**grand stellated hecatonicosachoron**. Its Schlaefli symbol is {5/2,5,5/2}, which, being palindromic, signals it as a self-dual regular star-polychoron. Its cells are 120 small stellated dodecahedra and its vertex figure is a great dodecahedron (the dual of its cell, of course, since the figure is self-dual). The cells are edge-stellations of the dodecahedral cells of the grand hecatonicosachoron. It would presently be too time-consuming to search all the stellachunks above the U stellayer to determine which belong to the {5/2,5,5/2} surchoron, even with the Heinz diagram in hand. There are simply too many of them.

Here is a picture of the 0.555 3D cross section of the chiral compound of five grand stellated hecatonicosachora, orthogonal to an axis of icosahedral symmetry, looking more or less down a fivefold symmetry axis, colored in the usual set of five colors. Each component has 120 vertices, and together they lie one per corner at the corners of a regular hecatonicosachoron. Coxeter’s symbol for it is {5,3,3}[

*5*{5/2,5,5/2}]{3,3,5}:

and here is a closeup:

As with the previous pictures, I believe this is “humanity’s first look” at this figure. Using Stella4D, I first constructed the compound of ten from its vertex figure (a pair of intersecting great dodecahedra). Then I deleted one set of five from the compound (and checked to make sure I had deleted the correct five). The section is a chiral compound of five differently colored congruent 3D cross sections of the individual grand stellated hecatonicosachora. And look at the little teeny five-sided whorl right on the symmetry axis

The great-dodecahedral vertex figures give the 4D points a sculptured appearance, clearly evident in the plethora of distorted intersecting great dodecahedra in this figure. To build this particular section as a real polyhedron model requires 860 nets, which despite the “high” complexity of the figure Stella4D manages to calculate. Most of the nets are small, having only one or two snivs, but three kinds (of which 20, 60, and 60 copies, respectively, are required) are pretty complicated cutouts in their own right.

The grand stellated hecatonicosachoron has the same vertices, edges, and faces (but not the same cells) as the great stellated hecatonicosachoron, which has no grooves in its points. So, theoretically, I could use Stella4D to construct a 0.555 cross section of the latter (which I was unable to model directly) by “reverse faceting” the compound seen here. Unfortunately, the cross section is too dense with edges for this to be feasible.