Non-Wythoffian Convex Uniform Polytopes in dimensions 5+?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Non-Wythoffian Convex Uniform Polytopes in dimensions 5+?

Postby ytrepus » Fri Feb 07, 2014 12:17 pm

Hello,

I was wondering if there are any known convex uniform polytopes in dimensions 5+ that can't be derived from the simplex, hypercube/orthoplex, demihypercube, or E6 to E8 groups, or from a prism of a non-Wythoffian polytope like the grand antiprism?

If there are not any known, has it been proven that they don't exist for any particular dimension?

EDIT: I'm not as interested in tilings/honeycombs at this point -- just discrete convex polytopes...thanks

Thanks a lot,
Rob
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Klitzing » Fri Feb 07, 2014 3:55 pm

Dont know whether you considered cartesian products of lower dimensional uniform polytopes already? E.g. in 4D there occur the n,m-duoprisms. In 6D you can have similar n,m,k-triprisms, etc. But also as crude things as the duoprism of a snub cube and a snub dodecahedron could be considered, or from the snub 24-cell and a heptagon, ... (both of which would be 6D figures). But multiprisms surely occur in every higher Dimension. And those all are surely not bluddy prisms!

Snubbing, as operation from (canonical) wythoffians into still uniforms, in higher dimensions becomes highly restricted quite soon. This is because of the task to relax simultanuously more and more independant types of usually differently sized edges (the mere outcome from alternated vertex faceting) back to unity. But the degree of freedom to do so does not hold up.

--- rk
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby ytrepus » Fri Feb 07, 2014 5:33 pm

Thanks for the reply!

Yes, I guess I know about the simplex and hypercube/orthoplex groups for all dimensions, the pentagonal polytopes up to dimension 4, the demihypercube groups that generate unique uniforms from dimension 5 and up, the 24-cell group, E6, E7, and E8 groups, and the prisms and Cartesian products that can be formed systematically, resulting in, like you said, things like a truncated tesseract-octahedron duoprism in 7 dimensions.

That leaves only a few uniforms left:
1) snub cube
2) snub dodecahedron
3) snub demitesseract/24-cell
4) snub dihedra (antiprisms in 3D)

I get why these can't occur in any higher dimensions, as they require a uniform snub in the dimension n-1.

and 5) my favorite polytope, the grand antiprism

I am wondering if there are other unique uniform polytopes (like the grand antiprism) in 5 dimensions or higher that seem to come out of nowhere (The grand antiprism was found by Conway when he did a systematic review of permissible 3d vertex figures for 4d polytopes). Has there been much research into other idiosyncratic cases like these? Have we ruled them out at all for 5D, for example?

Thanks a lot,
Rob
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby quickfur » Fri Feb 07, 2014 6:41 pm

ytrepus wrote:Thanks for the reply!

Yes, I guess I know about the simplex and hypercube/orthoplex groups for all dimensions, the pentagonal polytopes up to dimension 4, the demihypercube groups that generate unique uniforms from dimension 5 and up, the 24-cell group, E6, E7, and E8 groups, and the prisms and Cartesian products that can be formed systematically, resulting in, like you said, things like a truncated tesseract-octahedron duoprism in 7 dimensions.

That leaves only a few uniforms left:
1) snub cube
2) snub dodecahedron
3) snub demitesseract/24-cell
4) snub dihedra (antiprisms in 3D)

I get why these can't occur in any higher dimensions, as they require a uniform snub in the dimension n-1.

and 5) my favorite polytope, the grand antiprism

I am wondering if there are other unique uniform polytopes (like the grand antiprism) in 5 dimensions or higher that seem to come out of nowhere (The grand antiprism was found by Conway when he did a systematic review of permissible 3d vertex figures for 4d polytopes). Has there been much research into other idiosyncratic cases like these? Have we ruled them out at all for 5D, for example?
[...]

Far from coming "out of nowhere", the grand antiprism is just one of numerous 4D polytopes that have a multi-ring structure; it just happens to have uniform cells.

