet, I just was saying what I was going to innvestigate next, and when I'm done investigating (or stuck) I'll of course share my results with this forum.

I got something! When you take the o5o3x3o, cut it at the x5x3o's, delete the equatorial vertices, glue the x5x3o's together, there should occur some pentagonal prisms (the two pentagonal cupola's should get automatically relpaced with the other half of xxx5xoo&#x). Now if you delete a 10,10 edge of the x5x3o, a bilbiro should occur. Don't have much time to be clearer though.

Basically all info needed to build a bilbiro'd is included in this post. Any attempt I'l make to explain this is just saying the same in different words.

Bilbiroing means taking ids, and deleting some vertices such that you're left with two lunes. Then you glue these two lunes together and Tadaa!.
Now o5o3x3o has .2.-oriented ids at the equator, so some equatorial vertices should be deleted. These vertices are the vertices between the x5x3o-cuts of the rox. (Basically any bilbiroing deletes every vertex with |x|<phi/2, because the ids are perpendicular to a specific plane). This basically gives us the lace tower (something)||x5x3o||(the same something). The somethings are what is on the "pole" side of x5x3o in a normal rox . Now we can cut off an J91-pseudopyramid, (this is done by deleting an 10,10 edge of the x5x3o) resulting in a bilbiro. The maximal diminishing here is 10 cuts corresponding to an pentagonal antiprism inscribed in an id, but maybe a cut based on an inscribed octahedron is more interesting, and looks better if combined with the thawroing.
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student91
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### Re: Johnsonian Polytopes

Keiji wrote:Just going over some more Johnson polyhedra, I'm curious about the following three with threefold symmetry (similar to J63 and J92)...

*triaugmented triangular prism (J51)
*triaugmented hexagonal prism (J57)
*augmented truncated tetrahedron (J65)

In particular, might J51 || triangle, J57 || triangle and J65 || triangular prism exist as CRFs? looking at the vertex positions I don't think so, but just some shots in the dark

Well, the problem with first two is that triangular prism||triangle exists and so does hexagonal prism||triangle. So the extra augment vertices would have to have the same distance to vertices of this triangle and that doesn't seem very likely.
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### Re: Johnsonian Polytopes

I found some parameters for the ursulates. The numbers given here are the squares of the diameters of the top base.

d > 10.472 ... No ursulate exists
d > 5.236067... The 'valence faces' slope inwards, and thus the ursulate can be used as a capping, like a cupola.
d > 4.000000... no additional features to the ursulate.
d < 4.000000... The top face of an ursulate can be apiculated (a pyramid on top)
d < 3.618033... The first rectate also forms an ursulate.
....

The formula for the rectate of a triangle-based polytope, is c = (r+1)²b - 2r(r+1). r is the order of the rectate. One puts c=4*2.61803398875 and solves for b by r. This will give the order of 'having two rectates' etc.
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### Re: Johnsonian Polytopes

Last night I discovered another CRF, this time with tetrahedral symmetry, and with both J91 and J92 cells! Here is the lace tower:

Code: Select all
`x3o3of3x3oo3F3ox3x3fF3o3xx3o3fo3x3o`

Here's a projection of the near side (seen from the top of the lace tower):

The top begins with a single tetrahedral cell (yellow) surrounded by 4 J92's, with 4 tetrahedra (green) filling in the gaps between them. This is a patch of the rectified 120-cell's surface. Past this point, however, it begins to take on unique characteristics of its own: 4 octahedra (red outline) touch the 4 green tetrahedra, connected to 3 square pyramids and 3 pentagonal pyramids that fill in the gaps between the J92's, thus completing the near side of the polytope. Here, we have a very interesting situation where the projection envelope is a uniform icosidodecahedron EDIT: this is not an icosidodecahedron, but something else; see my other post below, but in 4D, it is actually a skew icosidodecahedron, because this polychoron does not have an icosidodecahedral cross-section! The points on this icosidodecahedral envelope are not coplanar.

Now we come to the far side, where things begin to get really interesting. The pentagonal faces from the near side are touching 6 J91s which surround an antipodal octahedron in alternated orientation:

For clarity, I didn't color the J91 nearest to the 3D viewpoint, but its outline should be obvious. The red outlined triangular faces show where 4 tridiminished icosahedra (J63) touch the octahedra on the near side of the polytope. They alternate with 4 other octahedra that surround the antipodal octahedron in a formation alternating with the J63's. The remaining gaps are filled by tetrahedra and triangular cupolae.

