quickfur wrote:Haven't been active on this forum for a while, but it is that time of the month again when I present a new Polytope of the Month. So this month, it's:
The rectified tesseract, a comparatively simple uniform polychoron, but nonetheless a mind-boggler when it comes to animations.
quickfur wrote:
...
In other words, this polytope has swirlprism symmetry, and is directly related to Jonathan Bowers' polytwisters. The rings "swirl" around the surface of the polytope in a chiral way. Every ring is equivalent to every other ring, and every cell in each ring is also equivalent. So the rings are transitive, and so are the cells.
So, this irregular-looking CRF polychoron turns out to have quite an awesome symmetry after all!
Keiji wrote:[...]
Can you do an animation of it rotating so that one ring becomes an equivalent ring and so on? With one ring highlighted.
Also can you do an animation looking down the original axis of symmetry you found, rotating through zw ("turning inside out")? With one group of three equivalent cells in the symmetry highlighted.
Not sure if that described it well enough, but they could help visualise
quickfur wrote:I can't say it looks very helpful, though. Any ideas on how to improve it?
Keiji wrote:quickfur wrote:I can't say it looks very helpful, though. Any ideas on how to improve it?
That's not exactly what I was looking for - if you take that orientation and rotate it 90 degrees around the y axis, then adjust the animation so it's "turning inside out" with the "middle" coming towards/away from the screen, that would be what I was imagining. Hopefully this is a better description
Keiji wrote:Okay, I give up - I have no idea what this polytope is supposed to be.
Keiji wrote:Okay, finally bothered to read the entire page... wish I'd done that sooner, it's a beautiful shape!
I would call it the bixylodiminished hydrochoron, though
A little more concise than bi-icositetradiminished six-hundred-cell...
I wonder if I can get my head around it enough to figure out what its FLD should be... It should be a fairly compact one, given that it's cell transitive... hmm...
quickfur wrote:Keiji wrote:Okay, finally bothered to read the entire page... wish I'd done that sooner, it's a beautiful shape!
I probably should redo that image so that it's more clear what the structure is. Maybe I can do two orthogonal rings, or the vertical ring plus one of the shallow rings. Putting everything in at once is kinda hard to see from our disadvantaged 3D viewpoint.
I would call it the bixylodiminished hydrochoron, though
A little more concise than bi-icositetradiminished six-hundred-cell...
Yeah, I'd adopt that too, except that then I'd lose search engine rank for not having the more common keywords. (Not that it matters in this case anyway -- I can't imagine a random web surfer typing in "biicositetradiminished" out of the blue -- but I'd like the names to be consistent with the more well-known objects, like "600-cell".)
Or maybe I should just come up with pet names for everything like Bowers does, and just reference the technical name somewhere on the page.
What's FLD?
Keiji wrote:quickfur wrote:Keiji wrote:Okay, finally bothered to read the entire page... wish I'd done that sooner, it's a beautiful shape!
I probably should redo that image so that it's more clear what the structure is. Maybe I can do two orthogonal rings, or the vertical ring plus one of the shallow rings. Putting everything in at once is kinda hard to see from our disadvantaged 3D viewpoint.
Actually, why not color the 8 rings accordingly: black/white, red/cyan, green/magenta, blue/yellow, where each ring is perpendicular to the ring of the inverse color. I'd like to see all cells highlighted in those colors. Maybe having this coloration do a Clifford rotation would be good? Can't know til you try.
I would call it the bixylodiminished hydrochoron, though
A little more concise than bi-icositetradiminished six-hundred-cell...
Yeah, I'd adopt that too, except that then I'd lose search engine rank for not having the more common keywords. (Not that it matters in this case anyway -- I can't imagine a random web surfer typing in "biicositetradiminished" out of the blue -- but I'd like the names to be consistent with the more well-known objects, like "600-cell".)
Or maybe I should just come up with pet names for everything like Bowers does, and just reference the technical name somewhere on the page.
Yes, I don't think Google rankings is the important thing here - just look how many of the top 10 slots we already have for crf polychora - the top two are here, and the next seven are your site (the last one is Wikipedia). Certainly for the topic of newly discovered polytopes, people will much more likely be searching for generic terms like that than names we've only just come up with.
