More 4D compounds now available

[Dinogeorge]

I was playing with Stella4D version 4.4 and was pleased to discover that it will now construct the 4D compounds of ten 600-cells and ten grand 600-cells from their vertex figures, which are compounds of two icosahedra and two great icosahedra, respectively. Earlier versions of Stella4D became lost trying to construct them (roundoff error?). And the compound of five octahedra now successfully yields the 4D compound of 75 16-cells. But I'm still awaiting constructions of some of the other compound stars, which continue to fail. I was also able to construct the chiral compounds of five 600-cells and five grand 600-cells by removing one subset of five components from the ten-compnent compounds. They all make for very intricate sectioning movies!

[Dinogeorge]

I continue to find more 4D compounds that Stella4D can construct. The latest is the uniform compound of 225 icositetrachora in a 120-cell, whose vertex figure is a compound of nine cubes. Extremely complicated!

[robertw]

Hmm, I don't know that anything changed in that part of the code between 4.3 and 4.4!

I think whether it works or not may depend on the internal order of the faces of the verf (vertex figure). If you create the verf via faceting, you get to create the faces in whatever order you want.

The problem with 4D uniform compounds is as follows. Compare the situation with the compound of 5 cubes in 3D. Vertex figure shown below:



Start at one vertex and use the verf to create the two cubes sharing that vertex. The opposite vertex is also matched to a verf easily but just reuses the existing two cubes. But all other new vertices created by these two cubes are on their own, ie only one cube so far. The problem is that the verf may match in two ways, each resulting in a new third cube, but with that cube being at a different angle. Only one orientation is correct, but it is not immediately obvious from the verf which way to match.

Similar ambiguity can occur for non-compound uniforms too, and some heuristics are used to make the right decision, but so far this hasn't worked consistently for compounds. If you have some suggestions for getting it right let me know

Rob.

[robertw]

I think whether it works or not may depend on the internal order of the faces of the verf (vertex figure).

Oh, so once you get it to work, make sure you save the 3D vertex figure you used. Then hopefully it will work again next time. If you didn't save it before generating the 4D compound, you can always hit undo (Ctrl+Z) to get back to it.

Rob.

[Dinogeorge]

I haven't tried these compounds since version 4.2 at earliest, so maybe the change occurred with 4.3. I can now also make the compound of 75 16-cells directly from its vertex figure (compound of five octahedra) and no longer obtain the "weirdo" compound pf 219 icositetrachora (remember that one?). I do save all the vertex figures as well as the 4D compounds they make.

The vertex-figure problem you describe comes up, for example, with the vertex figure of eight cubes, which should yield the compound of 200 icositetrachora in a 120-cell but fails. It is only after you add the ninth cube that the compound of 225 icositetachora in a 120-cell successfully pops up (which pleasantly surprised me). So to make the compound of 200, you have to go to the 225 and delete the correct 25 icositetrachora, something that I haven't had the time to do just yet.

I don't mind most of these kinds of workarounds. Nothing worthwhile in life is easy.

[Dinogeorge]

I finally built Stella4D models of the two uniform compounds of 200 icositetrachora. These are "strongly" uniform: Their vertices are transitive on the symmetry group, and so are the individual components. The harder of the two to build is the compound of 200 icositetrachora in a 120-cell, and the other is built, once you've gone to the trouble of building the former, by simply taking its dual. It is the compound of 200 icositetrachora about a 600-cell. They are both in Coxeter's table of "vertex-regular" and "cell-regular" 4D compounds in Regular Polytopes.

The uniform compound of 225 icositetrachora in a 120-cell, noted in previous posts, is only "weakly" uniform: Its vertices are transitive on the symmetry group, but its components, which are congruent icositetrachora, are not. They fall into two subsets: the 200 icositetrachora of the preceding compound, and the 25 (forming a less complicated uniform compound) that one must remove in order to leave the 200. So it is technically a compound of 200+25 icositetrachora. It has a dual compound of 225 icositetrachora, of course: 200+25 icositetrachora about a 600-cell. But this compound is not uniform, since it has vertices of two different kinds.

