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.