If you relax the requirement for the cells to be themselves uniform, but retain convexity, vertex-transitivity and unit edge lengths, then there is, among many things, bidex (bi-24-diminished 600-cell), which consists of 48 tridiminished icosahedra in 8 rings of 6 members each. In spite of the fact that its cells are non-uniform, the polytope as a whole is both vertex-transitive and cell-transitive, and has a chiral structure based on the Hopf fibration of the 3-sphere as applied to an octahedron (triangular antiprism). If you want to learn about the structure of bidex, check out my bi24dim600cell page.

If you further relax the requirement for cell-transitivity, then there's spidrox, a swirl-diminished rectified 600-cell, that has a marvelous structure consisting of 12 rings of alternating pentagonal prisms/antiprisms:

Image

which are interfaced by 20 rings of square pyramids which exhibit an internal 3-fold twist, 5 rings of which are shown below:

Image

Spidrox is convex, vertex-transitive, has unit edge lengths, and is chiral, and is also an expression of the Hopf fibration based on an icosidodecahedron. It is "almost" uniform, were it not for the square pyramids (which were, incidentally, regular octahedra before the diminishing).

There are all manner of marvelous things in 4D (and above!), but sadly they only occasionally pop up as uniform polytopes. Why that is, I can't explain, but I submit that these vertex-transitive, unit-edge polytopes are no less interesting than the uniform polytopes. ;) They routinely turn up in our ongoing search for 4D Johnsonian polytopes to awe us with their fascinatingly symmetric, yet non-uniform, structures.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Polyhedron Dude » Sat Feb 08, 2014 6:14 am

No known convex non-prismattic non-Wythoffians above 4-D to my knowledge - unless the Leech polytope turns out uniform, but I'm not sure on that one. I'm also unaware of any proofs done for 5-D and above, so it is still an open question. I am aware of non-Wythoffian star polytope regiments in 5 and 6 dimensions. There are the two Johnson antiprisms in 5-D "sidtaxhiap" and gadtaxhiap" as I call them. I also recently found "hosiap" which is a blend of 6 demipenteracts. In 6-D and all even dimensions you could blend several expanded hypercubes together. Their verfs will look like two simplex antipodiums blended together at two facets that are the verfs of the facet hypercubes - similar to ondip from uniform polychoron category 20.
Image.

There are all manner of marvelous things in 4D (and above!), but sadly they only occasionally pop up as uniform polytopes. Why that is, I can't explain, but I submit that these vertex-transitive, unit-edge polytopes are no less interesting than the uniform polytopes. ;) They routinely turn up in our ongoing search for 4D Johnsonian polytopes to awe us with their fascinatingly symmetric, yet non-uniform, structures.


One awesome set of convex polytopes are the 4-D fair dice http://www.polytope.net/hedrondude/dice4.htm - lots of bizarre symmetries show up here.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby wendy » Sat Feb 08, 2014 8:06 am

Of convex uniforms, not derived from Wythoff's ME construction, the following are known,

3D p-antiprism, snub cube, snub dodecahedron
4D snub 24choron, grand antiprism
5D+ none.

There are an infinite number of tilings that are non-wythoffian,

In hyperbolic space, there is one infinite class of nonwytioffian polychora, and about seven or eight miscellaneous groups.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby ytrepus » Sat Feb 08, 2014 10:10 am

Thanks everyone for the replies -- they more than answer my question.

Quickfur -- don't you worry, once I nail down the easy stuff (convex uniforms) and make sure I really understand the operations etc, I will start looking at the CRF polychora and the non-convex uniforms. The Johnson solids don't hugely appeal--other than the lovely fact that there are only 92, I find them a bit mechanical and boring other than the last few which are noncomposite forms like the hebesphenomegacorona etc. But it looks like these shapes can make beautiful and complex polychora, so I am looking forward to learning about them.

Polyhedron Dude -- thanks very much for pointing me to the fair dice -- those are very interesting, and I'm especially interested in the idea of the 'Catalan polychora' so I think I'll spend a bit of time looking at the uniform duals as well.

Wendy and Dr. Klitzing -- thanks also for your very helpful responses.

