*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.