Keiji wrote:I've just added the Bixylodiminished hydrochoron page, I'd appreciate if you checked it for accuracy, especially the element counts and how I arrive at them.
quickfur wrote:[...]And then things like Wendy's bistratic CRF with 4 tridiminished icosahedra, 5 tetrahedra, and 1 octahedron: I don't even know how to name such a thing except using Wendy's notation, but I find a bit hard to swallow that the title of the page would have to be "xfo3oox3ooo". There's no question her notation scheme is extremely powerful and expressive, but it doesn't give us pronunciable names.
[...]
Klitzing wrote:quickfur wrote:[...]And then things like Wendy's bistratic CRF with 4 tridiminished icosahedra, 5 tetrahedra, and 1 octahedron: I don't even know how to name such a thing except using Wendy's notation, but I find a bit hard to swallow that the title of the page would have to be "xfo3oox3ooo". There's no question her notation scheme is extremely powerful and expressive, but it doesn't give us pronunciable names.
[...]
Well, Wendy was just extrapolating into higher dimensions: xfo (or better xfo&#xt, to show what is meant really: a lace tower with unit edge (i.e. x) lacings (i.e. #)) is nothing but the regular pentagon. xfo3oox&#xt is nothing but the 3-diminished icosahedron. And then the next would be xfo3oox3ooo&#xt, which you have rendered, and now are asking for. So a pronouncable name could be "a 3-diminished icosahedron equivalent of 4D". Or you could get it even more funny, if you'd use Jonathans acronym instead: tridiminished icosahedron --> teddi: "Teddy in 4D"
Keiji wrote:I'll have a go at naming the shape... after you get the page up so that I can understand what it is.
PS. Quickfur - any ideas what's wrong with my edge count calculations? I'm not sure how you got 216 edges, but it's also the first time I've calculate number of edges the way I did - though I can't see any reason why my method wouldn't work.
quickfur wrote:[..] It consists of 4 tridiminished icosahedra in tetrahedral formation, arranged around the top cell which is a tetrahedron, and at the bottom is an octahedron with 4 tetrahedra filling in the gaps between the bases of the tridiminished icosahedra. A really cute little shape. Teddy. [...]
quickfur wrote:If you can do cross-eyed viewing, the 3D shape of the projection should be even more helpful.
It consists of 4 tridiminished icosahedra in tetrahedral formation, arranged around the top cell which is a tetrahedron, and at the bottom is an octahedron with 4 tetrahedra filling in the gaps between the bases of the tridiminished icosahedra. A really cute little shape. Teddy.
I didn't count the edges myself; I got 216 edges from the convex hull algorithm that I use to construct the model that my viewer uses to do the projection. I don't know what went wrong with your edge count calculations, it seems reasonable to me. But counting things in higher-dimensional polytopes has always been a tricky business. I mean, I have enough trouble with counting elements correctly in 3D polyhedra, let alone 4D.
Keiji wrote:quickfur wrote:If you can do cross-eyed viewing, the 3D shape of the projection should be even more helpful.
Oh come on, I'm practically the pioneer of cross-eyed viewing around here, didn't you know that?
[...] Well, these images suddenly made me realise something. The tridiminished icosahedron is in fact a member of an infinite family of polyhedra related to another family I'd been studying a while ago - the n-gonal trapezosemipyramids. The only difference is that in the latter family, the bottom face is not present, instead the triangles that would connect to it extend to a point to form kites. However, looking at the tridiminished icosahedron now, I see it is a type of antiprism, just with pentagons instead of the top set of triangles. So I'd like to call it the trigonal something-antiprism, which would make your "Teddy" the pentachoric (or pyrochoric) something-antiprism. Just a matter of figuring out what the something is, then you have my preferred name for it.
EDIT: Actually, seeing as the bottom cell in the "Teddy" is an octahedron, it's closer to the cupolae and rotundae than it is to the antiprisms. Perhaps we could just come up with another word in that vein?
I didn't count the edges myself; I got 216 edges from the convex hull algorithm that I use to construct the model that my viewer uses to do the projection. I don't know what went wrong with your edge count calculations, it seems reasonable to me. But counting things in higher-dimensional polytopes has always been a tricky business. I mean, I have enough trouble with counting elements correctly in 3D polyhedra, let alone 4D.
Is there any chance you could put something into your program to identify groups of transitive edges (or, more generally, elements of each dimension) and list those groups with the counts and number of cells surrounding them? This would likely help find the problem.