I'd like to point out that I love all of your websites and I have learned so much from all four of them in the past few weeks since I started to discover this topic, and I am very, very grateful. I can't believe I am talking to all of you in one thread!
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Keiji » Sat Feb 08, 2014 10:39 am

Polyhedron Dude wrote:One awesome set of convex polytopes are the 4-D fair dice http://www.polytope.net/hedrondude/dice4.htm - lots of bizarre symmetries show up here.


Ooh! Those are pretty amazing. I've never considered the idea of convex, cell-transitive 4D polytopes with no other restrictions. How did you find them? Have you done any work on 5D, or higher equivalents?
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Polyhedron Dude » Sat Feb 08, 2014 4:46 pm

Keiji wrote:
Polyhedron Dude wrote:One awesome set of convex polytopes are the 4-D fair dice http://www.polytope.net/hedrondude/dice4.htm - lots of bizarre symmetries show up here.


Ooh! Those are pretty amazing. I've never considered the idea of convex, cell-transitive 4D polytopes with no other restrictions. How did you find them? Have you done any work on 5D, or higher equivalents?


I first try to generate a set of congruent vertices in 4-D, then code POV-Ray to write out the coordinates in an OFF file that I open with Stella4D and then take the dual - great way of exploring 4-D fair die, I've found some real odd balls - like one with 250 sides that resemble the stone tablets of Moses that form two orthogonal girdles of 125, where each spirals into a corkscrew twisting 11 times. There is also one that has 72 square antiprisms (non-uniform, but symmetric with a twist). I have done some thoughts in higher dimensions and found some interesting cases. The step tegums (or as I like to call them - gyrotopes) make some amazing die in even dimensions. I also allow curved objects as fair die as long as they are transitive on the contact regions (part that touches the table), things like convex polytwisters, the rolly duoc, duospindle, and "coiloids" show up.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby quickfur » Sat Feb 08, 2014 5:07 pm

Polyhedron Dude wrote:
Keiji wrote:
Polyhedron Dude wrote:One awesome set of convex polytopes are the 4-D fair dice http://www.polytope.net/hedrondude/dice4.htm - lots of bizarre symmetries show up here.


Ooh! Those are pretty amazing. I've never considered the idea of convex, cell-transitive 4D polytopes with no other restrictions. How did you find them? Have you done any work on 5D, or higher equivalents?


I first try to generate a set of congruent vertices in 4-D, then code POV-Ray to write out the coordinates in an OFF file that I open with Stella4D and then take the dual - great way of exploring 4-D fair die, I've found some real odd balls - like one with 250 sides that resemble the stone tablets of Moses that form two orthogonal girdles of 125, where each spirals into a corkscrew twisting 11 times. There is also one that has 72 square antiprisms (non-uniform, but symmetric with a twist). I have done some thoughts in higher dimensions and found some interesting cases. The step tegums (or as I like to call them - gyrotopes) make some amazing die in even dimensions. I also allow curved objects as fair die as long as they are transitive on the contact regions (part that touches the table), things like convex polytwisters, the rolly duoc, duospindle, and "coiloids" show up.

I looked at your fair dice page. :o_o: Wow. Mind totally blown!! :o_o: :o_o: (And I'm only 1/3 of the way down the page!)

The Mobius skew dice are especially mind-blowing. I have a weakness for cell-transitive polychora, you see; one of these days I'm gonna hafta start exploring the totally wild world of unconstrained cell shapes. One thing that occurred to me: would it be possible to get an infinite number of fair die with polytwister symmetry? You just take a discrete, even partitioning of a regular polytwister, sorta like making a finite polygon out of a circle for each ring, and that should produce transitive cells, right?
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby quickfur » Sat Feb 08, 2014 5:17 pm

ytrepus wrote:Thanks everyone for the replies -- they more than answer my question.

Quickfur -- don't you worry, once I nail down the easy stuff (convex uniforms) and make sure I really understand the operations etc, I will start looking at the CRF polychora and the non-convex uniforms. The Johnson solids don't hugely appeal--other than the lovely fact that there are only 92, I find them a bit mechanical and boring other than the last few which are noncomposite forms like the hebesphenomegacorona etc. But it looks like these shapes can make beautiful and complex polychora, so I am looking forward to learning about them.
[...]

