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More 4D compounds now available
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robertw
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Joined: 10 Jan 2008
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Location: Melbourne, Australia

PostPosted: Thu Jan 17, 2008 2:54 am    Post subject: Reply with quote

Looks great!

Yeah, send me the .stel files. I'd like to watch the cross-sections animating if nothing else! Very Happy Might even add some to the standard library. I could at least upload a collection to this site so everyone can access them.

I expect the two .stel files that should act the same act different because their faces are ordered differently. It's usually pretty hard for to me determine exactly where these things go wrong, but it's due to the multiple matching choices as described earlier. I'd still like to find a way to be sure of choosing the right path. Any thoughts about heuristics for achieving this?

Rob.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Fri Jan 18, 2008 6:55 am    Post subject: Compounds of regular star-polychora Reply with quote

Well, assuming I've counted them correctly, I've managed to make Stella4D versions of all but two of the twelve compounds of ten regular pentagonal polychora (from which chiral compounds of five follow). The two that my version of Stella4D cannot yet reach are the compounds of ten {5,3,5/2} (grand 120-cells) and {5/2, 3, 5} (great stellated 120-cells). These are each other's duals, so I can't reach one and dualize it to get the other, as I've had to do with a couple of the others.

Ones that I cannot reach directly with the Stella4D "create from vertex figure" function are those whose vertex figures are compounds of two dodecahedra and two great stellated dodecahedra, namely ten {5/2,5,3} and ten {5,5/2,3}; I have to reach them by dualization. The fact that their vertex figures each have eight pairs of coincident vertices may have something to do with the problem.
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Dinogeorge



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PostPosted: Fri Jan 18, 2008 6:35 pm    Post subject: Five great grand stellated hecatonicosachora Reply with quote

Here is a 3D slice through this compound. The great grand stellated hecatonicosachoron has the narrowest possible "points" of any uniform polychoron, and this chiral compound features 3000 (=5x600) of them: 2400 as single points and 600 others gathered by fives into 120 more (2520 distinct corners altogether). There being two kinds of corners, this compound is not uniform. It is merely "cell-regular."



The section is cell first, halfway between the center and the circumsphere. It includes a rather large "swarm" of irregular tetrahedra (the cross sections of some of the 2400 single points, which are inclined at various different angles to the sectioning realm), among which one can spot the occasional compound of five tetrahedra (cross sections of some of the 120 fivefold points). In the sectioning movie, the swarm coalesces as the sectioning realm approaches the center, then disperses as the realm continues through to the other side of the compound. The central coalescence has already begun at this level.

Thanks to Stella4D, this is the "first look for humanity" at this figure. Now try to imagine a four-dimensional model maker building the full compound in four-space out of blocks of four-dimensional paper. It would take a lifetime, perhaps. But the finished model would be very beautiful! Laughing
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Dinogeorge



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PostPosted: Sat Jan 19, 2008 7:03 pm    Post subject: Ten grand hexacosichora Reply with quote

The cells of the grand hexacosichoron (600-cell) are 600 tetrahedra, and its vertex figure is a great icosahedron. In terms of the number of cellets in its surcell (and I have not counted them!), the grand hexacosichoron is the most complicated of the ten regular star-polychora. (This situation is much like that of its 3D analogue, the great icosahedron.) So of course I wanted to have a look at the monster vertex-regular compound of ten with Stella4D. Happily, she accommodated me. The vertex figure of the compound is a "tetrahedral pair" of great icosahedra, and the compound has the 600 corners of a regular hecatonicosachoron (120-cell). Here is a cell-first section a bit away from the center (at 0.555):



And a closeup of the same section from the same viewing direction:



Boy, look at all the little snivs! The sectioning movie is very intricate, as one might imagine, especially in closeup: something like watching lava flow. There are ten colors, one for each different component. This compound is the dual of the compound of ten great grand stellated hecatonicosachora, half of which appear as their own chiral compound in a previous post. Yes, of course there is a chiral compound of five grand 600-cells, too. See what Stella4D can do if she's pushed: another "first look for humanity." Whew Exclamation
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Jabe



