**compound of ten hecatonicosachora about a hexacosichoron**, displayed as the 0.555 section by a realm orthogonal to an icosahedral symmetry axis. I painted it in two colors, one for each chiral subset of five hecatonicsachora. All the compounds of ten regular pentagonal polychora and star-polychora are ferociously complicated, and coloring them in ten different colors doesn’t really help matters very much. As with the well-known compound of ten tetrahedra in a dodecahedron, which has external coplanar facelets, this compound has overlapping corealmic cellets, and their facelet cross sections are colored by Stella4D in a color intermediate between the two main colors, a pleasing effect, overall. This figure is a compound of ten congruent chiral polyhedra in left- and right-handed forms. Stella4D will print the 1442 nets, some complicated but mostly just snivs and sniv pairs, needed to assemble a model:

Coxeter’s symbol for this figure is [

*10*{5,3,3}]

*2*{3,3,5}; I think Jonathan Bowers might call it a tenhi. The lack of a leading Schlaefli symbol in Coxeter’s notation signals that the vertices do not lie at the corners of a regular polychoron, so it is not vertex-regular. I have described the set of 2520 vertices of the compound of five hecatonicosachora in a previous post. The compound of ten uses the same vertices, only twice over, so that there are 120 vertices where the vertex figure is a compound of ten tetrahedra, and 2400 vertices where the vertex figure is merely a compound of two tetrahedra: two tetrahedra from the compound of ten that have a pair of overlapping coplanar faces. This follows from the “2” in Coxeter’s notation.

Incidentally, Stella4D will use that vertex figure of two tetrahedra with square faces to find the compound of five tesseracts in a tall dodecahedral prism (or a uniform small or great ditrigonary icosidodecahedral prism). It is the uniform compound you get when you make a 4D prism based on the well-known compound of five cubes in a dodecahedron.

The 1200 dodecahedral cells of the ten hecatonicosachora lie by pairs in the 600 cell realms of a regular hexacosichoron. The pairs of dodecahedral cells are octahedrally symmetric compounds (although only the tetrahedral subgroup of order 24 is used in the 4D compound). Here is what they look like:

The polyhedron common to both dodecahedra (the convex core) is a kind of tetrakis cube, dual to a quasiuniform “golden truncated” octahedron. The dual of the dodecahedra pair is a compound of two icosahedra, inscribable in that “golden truncated” octahedron, which Stella4D can use as a vertex figure to create the compound of ten hexacosichora in a hecatonicosachoron (to be displayed in a future post). Once that compound is created, the chiral compound of five hexacosichora (displayed in an earlier post) follows by removing five of the ten hexacosichora; the compound of ten hecatonicoachora (displayed here) follows by dualizing the compound of ten hexacosichora; and the compound of five hecatonicosachora (also displayed in an earlier post) follows by either dualization or deletion. In the 4D model being displayed, the 120 tetrakis-cube cores would be colored with the blend of light yellow and teal, and cross sections of their visible (external) cellets appear colored this way in the picture.