$ ./polyview -p data/crf/bi24dim600cell.def
Polytope bi_icositetradim_600cell loaded
Dimension: 4
72 vertices
216 edges
192 faces
48 cells
bi_icositetradim_600cell> calculate cell neighbours of edges
144 edges with 3 cells: all except 0, 2, 7, 10, 15-16, 18, 20, 25-26, 28-29, 37, 39-40, 46, 48-49, 51, 57, 59, 63, 68-69, 71, 73, 75, 77, 81, 88, 90-91, 97, 101, 104, 108, 111, 113, 117-118, 124, 126, 132-133, 138-139, 144, 149-151, 158-159, 161, 165, 168, 171, 176-177, 181, 183, 188, 190-192, 196-197, 199, 203-204, 207, 209, 212
72 edges with 4 cells: 0, 2, 7, 10, 15-16, 18, 20, 25-26, 28-29, 37, 39-40, 46, 48-49, 51, 57, 59, 63, 68-69, 71, 73, 75, 77, 81, 88, 90-91, 97, 101, 104, 108, 111, 113, 117-118, 124, 126, 132-133, 138-139, 144, 149-151, 158-159, 161, 165, 168, 171, 176-177, 181, 183, 188, 190-192, 196-197, 199, 203-204, 207, 209, 212
bi_icositetradim_600cell>
quickfur wrote:I have this suspicion that this construction is general, and there should be an analogous construction in 5D where you'll end up with a 5D polyteron consisting of 1 5-cell on top, 5 5-cells surrounding a rectified 5-cell at the bottom, with 5 Teddy's around the side. Topologically, at least, this construction is valid, but I'm not 100% sure it will be CRF, because the edge lengths and diteral angles may not line up correctly. If this shape is indeed a valid 5D CRF, then I shall have to call it the Grand Teddy, and if there's a CRF 6D analogue, then it will have to be the Great Grand Teddy. And it's Teddies all the way up! (You may groan now.)
quickfur wrote:wendy wrote:You can of course derive snubs by alternating the vertices of a omnitruncate.
Yes, I understand that it is a general operation that works for any polytope (since an omnitruncate is always even).
The snub-faces form by the removal of alternate vertices = "diminishing", which typically form simplexes. The degrees of freedom correspond to the number of dimensions (ie the marked dots in the dynkin graph), while the variables to set correspond to the edges of the triangle thus formed. When N=4, E=6, so you are attempting to solve four variables in six equations.
None the same, you can make vertices "the same" in symmetry, and one has to reckon only those edges which link different vertices, eg consider the {3,3,3}, where nodes become a3b3B3A. You have vertices types a=A, b=B,and edge types aA, bB, and aB=Ba. This is three variables in two unknowns, which is usually not solved. This is a shame, because the faces of this creature consist of ten icosahedra, twenty octahedra, and sixty tetrahedra, but only topological.
I'm more interested in actual uniform snubs, though. What is the general approach for deriving the coordinates of, say, the uniform snub cube? And for polytopes that do not have uniform snubs, is there a way to derive the "most symmetric" form of the snub generated by the alternation of its omnitruncate?
You can alternate some of the vertices, as long as an even (or unmarked = 2), branch connects marked and unmarked vertices. This gives,
s3s3s4o = s3s4o3o = snub 24choron, s4o3o3o = x3o3o4o.
You can apply stott-style addition to these things too, which gives the rather interesting, although not uniform s3s4o3x. The faces of this figure consist of 24 truncated tetrahedra, 24 icosahedra, and 96 triangular cupola (cuboctahedra cut across a hexagon). Allowing for non-uniform faces, one can have the equalateral figure formed by 3xo*xx2%o, or, as Richard Klitzing writes, cube || icosahedorn. This is a lace-prism, formed by a parallel planes containing a cube and an icosahedron, with assorted triangular prisms, tetrahedra, and square pyramids forming the lacing faces.
This is interesting; recently on Wikipedia somebody came up with 4D cupolas which are generated by expanding (runcinating) 4D pyramids.
I wonder how many Johnson polychora there are, whose facets, to retain consistency with the generalization of the Archimedean polyhedra, are allowed to be any Johnson solid.
In particular, I'm quite curious as to whether there are vertex-transitive polytopes whose facets are not uniform (this is a possibility, e.g., if you take a Johnson solid whose vertices aren't transitive, say they are of two types X and Y, you can, when forming the polychoron, join the type X vertices of one facet with the type Y vertices of another, and thereby make the result vertex transitive---at least, this is why I think it may be possible, but I don't know if there are actual examples of such polytopes).
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