The ultimate regular aggrandizement of the hecatonicosachoron is constructed from the great grand hecatonicosachoron {5,5/2,3} by stellating its already huge great-dodecahedral cells into relatively enormous great stellated dodecahedra. The resulting figure, which has 600 vertices instead of the 120 of all the other regular star-polychora, has the Schlaefli symbol {5/2,3,3}, and is commonly called the
great grand stellated hecatonicosachoron. If its great stellated dodecahedral cells are replaced by the dodecahedra with the same vertices, their pentagonal faces will not meet, so the sequence of aggrandized regular star-polychora that began with the hecatonicosachoron ends with this figure. The great grand stellated hecatonicosachoron’s tetrahedral-pentagrammatic points are the narrowest of any uniform polychoron.
Just incidentally, there are several neat 600-vertex star-polychora that have those 120 dodecahedra among their cells. Since the dodecahedra do not meet at their pentagonal faces, at least one other set of cells is required to close the figure. Several such symmetric sets of “auxiliary” cells exist, and the resulting uniform star-polychora all belong to the
dattady regiment (Jonathan Bowers’s name). One I find particularly elegant and easy to comprehend is the
dattathi, whose cells are the 120 dodecahedra along with 120 ditrigonary dodecadodecahedra and 120 great stellated dodecahedra. The pentagons of the dodecadodecahedra adjoin the pentagons of the dodecahedra, leaving the dodecadodecahedral pentagrams free. But these are in turn taken up by the pentagrams of the great stellated dodecahedra, closing the polychoron. These latter cells are inscribed in the cells of the circumscribing hecatonicosachoron and lie entirely in the surchoron of the figure. The great stellated dodecahedra touch one another only at their corners. Here, courtesy of Stella4D, is a picture of the usual 0.555 3D icosahedrally symmetric cross section of dattathi:
If, on the other hand, you want to close the 120 deep dodecahedra with just one other kind of auxiliary cell, you can do it with 120 great ditrigonary icosidodecahedra. These adjoin the dodecahedra along their pentagonal faces and adjoin one another along their equit faces, leaving no free faces. The resulting star-polychoron is the
giddatady. Here is a picture of the 0.555 section:
Finally, these two polychora can be
blended together: merged so that their vertices coincide, with common cells deleted entirely or blended into different cells. Here the dodecahedra vanish, the two kinds of ditrigonary cells blend into 120 small ditrigonary icosidodecahedra, and the 120 great stellated dodecahedra remain untouched. The resulting uniform polychoron is the
dattady itself, the colonel of the regiment. Jonathan Bowers and I discovered this regiment independently more than a decade ago (how time flies
); he likely beat me to the punch by a year or two. Here is a picture of the 0.555 cross section:
The cross sections of the cells and cellets that form the faces and facelets in the pictures of these polychora are color coded: the great stellated dodecahedra are teal, the dodecahedra are maroon, the ditrigonary dodecadodecahedra are light yellow, the great ditrigonary icosidodecahedra are red, and the small ditrigonary icosidodecahedra, being blends of the latter two, are orange.
Many uniform polytopes come in
conjugate forms, that is, topologically equivalent forms in which star faces are exchanged for non-star faces that have the same number of sides (for example, pentagons and pentagrams) and
vice versa. Dattady and giddatady are conjugates, and their blend dattathi is self-conjugate. The fact that this regiment contains a self-conjugate form is sufficient to make the entire regiment self-conjugate, that is, each uniform polychoron in it also has its conjugate in it. (On the other hand, some pairs of regiments will contain each other’s conjugates.) The small and great ditrigonary icosidodecahedra are conjugate uniform polyhedra, the dodecahedron and the great stellated dodecahedron are conjugate regular polyhedra, and the ditrigonary dodecadodecahedron is a self-conjugate uniform polyhedron. The great grand stellated hecatonicosachoron and the ordinary hecatonicosachoron are each other’s conjugates. End of digression
The regular polychora with 120 vertices make vertex-regular compounds of five and ten when their vertices are placed at the 600 corners of a regular hecatonicosachoron. But the compounds of five and ten great grand stellated hecatonicosachora have too many vertices for this kind of neat arrangement. Instead, the totals of 3000 and 6000 vertices occupy the corners of a polychoron that has two different kinds of corners, so the compounds are not even uniform (they’re biform): the 2400 corners of a
small diprismatohexacosihecatonicosachoron (Jonathan Bowers’s acronym:
sidpixhi), together with the 120 corners of a concentric hexacosichoron that has the same circumglome, located above the centers of the 120 sidpixhi dodecahedral cells. In the compound of five, 600 of the 3000 vertices come together by fives at the 120 hexacosichoric vertices, and the remaining 2400 are distributed one per sidpixhi corner. In the compound of ten, 1200 of the 6000 vertices come together by tens at the hexachoric vertices, and the remaining 4800 are distributed pairwise at the sidpixhi corners. These compounds cannot (yet) be made directly by Stella4D; but they may be constructed as duals of the compounds of five and ten grand hexacosichora {3,3,5/2}, which Stella4D
does (now) make directly. Coxeter’s notations for these compounds are [
5{5/2,3,3}]{3,3,5} and [
10{5/2,3,3}]
2{3,3,5}. The absence of a leading regular-polychoron Schlaefli symbol signals that these are cell-regular compounds but not vertex-regular. The five and ten times120 cell realms belong to a tiny central hexacosichoron buried inside the figure (twice over, in the case of the compound of ten). Coxeter did not specify the distribution of the vertices of these compounds, so this post helps to rectify that situation.
The compound of five has already appeared in a previous post, so the picture here is no longer “humanity’s first look” at it. It is the usual 0.555 3D cross section orthogonal to an icosahedral symmetry axis, in line with the preceding posts:
Stella4D says this particular section has “extreme” complexity, despite which she still manages to calculate its nets. There are only 440 of them, most quite complicated but a couple of kinds comprising just one or two snivs. The fivefold rosette near the center of the picture is a cross section near one of the 120 corners where the points of five great grand stellated hecatonicosachora come together. (Recall the "whorls" in the two preceding compounds. The actual corner is outside this sectioning realm.) At this close a sectioning realm to the center, there are no “loose” pieces of the section; it all hangs together as a compound of five polyhedra. colored in the usual five colors.
