Uniform compounds of icositetrachora

[Dinogeorge]

There are lots of symmetric compounds of icositetrachora (including at least one infinite set, discovered, if I remember correctly, by Jonathan Bowers), many, but by no means all, of which are vertex-regular, cell-regular, or uniform. The simplest is the compound of the icositetrachoron and its dual, which is strictly regular, that is, its symmetry group (the double icositetrachoric group) is transitive on all its flags. (A flag of a polytope is any of its vertices, plus any edge incident at that vertex, plus any face incident with that edge, plus any cell incident with that face, and so on, to any ridge incident with the preceding peak, plus any facet incident with that ridge, plus the body of the polytope itself. This definition extends trivially to compound polytopes.) It is very easy to construct this compound using Stella4D, so I won’t waste much space describing it here. Not many compounds are strictly regular; in three-space only the Stella Octangula (two tetrahedra in dual position) is. Here is a picture of the 0.555 3D cross section of the compound of two icositetrachora in dual position:



The orientation of the section is such that the components always make two congruent cross-section polyhedra, along the entire sectioning axis from “top” to “bottom.” Just do a quarter-turn about a particular axis in the section followed by a half-turn about a perpendicular axis at any time to interchange the two components. Although the compound is strictly regular, its 48 corners and 48 cell-realms are not those of any regular polychoron, and it is neither vertex-regular nor cell-regular. Its corners are those of a Catalan diacosiogdoecontaoctachoron, whose cells are 288 identical rather flattened disphenoids, the dual of the bitruncated icositetrachoron or uniform tetracontaoctachoron. The latter's 48 cells are all truncated cubes, and their realms are the ones in which the cells of the self-dual icositetrachoric two-compound lie. Just as a great stellated dodecahedron can only stand on five corners, so a four-dimensional model maker could only stand a model of this compound on two nearby skew edfes.

One of the first compounds I tried to construct after I discovered that Stella4D had edged further into the business of compound making was the “scrunched” compound of 200 icositetrachora in a hecatonicosachoron. (There are two compounds of 200 icositetrachora in Coxeter’s table of “partially regular” compounds, duals of each other. One has the 600 corners of a hecatonicosachoron, with the vertices of eight icositetrachora at each corner, the other has the 2400 corners of a diprismatohexacosihecatonicosachoron, or runcinated hexacosihecatonicosachoron, with the vertices of two icositetrachora at each corner. The former is the “scrunched” compound, the latter is the “strewn” compound.) These compounds arise when ten “scrunched” or “strewn” compounds of 25 icositetrachora, which have the corners of a hexacosichoron and a hecatonicosachoron, respectively, are substituted for corresponding components of the pentagonal-polychoric ten-compounds reviewed previously at this forum (and any coincident 25-compounds are removed).

I figured that since Stella4D was happy using the various paired pentagonal polyhedra as vertex figures, she might also accommodate the vertex figure of eight cubes, of the “scrunched” 200-compound. This vertex figure is constructed by replacing the two dodecahedra in a compound dodecahedral pair by the compounds of five cubes that have the same corners. This merger results in a compound of ten cubes, but two of the cubes are special and coincide, so these are removed, leaving a biform eight-cube compound. The octahedral symmetry group of the compound is transitive on the eight cubes; that is, there is a symmetry of the compound that will carry any cube into any other. (There isn’t if those two cubes are not removed.)

Well, unfortunately, Stella4D still fails to construct the 200-compound whose vertex figure the eight-cube compound is. But I decided to try adding back one of the cubes that was removed, to make a compound of nine cubes. This is the vertex figure of a symmetric “scrunched” compound of 225 icositetrachora, the extra 25 icositetrachora resulting from the extra ninth cube. Although it is vertex-regular, it is not in Coxeter’s list, presumably because no symmetry of the compound carries one of the 25 icositetrachora into one of the 200: The 225 icositetrachora fall into two symmetry classes, and so the compound is not regular (and only weakly uniform), even though at first sight it seems to fulfill Coxeter’s definition of a vertex-regular compound (see page 47 of Regular Polytopes, Dover edition).