Joined: 12 Jan 2008
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Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Sun Jan 20, 2008 6:42 am    Post subject: Reply with quote

This "ten-gax" looks incredible, I hope to one day see the swirl ten-gax which belongs to the infinite gax swirl compound family - to obtain it, start with gax (vertex oriented), swirl it 7.2 degrees (rotate xy plane 7.2 degrees and rotate zw plane 7.2 degrees) add another gax (in original orientation) swirl both gaxes 7.2 degrees, and continue until there are ten gaxes. To make this Stella4D would need an ability to rotate polychora at a typed in angle (i.e rotate xy 70 degrees, rotate xz 50 degrees), and the ability to compound with a saved object in the original position - or possibly a "duplicate object at an angle" feature.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Sun Jan 20, 2008 10:08 pm    Post subject: Siamese prismosauri Reply with quote

Once I got through making as many compounds of the regular star-polychora as I could with the latest version of Stella4D, I began to try making similar kinds of compounds of other uniform polychora. For example, there are compounds of five and ten small and great swirlprisms, but I can’t yet tell whether or not they are uniform, because there are several different ways to compound their vertex figures and more than one way might appear in a compound. (If not, they will lead to several different uniform compounds.) Stella4D vehemently rejects my efforts to model them, but I cannot tell whether multiple vertex figures is the reason, or whether she is just getting confused.

The compounds of the small and great prismosauri, however, add a different kind of wrinkle. When I doubled their vertex figures in preparing to build compounds of ten in a 120-cell, Stella4D warned me that some of the vertex-figure faces were coplanar. Well, OK, I blended them out and made two single (noncompound) vertex-figure polyhedra of 96 faces. No two faces actually coincided. Rather, 48 of the 120 faces (isosceles triangles, or isots) in both vertex figures blended in pairs with one common leg to make 24 butterfly faces. The other 72 faces remained unblended and untouched. After doing the arithmetic, I concluded that we now have two new scaliform polychora, although not everyone will agree that they are “true” polychora, because they have some coincident edges. The cells of the small Siamese prismosaurus are 3600 Siamese pentagrammatic prisms; the cells of the great Siamese prismosaurus are 3600 Siamese pentagonal prisms. In a Siamese prism, two identical concentric Archimedean prisms are joined at a square face (or two opposite square faces), with one prism rotated pi/2 relative to the other. This makes for a polyhedron whose faces are all regular polygons (the four bases and the remaining mantle faces of the two prisms). A whopping 96 of these kinds of cells come together at each of the 600 vertices of either Siamese prismosaurus (they’re conjugate polychora).

Here are the Siamese-prism cells of the small and great Siamese prismosauri, pentagrammatic and pentagonal, respectively:




and here are their vertex figures:




In the vertex figures, the butterfly faces are red, and the untouched isots are light yellow, maroon, and teal (three different symmetry classes). The teal faces are symmetrically disposed, while the light yellow faces are enantiomorphically disposed mirror images of the maroon faces.

Unfortunately, Stella4D tells me the 4D figures don’t close, but this must be because of their numerous compound edges, as reflected in the compound vertices of their vertex figures. The Siamese polychora must close, because the original figures are closed and they are properly blended with no remaining unblended coincident faces or cells. It is interesting that all 7200 cells of the two component prismosauri blend in pairs with none left over.

The chiral compounds of five small and great prismosauri exist and have no corealmic cells, but they’re thus far unreachable with Stella4D. I tried to make them with a compound vertex figure made up of one small and one great prismosaurus vertex figure, but Stella4D resisted this strategy and became confused, likely because of the compound vertices and edges of the vertex figure. (Being in the same regiment, the two different prismosauri happen to share the same set of squares.) So, no 4D section pix with this post. Crying or Very sad
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Jabe



Joined: 12 Jan 2008
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Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Mon Jan 21, 2008 5:39 am    Post subject: Reply with quote

AH I remember these scaliforms, I call these sistakix and gastakix (short for small and great snub 3600choron), found these in Oct 2005 - sorry not new Wink but no sections have been seen yet - hope that Stella can one day take them on.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Mon Jan 21, 2008 5:58 am    Post subject: "New" scaliforms Reply with quote

Jabe wrote:
AH I remember these scaliforms, I call these sistakix and gastakix (short for small and great snub 3600choron), found these in Oct 2005 - sorry not new Wink but no sections have been seen yet - hope that Stella can one day take them on.