Speaking of which, just in the past week I managed to discover two completely unexpected CRFs containing cells from the last few of Johnson's list: J91 - bilunabirotunda, and J92 - triangular hebesphenorotunda. The former is still somewhat "tame", having regular icosahedral symmetry, but the latter has thus far defied all classification. Well, OK, it has trigonal symmetry, but that barely even begins to describe what it is. Even I myself have trouble describing just what it is. :lol:

ytrepus wrote:[...]
I'd like to point out that I love all of your websites and I have learned so much from all four of them in the past few weeks since I started to discover this topic, and I am very, very grateful. I can't believe I am talking to all of you in one thread!

I should mention that that only barely scratches the surface of what this forum has to offer. If you're into torus-like shapes, the toratopes section of this forum has many threads describing some of the most mind-blowing higher-dimensional torus equivalents, including the infamous "tiger" (a pun on "tora-", which means "tiger" in Japanese), a toroidal 4D manifold with properties that defy all analogy with lower dimensions.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Polyhedron Dude » Sun Feb 09, 2014 6:40 am

quickfur wrote:I looked at your fair dice page. :o_o: Wow. Mind totally blown!! :o_o: :o_o: (And I'm only 1/3 of the way down the page!)

The Mobius skew dice are especially mind-blowing. I have a weakness for cell-transitive polychora, you see; one of these days I'm gonna hafta start exploring the totally wild world of unconstrained cell shapes. One thing that occurred to me: would it be possible to get an infinite number of fair die with polytwister symmetry? You just take a discrete, even partitioning of a regular polytwister, sorta like making a finite polygon out of a circle for each ring, and that should produce transitive cells, right?


As a matter of fact there are an infinite number of swirl dice that approach the polytwisters, the one with 72 square antiprisms is one of them. However I have yet to find a way to make any partition work. Take the cubetwister set. If we subdivide the 6 twisters into 8 pieces, you get cont (the octagonny), if you divide the twisters into 12 pieces, you get the one with square antiprisms, we can get dice by subdividing the square twisters in any multiple of 4 - but I have yet to figure out how to subdivide the twisters into 5 or 7 pieces, so far my attempts lead to non-dice - I suspect that the rings will need to be spun a bit in a skewed way to make them work. However there are known dice that split the dysters in odd ways, there has got to be some way to do the polytwisters in the same way.

By the way, I have actually built models of many of the "dice cells" out of poster board :mrgreen: , got a bag full of 'em. :nod:
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby wendy » Sun Feb 09, 2014 6:52 am

The Catalans (uniform margin angles + face-transitive), are mostly by Wythoff Mirror-Margin, with a motley collection corresponding to the non-wythoff uniforms.

The dual of a wythoff mirror-edge is to replace the 'x' with 'm', eg the dual of o3x4x is o3m4m.

The actual operation is this. The symmetry cell is considered 'capped' by a cover, connected to each of the corners of the cell. By adjusting the radius on each arm, you set the opposite margin-angle ("dihedral angle in 3d"). An 'o' node means there is no wall opposite, and therefore the radius must be set so that the margin angle is c60 (a half circle or pi). An x means that there is a wall retained, so that the wall forms on the mirror between the current and the next cell.

There are also tegum-products of catalans can give a catalan. Tegum product is the dual of the prism product.

Outside of this, there are a motley crew of duals of non-wythoffian figures, eg p-gonal antitegum, and the snub-duals of the C, D, and 24ch. The gap also producrs a wythoff dual.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Marek14 » Sun Feb 09, 2014 6:47 pm

quickfur wrote:I should mention that that only barely scratches the surface of what this forum has to offer. If you're into torus-like shapes, the toratopes section of this forum has many threads describing some of the most mind-blowing higher-dimensional torus equivalents, including the infamous "tiger" (a pun on "tora-", which means "tiger" in Japanese), a toroidal 4D manifold with properties that defy all analogy with lower dimensions.