Stella4D will instantly reveal the cases and cores of these compounds. The compound of 200 icos about a 600-cell has a small diprismatohexacosihecatonicosachoron, or "sidpixhi," as its convex hull, while its dual, the compound of 200 icos in a 120-cell, has, as might be expected, the dual of that uniform polychoron as its convex core. That core is the "Catalan polychoron" whose cells are 2400 equit (equilateral-triangular) unequal-antibipyramids (one apex is taller than the opposite apex).

The nonregular case (= convex hull) and ccre of the 225-compounds are a bit tougher to describe, but they do make sense. The core of the 225 icos in a 120-cell is that Catalan polychoron with its 120 icosahedral vertices truncated away, leaving 120 little icosahedra and 2400 antibipyramids with their taller apices truncated as its cells. The case of the other compound is the dual of the core of the former: a sidpixhi with a low dodecahedral pyramid joined onto and thus replacing each of its 120 dodecahedral cells.

Whew.

[Dinogeorge]

Here are Stella4D pictures of central 3D cross-sections of the compound of 200 icositetrachora in a 120-cell:



and its dual compound of 200 icos about a 600-cell:



(This is my first attempt to post pictures, so I hope they appear OK. They look OK in the Preview.)

Astonishingly complicated! And Stella4D will display section movies of the 4D compounds, of which the above are single frames.

[Dinogeorge]

I forgot to mention that, thanks in part to Stella4D, the pictures above are almost certainly humanity's first actual look at these monster compounds.

[Jabe]

I forgot to mention that, thanks in part to Stella4D, the pictures above are almost certainly humanity's first actual look at these monster compounds.

These compounds look GREAT.

[Dinogeorge]

Ah--I recognize Jonathan B. "by his paw."

[Jabe]

Ah--I recognize Jonathan B. "by his paw."

Uh Oh, I got BUSTED

[Dinogeorge]

I found that the "tetrahedral" compound of two great dodecahedra used as a vertex figure will generate the compound of ten grand stellated 120-cells {5/2,5,5/2} via Stella4D. So I was also able to construct the chiral compound of five grand stellated 120-cells by deleting the correct choice of five components from the compound of ten. Both compounds are self-dual, so making their duals doesn't add to the list of Stella4D-constructible star-compounds. The same vertex figure should also generate the compound of ten icosahedral 120-cells {3,5,5/2} but alas Stella4D gets lost.

[Dinogeorge]

Here are two views of the uniform chiral compound of five quasitruncated grand stellated 120-cells (hecatonicosachora). Stella4D makes these with her truncate function. The cells of a component are 120 quasitruncated stellated dodecahedra and 120 great dodecahedra; the vertex figure is a pentagrammatic pyramid. The compound has 7200 vertices. First is a shot of the whole central section of the compound:



and here is a random closeup of the surface of a somewhat off-center section (0.476):



The five components are colored red, white, light yellow, teal, and maroon. There are probably several million pieces ("cellets") that make up the external surcell of the compound; each little fragment ("sniv") in the picture is a cross-section of a distinct cellet. This is almost certainly humanity's first look at this beast. Too bad I can't show the whole sectioning movie here. What fun, eh?

[Dinogeorge]

The uniform compound of five quasitruncated grand stellated hecatonicosachora is chiral, but its set of vertices is not. So when the compound is combined with its mirror image into a compound of ten, their combined 14400 vertices double up, and the convex hull of the ten-compound is the same figure as the convex hull of either five-compound. The vertex figure of the ten-compound is a pair of interpenetrating pentagrammatic pyramids.

As one might imagine, the sections of the ten-compound are even wilder-looking than those of the five-compound, but I haven't had the time to post pictures yet.

[Dinogeorge]

I have two Stella4D versions of the vertex figure for the uniform compound of ten {3,5/2,5}. The first version failed to generate the compound (it got lost), but the second one, created yesterday, found it correctly(!), to my surprise and delight. (Note to Robert W.: I can send you the .stel files if you're interested. Then you might be able to say why one works and the other doesn't. They do have slightly different 3D pictures, so the problem may lie with how Stella4D takes their faces.) I selected five of the ten for the chiral compound, and here is a picture of an orthogonal shadow of it (not a section) generated by Stella4D. The five components are, again, colored white, light yellow, red, teal, and maroon.



Once one creates the figure, Stella4D will truncate, rectify, dualize, and expand the components to yield further interesting and intricate compounds. As with the preceding figures, this is humanity's first look at this compound!