When I added the ninth cube and requested the 4D figure with that vertex figure, the “scrunched” compound of 225 icositetrachora suddenly popped up on the screen. But for some stupid reason, I overwrote the vertex figure, and now no assembly of the compound of nine cubes will bring up the 225-compound again! I did save the 225-compound itself, of course, so everything displayed from here on stems from that one lucky break. I have no idea what I did right. Even the vertex figure of the 225-compound itself fails to reproduce the compound! Go figure. (But this makes me think that there may be some combination of vertex-figure assembly procedures that will eventually give me the compound of ten great stellated hecatonicosachora or grand hecatonicosachora, which were missing from my string of posts on the ten-compounds, as well as get the 225-compound back.) Here is the vertex figure of nine cubes:



And here is how 4800 of the 5400 octahedral cells of the 225 icositetrachora pair up in their 2400 cell-realms:



The other 600 octahedra, which all belong to the 25 “extra” icositetrachora, lie in 120 sets of five per cell realm, the familiar compound of five octahedra. This shows that the extra 25 icositetrachora form the “strewn” compound of 25, there being just one vertex of the 25 at each corner of the convex hull hecatonicosachoron, and not five vertices "scrunched" at each corner of a hexacosichoron.

Perhaps the most important icositetrachoric compounds are the “scrunched” and “strewn” compounds of 25. These have the corners of a regular hexacosichoron and hecatonicosachoron, respectively, and are each other’s duals. A wide range of other symmetric icositetrachoric compounds may be derived from them by removing sets of five icositetrachora that also have the corners of a hexacosichoron or hecatonicosachoron. Stella4D constructs the “scrunched” compound of 25 isositetrachora with no problem directly from its vertex figure of five cubes in a dodecahedron. The “strewn” 25-compound (whose vertex figure is a single cube) then follows by dualization. We may remove “strewn” and “scrunched” 25-compounds of icositetrachora from the “scrunched” and “strewn” compounds of 225, respectively, to construct the “scrunched” and “strewn” 200-compounds, which Coxeter does list as vertex-regular and cell-regular icositetrachoric compounds. The reason we must remove the icositetrachora by fives and 25s is to preserve or induce uniformity of the vertex figures. After all, we can pile icositetrachora together almost endlessly (for example, by removing one icositetrachoron at a time from the compounds of 225) to construct compounds of little geometric significance.

Here is “humanity’s first look” at the “scrunched” 225-compound of icositetrachora, as the usual 0.555 section orthogonal to an icosahedral symmetry axis. The compound has no Coxeter notation, having been excluded from his list of vertex-regular compounds, even though it has the 600 corners of a regular hecatonicosachoron. I presume he excluded it because the symmetry group is not transitive on all the components, but he does not state that in Regular Polytopes as far as I can read. I can contrive a symbol for it that conforms to other of Coxeter’s usages: (8+1){5,3,3}[(200+25){3,4,3}]. This symbol emphasizes the number and kinds of components in the compound.



Here is a closeup of the surhedron of the section, which is automatically colored in 225 colors by Stella4D. It would be nice to color it in just two or three colors, emphasizing the special set of 25 icositetrachora and the two chiral sets of 100, but finding the right components to color in the welter of facelets is impractically time-consuming:



It is equally impractical to try to locate the 25 icositetrachora for removal to make the vertex-regular compound of 200 icositetrachora, which Stella4D cannot create directly from its vertex figure of eight cubes. It is, however, easy to do using a different method, but I will leave that for a future post.

[Dinogeorge]

The dual of the “scrunched” compound of 225 icositetrachora is the “strewn” compound. The 225 “scrunched” icositetrachora dualize into 225 like polychora that are distributed differently to make up the “strewn” compound. Specifically, there are more corners in the “strewn” compound, with fewer icositetrachoric vertices at each corner. And correspondingly, there are fewer cell-realms but more cells in each.

The “scrunched” compound has nine icositetrachoric vertices at every corner, and the 600 corners are those of a regular hecatonicosachoron. Dually, the “strewn” compound has nine octahedra in each cell-realm, and the 2520 corners familiar from some of the star-polychoric compounds I have described elsewhere on this forum. The fact that 120 of the corners, where five icositetrachoric vertices come together, lie at the vertices of a regular hexacosichoron makes it pretty easy to find and delete the “extra” 25 icositetrachora whose vertices those are. This immediately gives us the “strewn” cell-regular compound of 200 icositetrachora in Stella4D, and redualizing that quickly gives us the vertex-regular “scrunched” compound of 200 icositetrachora. This is a classic instance of a tangled problem simplified by going to the dual situation. In the "strewn" 200-compound, there are just two icositetrachoric vertices per corner rather than nine, which also makes is easy to sort the 200 icositetrachora into their two chiral subsets of 100. Both compounds of 200, and both compounds of 100, are listed among the “partially regular” compound polychora in Coxeter’s Regular Polytopes. I will get to them in due time on this forum.