Here are the coordinates of this beauty ('~' means concatenation, it's a nice way to be able to use Wendy's permutation operators on an initial segment of the vector):
Code: Select all
`# x3o3o:apecs<1/√2, 1/√2, 1/√2> ~ <0># f3x3o:apecs<phi/√2, phi/√2, (phi+2)/√2> ~ <1/(phi*√2)># o3F3o:apacs<phi^2*√2, 0, 0> ~ <1/√2># x3x3f:apecs<1/(phi*√2), phi^2/√2, -(3+phi)/√2> ~ <phi/√2># F3o3x:apecs<(phi+2)/√2, phi/√2, (phi+2)/√2> ~ <phi^2/√2># f3o3x:apecs<phi^2/√2, phi^2/√2, -1/(phi*√2)> ~ <(phi+2)/√2># o3x3o:apacs<0, 0, √2> ~ <phi^3/√2>`

I'll post the .def and .off files on the wiki.

EDIT: Wiki page is up: CJ4.8

EDIT2: fixed reversed CD symbol in the lace tower at the top of this post.
Last edited by quickfur on Wed Feb 26, 2014 11:57 pm, edited 2 times in total.
quickfur
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### Re: Johnsonian Polytopes

Looks very interesting, one side with J91's, one side with J92's. Like Jekyll and Hyde.
Marek14
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### Re: Johnsonian Polytopes

Marek14 wrote:Looks very interesting, one side with J91's, one side with J92's. Like Jekyll and Hyde.

Haha, maybe we should call it the Jekyll and Hyde polychoron, or JH-choron. (You intended that, didn't you? )
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### Re: Johnsonian Polytopes

quickfur wrote:
Marek14 wrote:Looks very interesting, one side with J91's, one side with J92's. Like Jekyll and Hyde.

Haha, maybe we should call it the Jekyll and Hyde polychoron, or JK-choron. (You intended that, didn't you? )

Wouldn't it be a JH-choron?
Marek14
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### Re: Johnsonian Polytopes

Marek14 wrote:[...] Wouldn't it be a JH-choron?

Haha, fixed the typo just as you replied.

But on a more serious note, what fascinates me is that skew icosidodecahedron. The images above are done with perspective projection, where the skew is obvious; when I was building the polytope with parallel projections, it looked like an actual icosidodecahedron, and the only hint that it wasn't quite what it seemed was that the convex hull algo produced a cuboctahedron with F-edges on the far side of the polychoron (after the F3o3x layer) instead of an icosidodecahedral cell, so I knew the points on the icosidodecahedron are not coplanar. (Well in retrospect it's pretty obvious, since the vertex layers never included o5x3o!)

EDIT: Hey wait a minute... that's not an icosidodecahedron!! That's some kind of strange skew polyhedron with a mixture of hexagonal faces and pentagonal faces...
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### Re: Johnsonian Polytopes

Interesting... here's a (parallel) projection of this skew polyhedron:

Contrary to what I said, it does still look skewed in parallel projection. Looks to be some kind of modified icosidodecahedron where some pentagons are substituted with hexagons. It has some regular decagonal sections, but also a bunch of weird swirling skew dodecagons. On the far side you can see an interesting lune consisting of a triangle, a pentagon, 2 triangles, a hexagon, then 2 triangles and a pentagon again, and ending in a triangle, all lying between two intersecting skew dodecagons. Really fascinating structure!
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### Re: Johnsonian Polytopes

quickfur wrote:Interesting... here's a (parallel) projection of this skew polyhedron:

Contrary to what I said, it does still look skewed in parallel projection. Looks to be some kind of modified icosidodecahedron where some pentagons are substituted with hexagons. It has some regular decagonal sections, but also a bunch of weird swirling skew dodecagons. On the far side you can see an interesting lune consisting of a triangle, a pentagon, 2 triangles, a hexagon, then 2 triangles and a pentagon again, and ending in a triangle, all lying between two intersecting skew dodecagons. Really fascinating structure!

You know, where else on the internet can you find a forum thread which would so often blow your mind?

Maybe we should send a message to Polyhedron list telling them about the recent discoveries during the CRFebruary.
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### Re: Johnsonian Polytopes

Marek14 wrote:[...]
You know, where else on the internet can you find a forum thread which would so often blow your mind?

Why else do you think I'm here?

Maybe we should send a message to Polyhedron list telling them about the recent discoveries during the CRFebruary.

Well, I'm not on that list (only ever heard about it). I believe Klitzing and Polyhedron Dude are. Are you on too? It would be nice to have these discoveries made known to more people interested in the subject.

