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?
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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.
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.
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?
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.
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.
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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!
quickfur wrote:I looked at your fair dice page.Wow. Mind totally blown!!
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(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?
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.
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.
By the way, I have actually built models of many of the "dice cells" out of poster board, got a bag full of 'em.
quickfur wrote:Oooh... there's a convex twister dice with square antiprisms? Like, uniform square antiprisms? I wanna know how to construct that!!
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".
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.
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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