Boy, no matter where you walk in four-space, you run across Jonathan B.'s footprints! Smile
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Jabe



Joined: 12 Jan 2008
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Location: Somewhere between Texas and the Fourth Dimension

PostPosted: Mon Jan 21, 2008 6:15 am    Post subject: Re: "New" scaliforms Reply with quote

Dinogeorge wrote:
Jabe wrote:
AH I remember these scaliforms, I call these sistakix and gastakix (short for small and great snub 3600choron), found these in Oct 2005 - sorry not new Wink but no sections have been seen yet - hope that Stella can one day take them on.


Boy, no matter where you walk in four-space, you run across Jonathan B.'s footprints! Smile


Four-space makes great stomping grounds - I suspect there's still a few untouched regions - I better head for those quick before George moves in Razz
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Dinogeorge



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PostPosted: Mon Jan 21, 2008 11:24 pm    Post subject: Scaliforms Reply with quote

Jabe wrote:
AH I remember these scaliforms, I call these sistakix and gastakix (short for small and great snub 3600choron), found these in Oct 2005 - sorry not new Wink but no sections have been seen yet - hope that Stella can one day take them on.


These two are not yet in the Stella4D library as far as I can tell. This probably has a lot to do with why I thought they were new. I went through the Stella4D scaliforms some time ago and believed that was all that were known. (Maybe it was, when the library was put together.)
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robertw
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PostPosted: Tue Jan 22, 2008 2:40 am    Post subject: Reply with quote

No, Stella4D inclused all known uniform polychora, but not all the scaliforms, so please send them through if you make any Wink

Rob.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Tue Jan 22, 2008 4:46 am    Post subject: Scaliforms Reply with quote

robertw wrote:
No, Stella4D inclused all known uniform polychora, but not all the scaliforms, so please send them through if you make any Wink


Well, I have the two vertex figures, but my copy of Stella4D falls short of constructing the corresponding polychora. Crying or Very sad

The polychora do have coincident edges, so maybe that's the problem Question
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robertw
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PostPosted: Tue Jan 22, 2008 6:12 am    Post subject: Re: Scaliforms Reply with quote

Dinogeorge wrote:
The polychora do have coincident edges, so maybe that's the problem Question


Maybe, although I thought there were other ones like that which worked (been a while since I looked at this stuff). Send me the .stel files anyway and I'll have a look.

Rob.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Tue Jan 22, 2008 7:12 am    Post subject: Re: Scaliforms Reply with quote

robertw wrote:
Dinogeorge wrote:
The polychora do have coincident edges, so maybe that's the problem Question


Maybe, although I thought there were other ones like that which worked (been a while since I looked at this stuff). Send me the .stel files anyway and I'll have a look.

Rob.


OK, .stel files are on their way. Maybe coincident edges are only a problem when there are lots of vertices? Anyway, hope you can figure out what's going wrong and can fix it.
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Dinogeorge



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PostPosted: Tue Jan 22, 2008 7:30 am    Post subject: Five great grand hecatonicosachora Reply with quote

Here are a shot and a closeup of a 3D vertex-first (same as cell-first) section of the chiral compound of five great grand hecatonicosachora, {5,5/2,3}. The figure has the 600 corners of a hecatonicosachoron (120-cell), and a total of 600 great-dodecahedral cells. The vertex figure is a small stellated dodecahedron. Coxeter's notation for this figure is {5,3,3}[5{5,5/2,3}]{3,3,5}





The closeup is from the same viewing direction as for the full shot, which is for sectioning level 0.555. Sections close to the center but not actually passing through it are usually more intricate than central sections, since they avoid coincident elements. Smile
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