I'm waiting for someone to generalize Stewart toroids and make a tiger-shaped polychoron with regular faces :)
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby quickfur » Mon Feb 10, 2014 5:22 am

Polyhedron Dude wrote:[...]
As a matter of fact there are an infinite number of swirl dice that approach the polytwisters, the one with 72 square antiprisms is one of them. However I have yet to find a way to make any partition work. Take the cubetwister set. If we subdivide the 6 twisters into 8 pieces, you get cont (the octagonny), if you divide the twisters into 12 pieces, you get the one with square antiprisms, we can get dice by subdividing the square twisters in any multiple of 4 - but I have yet to figure out how to subdivide the twisters into 5 or 7 pieces, so far my attempts lead to non-dice - I suspect that the rings will need to be spun a bit in a skewed way to make them work. However there are known dice that split the dysters in odd ways, there has got to be some way to do the polytwisters in the same way.

Oooh... there's a convex twister dice with square antiprisms? Like, uniform square antiprisms? I wanna know how to construct that!!

By the way, I have actually built models of many of the "dice cells" out of poster board :mrgreen: , got a bag full of 'em. :nod:

One of these days, I'd like to figure out how to make physical models of my projections (what Wendy calls "bubble" or "foam" projections) with glass (or some translucent material) cells so that you can see the entire structure of the projection. Maybe a 3D printer might be able to do it?

I used to build paper models of the Platonic solids with little extra "tabs" on the exposed edges that can interlock each other, so that it can hold its shape without any glue. It doesn't work very well with more complicated shapes, though, because the foldouts have polygons that sit too close to each other, and doesn't give enough room to make a strong enough "tab".
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby Polyhedron Dude » Mon Feb 10, 2014 6:44 am

quickfur wrote:Oooh... there's a convex twister dice with square antiprisms? Like, uniform square antiprisms? I wanna know how to construct that!!


The antiprisms aren't uniform, but they do have square rotational symmetry, they are somewhat twisted.

One of these days, I'd like to figure out how to make physical models of my projections (what Wendy calls "bubble" or "foam" projections) with glass (or some translucent material) cells so that you can see the entire structure of the projection. Maybe a 3D printer might be able to do it?

I used to build paper models of the Platonic solids with little extra "tabs" on the exposed edges that can interlock each other, so that it can hold its shape without any glue. It doesn't work very well with more complicated shapes, though, because the foldouts have polygons that sit too close to each other, and doesn't give enough room to make a strong enough "tab".


I wonder if plexiglass, clear silicone, and colored water to fill the solids could work - I imagine it will take a lot of patience to build.
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Re: Non-Wythoffian Convex Uniform Polytopes in dimensions 5+

Postby quickfur » Mon Feb 10, 2014 7:25 am

Polyhedron Dude wrote:
quickfur wrote:Oooh... there's a convex twister dice with square antiprisms? Like, uniform square antiprisms? I wanna know how to construct that!!


The antiprisms aren't uniform, but they do have square rotational symmetry, they are somewhat twisted.

Oh. Darn, I thought they'd be uniform. :\ But still, it's pretty cool.


One of these days, I'd like to figure out how to make physical models of my projections (what Wendy calls "bubble" or "foam" projections) with glass (or some translucent material) cells so that you can see the entire structure of the projection. Maybe a 3D printer might be able to do it?

I used to build paper models of the Platonic solids with little extra "tabs" on the exposed edges that can interlock each other, so that it can hold its shape without any glue. It doesn't work very well with more complicated shapes, though, because the foldouts have polygons that sit too close to each other, and doesn't give enough room to make a strong enough "tab".


I wonder if plexiglass, clear silicone, and colored water to fill the solids could work - I imagine it will take a lot of patience to build.

Hmm. I'm thinking that if enough people want something like this, I could make a puzzle out of it: with a highly-symmetric projection like the dodecahedron-centered projection of the 120-cell, you only have a few distinct shapes, so if I make them somehow detachable, then I could mass-produce many copies of those distinct shapes which should be cheaper than building each one by hand (!), then package them into sets that comprise one projection each. Then it can be sold as build-your-own-polytope kits where you get to assemble the pieces to make the full projection. :lol:

Alternatively, I heard that some 3D printers are quite flexible in what kind of materials they print with: you could have different-colored plastics (i.e., same polymer base mixed with different dyes) in different tubes, and you could print a single solid made of multiple materials that way. Might be really expensive to do it this way, though!
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