And heh, yeah, this month has been quite the month for CRFs, hasn't it? CRFebruary indeed. (Pity it doesn't sound as good pronounced out loud than it looks in written form. ) We've been dreaming about CRF crown jewels for so long, even years IIRC, not knowing how to even start finding them, and now all of a sudden we're finding lots of them almost everywhere we look.

But on that note, I again observe that so far, our discoveries have been limited to the Johnson solids that are closely related to the icosahedral polyhedra, thus there is a natural association, I would even say affinity, for structures derived from, or close to, the 120-cell family uniform polychora. And while computing the coordinates for CJ4.8 last night, I got the feeling that there are probably other ways to CRF-ify the initial tetrahedron + 4*J92 cell complex. (I'm thinking I should explore those other possibilities as well, some time.) But there hasn't been anything truly novel yet, like the analogue of the snub square antiprism or the snub disphenoid. This is why I came up with the idea of polytope complexity, which led to our definition of CVP, the idea being that even though J91 and J92 (and the icosahedron diminishings, etc.) are 3D crown jewels, they are "soft crown jewels" in the sense of only being CVP 2, whereas truly unique structures like snub disphenoid or snub square antiprism are CVP 3, and may be regarded as "hard" crown jewels. So in a sense, the crown jewels we've found so far are still in the "easy" class, whereas the 4D CVP 3 CRFs, if they exist, will probably the hardest to find (and also the most gratifying when we do find them!). I still don't know how to even begin searching for them, though.
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### Re: Johnsonian Polytopes

Yes, I am on the list.

As for soft and hard -- snub square antiprism is also only "soft" crown jewel -- snub disphenoid, icosahedron and snub square antiprism form a natural progression.

BTW, I was thinking about snub cube the other day. Snub cube can be thought of as a "broken" rhombicuboctahedron -- the "side" squares are broken in two triangles and then the whole thing is put back together. But in 4D, there is not as much opportunities to do similar things.

So I was thinking: Take an icositetrachoron. It has 24 octahedra that can be split in 3 groups of 8. Now... if you'd take some of those octahedra and broke them in two square pyramids, could there be a way to introduce a global "fault" in the figure that could cause it to come together with non-180 dichoral angle on square faces? A broken icositetrachoron, of sorts?
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### Re: Johnsonian Polytopes

Marek14 wrote:Yes, I am on the list.

As for soft and hard -- snub square antiprism is also only "soft" crown jewel -- snub disphenoid, icosahedron and snub square antiprism form a natural progression.

Are you sure the snub square antiprism is soft? Didn't we establish that it has CVP 3? As for natural progressions, going from 6,6-duoprism (CVP 2) to 7,7-duoprism is a pretty natural progression to me, but it doesn't stop heptagons in the 7,7-duoprism from being CVP 3.

BTW, I was thinking about snub cube the other day. Snub cube can be thought of as a "broken" rhombicuboctahedron -- the "side" squares are broken in two triangles and then the whole thing is put back together. But in 4D, there is not as much opportunities to do similar things.

So I was thinking: Take an icositetrachoron. It has 24 octahedra that can be split in 3 groups of 8. Now... if you'd take some of those octahedra and broke them in two square pyramids, could there be a way to introduce a global "fault" in the figure that could cause it to come together with non-180 dichoral angle on square faces? A broken icositetrachoron, of sorts?

I've tried these kinds of analogies before, but I found that these kinds of deforming operations don't carry over to 4D very well. So far, I've had much better success with tackling 4D constructions directly.

Having said that, though, the snub 24-cell has some similarities to the snub cube, it has two kinds of tetrahedra, one interfacing with the icosahedra, and the other sharing faces only with other tetrahedra. So your idea of deforming the 24-cell while allowing non-uniform Johnson cells (square pyramids) might lead somewhere interesting, I don't know.

I've also been thinking that so far, we've been working mostly with rigid constructions -- diminish some vertices, substitute some cells / cross sections, etc.. But another way to approach this is to construct the polychoron topologically -- figure out what types of cell shapes can close up topologically, then after we've figured out a topologically-closed cell complex, we can check to see if it can be embedded in 4-space in a CRF way. Working topologically also means you won't work yourself into an impossible corner -- where no possible cell combination can close the shape. Maybe we should figure out what kind of cells might result from your "broken" 24-cell, then see if it can be made CRF?
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### Re: Johnsonian Polytopes

quickfur wrote:
Marek14 wrote:Yes, I am on the list.

As for soft and hard -- snub square antiprism is also only "soft" crown jewel -- snub disphenoid, icosahedron and snub square antiprism form a natural progression.