Meanwhile, here are the nine octahedra as they lie in each of the 600 cell-realms of the “strewn” compound of 225 icositetrachora:



Here the “special” ninth octahedron is colored teal; it is the dual of the “extra” cube in the vertex figure of the “scrunched” icositetrachoric 225-compound. The light yellow and red octahedra are the cells of the left-handed and right-handed “strewn” icositetrachoric 100-compounds that together make up the “strewn” icositetrachoric 200-compound. Here is “humanity’s first look” at the “strewn” 225-compound of icositetrachora, as the usual 0.555 icosahedrally symmetric 3D cross section. It is colored to correspond with the compound of nine octahedra, and it makes a striking Stella4D sectioning movie:



The “strewn” 225-compound is not in Coxeter’s list of cell-regular compounds despite having the 600 cell-realms of a regular hecatonicosachoron, presumably because it contains the “scrunched” compound of 25 teal icositetrachora, which no symmetry of the compound will carry into any 25 of the other 200 components. It is a homomeric (Greek: “like parts”) compound that is not isomeric (Greek: “same parts”). (Compounds that have several different kinds of components are polymeric, and the “poly-“ prefix nay be replaced by a Greek numerical prefix as needed.) My symbol for it turns around the symbol for the "scrunched" 225-compound: [(200+25){3,4,3}](8+1){3,3,5}.

The “scrunched” and “strewn” compounds of 225 icositetrachora lead to a number of additional interesting symmetric icositetrachoric “monster” compounds constructible with the present version of Stella4D. Here is a short list (to which I may add if I find any others, or from which I may remove if they don't check out):

450 icositetrachora: 225 “scrunched” with 225 “strewn” in dual position
425 icositetrachora "scrunched"
425 icositetrachora "strewn" (dual of preceding)
400 icositetrachora: 200 “scrunched” with 200 “strewn” in dual position
200 icositetrchora “scrunched” (in Coxeter’s vertex-regular list)
200 icositetrachora “strewn” (in Coxeter’s cell-regular list)
100 icositetrachora “scrunched” (chiral; in Coxeter’s vertex-regular list)
100 icositetrachora “strewn” (chiral; in Coxeter’s cell-regular list)
200 icositetrachora “scrunched” and “strewn” in dual position (chiral)
125 icositetrachora “scrunched” (chiral)
125 icositetrachora “strewn” (chiral): dual of the preceding

I’ll discuss these individually in more detail in future forum posts. Some have much nicer-looking sections than others. The 100’s and 125’s are probably the nicest: complicated but not so intricate as to be totally confusing.

[Dinogeorge]

This is the compound of the “scrunched” and “strewn” compounds of 225 icositetrachora each, in dual position. As such, it has the 600 corners of the “scrunched” compound, where nine icositetrachoric vertices come together, along with the 2520 corners of the “strewn” compound. Of those latter, 2400 have two icositetrachoric vertices, with a pair of cubes as vertex figure, and the remaining 120 have five icositetrachoric vertices, with the compound of five cubes as vertex figure. The whole compound is thus homomeric and triform. No two components coincide. Stella4D’s element counters do not recognize the differences among the corners and simply total them up vertex by vertex: 10800 vertices altogether for the 450 icositetrachora. The 120 corners with five vertices are the corners of the “scrunched” compound of 25 icositetrachora in a hexacosichoron, which Stella4D will instantly assemble given the five-cubes compound as vertex figure. If we remove this particular “subcompound” from the compound of 450, the “scrunched” compound of 425 icositetrachora will remain. Its dual is the “strewn” compound of 425 icositetrachora (both to be profiled in later posts), which may also be constructed by removing a “strewn” compound of 25 icositetrachora from among the 600 enneavalent corners of the compound of 450, leaving them octavalent. I’ll have more to say about these compounds in subsequent posts.

The compound of 450 icositetrachora is the “master compound” of icositetrachora, in the sense that all the other hexacosichorically symmetric icositetrachoric compounds may be found as “subcompounds” in it. Even the compound of one icositetrachoron with its dual is represented, 225 times. Like the strictly regular compound of two, the compound of450 has no Coxeter symbol, and it is neither cell-regular nor vertex-regular, but of course it is self-dual and it has all 14400 symmetries of the hexacosichoric symmetry group. Here is a picture of the usual 0.555-level icosahedrally symmetric 3D cross section of it. It closely resembles the preceding compound of 225, except that it has regions of light yellow "fungus" on it. This realm passes through all 450 components, so the 3D cross section is a compound of 450 polyhedra, each a cross section of an icositetrachoron. Some of the snivs are preposterously small; Stella4D cannot make its nets:



The various components are painted in five different colors: The teal parts are cross sections of the “scrunched” subcompound of 25 icostetrachora; the light yellow "fungus" parts are cross sections of the “scrunched” subcompound of 200 icositetrachora, and interspersed among these are the maroon cross sections of the “strewn” subcompound of 25 icositetrachora; and the light blue and red parts are cross sections of the “strewn” subcompound of 200, either color being one of the chiral subcompounds of 100 of those. Where the latter overlap they color the region a shade of taupe. Here is a closeup:



One can surely see how deleting various subcompounds from the "master" compound of 450 will generate smaller icositetrachoric compounds. Using Stella4D in this manner, I’ve already modeled literally dozens of different compounds of icositetrachora, and it’s getting difficult to keep everything organized Some of the compounds are uniform, some are vertex-regular, some are cell-regular, and some are none of the above, just hexacosichorically symmetric. Some are chiral, some doubly so, with two independently chiral subcompounds. It will take several months of writing to get through this lot, let me tell you.

[Dinogeorge]

Coxeter’s list of “regular” icositetrachoric compounds in Regular Polytopes altogether includes seven: the (incompletely) self-dual compound of five, two mutually dual compounds of 25 (“scrunched” and “strewn”), two mutually dual compounds of 100 (“scrunched” and “strewn”), and two mutually dual compounds of 200 (likewise “scrunched” and “strewn”). I have been able to model all seven, and many others not quite so “regular” but nevertheless quite symmetric, using Stella4D. Indeed, since I have a Stella4D model of the “master compound” of 450, all the interesting compounds (and a host of “uninteresting” ones) can in theory be modeled with Stella4D by simply keeping or deleting appropriate component subsets from the “master compound,” provided one has the patience to find them.

Stella4D readily constructs the “scrunched” compound of 25 icositetrachora in a hexacosichoron when given its vertex figure, the well-known compound of five cubes in a dodecahedron. The Coxeter symbol is 5{3,3,5}[25{3,4,3}]{3,3,5}. The dual “strewn” compound of 25 icositetrachora in a hecatonicosachoron follows at once by dualizing the “scrunched” compound. Its Coxeter symbol is {5,3,3}[25{3,4,3}]5{5,3,3}. One cannot yet make it from its vertex figure in Stella4D, because the vertex figure is the same as that of a single 24-cell, namely, a cube. Here are pictures of the icosahedrally symmetric 0.555 3D cross sections of these two compounds:



One may see why I call the compound above “scrunched” and the compound below “strewn”: the vertices of the 24-cells are all "scrunched" together at the corners of the convex-hull hexacosichoron, but "strewn" apart at the corners of the convex-hull hecatonicosachoron. Both compounds have the full hexacosichoric symmetry group of order 14400. The compound above can be built from 3420 Stella4D nets (one a sniv), the compound below from 840 (one a sniv). In the "scrunched" compound, the octahedral cells do not overlap but occupy the 600 cell-realms of a core hexacosichoron; in the "strewn" compound, the octahedral cells lie by fives in the 120 cell-realms of a core hecatonicosachoron.



The above compound has a cube as its vertex figure, but only half of the pyritohedral subgroup of 24 (that is, the rotational tetrahedral group of oder 12) of its 48 symmetries are used in it. Each compound comprises five copies of the self-dual compound of five icositetrachora, and the pictures are colored in five colors, one for each subcompound. Here is what the compound of five looks like all by itself, as the usual 0.555 3D cross section, colored in five colors, one for each icositetrachoron:



It is chiral and incompletely self-dual, and has only 1/10 of the symmetries, that is, 1440, of the full hexacosichoric group. This is five times the number of symmetries in the rotational icositetrachoric group, namely 576, divided by two, which explains its chirality (the reflections are all absent). Stella4D will create 74 nets from which one may build this model. One net is a teeny sniv. The compound has no remaining icosahedral symmetry axes but does have tetrahedral symmetry axes, so the section is taken orthogonal to a tetrahedral symmetry axis, and viewed down a triangular symmetry axis (along which it has two different views, front and back). One may compound five of these compounds into either a "scrunched" or a "strewn" 25-compound. Then the chirality vanishes, and the full hexacosichoric symmetry group of order 14400 is restored.

Coxeter spent two pages describing this remarkable figure. which one may construct using Stella4D by keeping a suitable subset of five icositetrachora of the “scrunched” (or “strewn”) compound of 25. There are many choices of five, but only two (enantiomorphic) will uniformly retain all 120 corners of the “scrunched” 25-compound. The choice pattern is not too difficult to discover once one gets the hang of working with the figure.