Are you sure the snub square antiprism is soft? Didn't we establish that it has CVP 3? As for natural progressions, going from 6,6-duoprism (CVP 2) to 7,7-duoprism is a pretty natural progression to me, but it doesn't stop heptagons in the 7,7-duoprism from being CVP 3.

BTW, I was thinking about snub cube the other day. Snub cube can be thought of as a "broken" rhombicuboctahedron -- the "side" squares are broken in two triangles and then the whole thing is put back together. But in 4D, there is not as much opportunities to do similar things.

So I was thinking: Take an icositetrachoron. It has 24 octahedra that can be split in 3 groups of 8. Now... if you'd take some of those octahedra and broke them in two square pyramids, could there be a way to introduce a global "fault" in the figure that could cause it to come together with non-180 dichoral angle on square faces? A broken icositetrachoron, of sorts?

I've tried these kinds of analogies before, but I found that these kinds of deforming operations don't carry over to 4D very well. So far, I've had much better success with tackling 4D constructions directly.

Having said that, though, the snub 24-cell has some similarities to the snub cube, it has two kinds of tetrahedra, one interfacing with the icosahedra, and the other sharing faces only with other tetrahedra. So your idea of deforming the 24-cell while allowing non-uniform Johnson cells (square pyramids) might lead somewhere interesting, I don't know.

I've also been thinking that so far, we've been working mostly with rigid constructions -- diminish some vertices, substitute some cells / cross sections, etc.. But another way to approach this is to construct the polychoron topologically -- figure out what types of cell shapes can close up topologically, then after we've figured out a topologically-closed cell complex, we can check to see if it can be embedded in 4-space in a CRF way. Working topologically also means you won't work yourself into an impossible corner -- where no possible cell combination can close the shape. Maybe we should figure out what kind of cells might result from your "broken" 24-cell, then see if it can be made CRF?

Well, another interesting breaking would be breaking of o5o3x3o or o5o3x3x which contain icosahedra, since icosahedron can be broken in many different ways.
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### Re: Johnsonian Polytopes

quickfur wrote:Last night I discovered another CRF, this time with tetrahedral symmetry, and with both J91 and J92 cells! Here is the lace tower:

Code: Select all
`x3o3of3x3oo3F3ox3x3fF3o3xf3o3xo3x3o`

Cool CRF, really cool CRF!! . I think you made a typo, shouldn't the f3o3x be x3o3f? This is a really cool polytope, it has both J91 and J92, and it isn't directly related to an 120-uniform, so It's even cooler than the J92-rhombochoron!! .

Now a bit of analysis on it's structure:
Here's a projection of the near side (seen from the top of the lace tower):

The top begins with a single tetrahedral cell (yellow) surrounded by 4 J92's, with 4 tetrahedra (green) filling in the gaps between them. This is a patch of the rectified 120-cell's surface. Past this point, however, it begins to take on unique characteristics of its own: 4 octahedra (red outline) touch the 4 green tetrahedra, connected to 3 square pyramids and 3 pentagonal pyramids that fill in the gaps between the J92's, thus completing the near side of the polytope. Here, we have a very interesting situation where the projection envelope is a uniform icosidodecahedron EDIT: this is not an icosidodecahedron, but something else; see my other post below, but in 4D, it is actually a skew icosidodecahedron, because this polychoron does not have an icosidodecahedral cross-section! The points on this icosidodecahedral envelope are not coplanar.

This seems to be a tetrahedron-oriented thawroing of o5x3o3o. The thawroings we've seen before were all icosahedron-oriented. Doing a tetraheron-oriented one is really revolutionary
Now we come to the far side, where things begin to get really interesting. The pentagonal faces from the near side are touching 6 J91s which surround an antipodal octahedron in alternated orientation:

For clarity, I didn't color the J91 nearest to the 3D viewpoint, but its outline should be obvious. The red outlined triangular faces show where 4 tridiminished icosahedra (J63) touch the octahedra on the near side of the polytope. They alternate with 4 other octahedra that surround the antipodal octahedron in a formation alternating with the J63's. The remaining gaps are filled by tetrahedra and triangular cupolae.
[...]

this side looks like a tetrahedron-oriented bilbiroing of the o5o3x3o. Apparantely, a tetrahedron-oriented bilbiroing isn't happening at the equator, but at the second set of ids. The tetrahedron-oriented bilbiro-ing resp. thawro-ing certainly opens new possibilities , great find
It's especially cool that you integrated a thawro-ing and a bilbiro-ing of two different 120-uniforms, this one definately classifies as exceptional crown jewel
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