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Five grand stellated hecatonicosachora

Posted: Fri Feb 08, 2008 11:56 pm
by Dinogeorge
The next regular aggrandizement of the hecatonicosachoron arises by "stellation" (edge-stellation) of the grand hecatonicosachoron, so it is the “stellated grand hecatonicosachoron” or, in keeping with the tradition established by Arthur Cayley in naming the Kepler-Poinsot polyhedra, the 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. 8)

Five great grand hecatonicosachora

Posted: Sun Feb 10, 2008 3:55 am
by Dinogeorge
The terms “stellated,” “great,” and “grand” in the names of the regular star-polychora are commutative: It doesn’t matter in what order they are applied. In the context of the regular star-polytopes (which exist in two, three, and four dimensions), “stellated” refers to the edge-stellation of a regular polygon (or star-polygon): The edges of the polygon are prolonged in the plane of the polygon until they meet other likewise prolonged edges to form a larger regular star-polygon. Then the points of intersection become the new vertices, and the old vertices become re-entrant “false” vertices. Going up one dimension, “great” refers to the extension of a regular face of a polyhedron in its plane until it meets other face planes likewise extended, to make similar larger regular faces. The lines of intersection of the face planes include the edges of the new faces, while the old edges become re-entrant “false” edges. Finally, going up one more dimension, “grand” refers to the expansion of a regular cell of a polychoron in its realm until it meets other cell realms likewise extended, to make similar larger regular cells. The planes of intersection of the realms include the faces of the new cells, while the old faces become re-entrant “false” faces. The three operations are perfectly analogous among themselves for the star-polytopes in all n dimensions, n = 2, 3, or 4.

There is just one way to stellate a regular pentagon, so in all instances in which the name includes the adjective “stellated,” the underlying polytope has had its regular pentagons stellated into pentagrams to create the stellated polytope. Stellations exist for all convex regular polygons with five or more sides, although in the case of the hexagon, the one stellation results in a compound of two equits (the Mogen David). Among regular polyhedra, greatenings can take place with equits, regular pentagons, and pentagrams. Thus we have the great icosahedron from the regular icosahedron, the great dodecahedron from the regular dodecahedron, and the great stellated dodecahedron from the (small) stellated dodecahedron. This accounts for all six regular polyhedra with icosahedral symmetry. Note that the great stellated dodecahedron is the same figure as the “stellated great dodecahedron,” establishing the commutativity of the adjectives “great” and “stellated.”

In four-space, five kinds of cells will aggrandize to yield “grand” regular star-polychora: regular tetrahedra, regular dodecahedra, small stellated dodecahedra, great dodecahedra, and great stellated dodecahedra. These account for ten of the twelve regular polychora with hexacosichoric symmetry. The other two, which have icosahedra and great icosahedra for cells, cannot aggrandize in the restricted sense being discussed here (wherein the “grand” new cell must be similar to the old cell); the latter is merely a greatening of the former. But under general aggrandizement (wherein the new cell need not be similar to the old cell but may be anything that makes a closed polychoron), nine of the regular star-polychora are aggrandizements of the hecatonicosachoron, and the tenth is an aggrandizement of the hexacosichoron.

Only one regular star-polyhedron exemplifies the commutativity of the terms “stellated,” “great,” and “grand.” But in four-space, four regular star-polychora do that, namely, the great stellated, grand stellated, great grand, and great grand stellated hecatonicosachora. The adjectives may be written in any order before “hecatonicosachoron” in naming the figure. The order that I prefer stems from Cayley’s 1859 names for the Kepler-Poinsot polyhedra (that is, putting the adjective “stellated” last) and Coxeter’s comment to me that he enjoyed the combination “great grand” because it reminded him of “great-grandfather.” I believe Coxeter, in his book Regular Complex Polytopes, acknowledges John Horton Conway as the discoverer of this bit of polytopic trivia.

Thus the great grand hecatonicosachoron, in which the dodecahedra are first aggrandized into the larger dodecahedra of a grand hecatonicosachoron and then greatened into even larger great dodecahedra, is also the “grand great hecatonicosachoron,” in which the dodecahedra are first greatened into larger great dodecahedra and these are aggrandized into the even larger great dodecahedra. It’s Schlaefli symbol is {5,5/2,3}. It is constructed by replacing the 120 small stellated dodecahedra of {5/2,5,5/2} with the 120 great dodecahedra that have the same vertices. This also stellates the great dodecahedral vertex figure of {5/2,5,5/2} from a great dodecahedron into a great stellated dodecahedron.

Here is a picture of the 0.555 3D cross section of the compound of five great grand hecatonicosachora. The equit depressions (“grooves”) in the points of {5/2,5,5/2}, whose cross sections are the great-dodecahedral-like dimples all over the preceding cross section of the grand stellated hecatonicosachoron, are replaced by equit crests (because the dimples of the great dodecahedron vertex figure become the points of the great stellated dodecahedron vertex figure developed from it), whose sections form the triangular peaks all over this figure:


and here is a closeup:


I’ve already displayed this compound in a previous post, so this is no longer “humanity’s first look,” but these pictures use the “standard pose” of my last few posts (0.555 cross section orthogonal to an axis of icosahedral symmetry). This particular section, which Stella4D notes has “high” complexity, requires 2720 nets, several of which are fairly complicated but most of which comprise just one or two small snivs. The “teeny whorl” near the center of the previous section has evolved into a somewhat more extensive whorl in this one. :)

Five great grand stellated hecatonicosachora

Posted: Sun Feb 10, 2008 11:58 pm
by Dinogeorge
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. 8)

Five great icosahedral hecatonicosachora

Posted: Mon Feb 11, 2008 5:59 am
by Dinogeorge
There are two cute ways to construct the regular star-polychoron {3,5/2,5}. One is to greaten each of the 120 icosahedral cells of {3,5,5/2}, the icosahedral hecatonicosachoron, into relatively huge great icosahedra. Then the 20 equit faces of each icosahedral cell naturally expand (“greaten”) into the 20 much larger equit faces of each great-icosahedral cell. The other way is to substitute for each of the 120 small stellated dodecahedra of a grand stellated hecatonicosachoron the great icosahedra that have the same vertices and edges. In the first way, it is clear that the polychoron closes, because the great star-polychoron has the same shared face planes as {3,5,5/2}, which we already know exists. The second way requires a bit of extra work to show that the inscribed great icosahedra do indeed share each equit with just one of the other great icosahedra. Either way leads to the name great icosahedral hecatonicosachoron for the resulting regular star-polychoron: the first way greatens the underlying icosahedral hecatonicosachoron, the second way simply makes all its cells great icosahedra; they\re “pre-greatened,” so to speak. Hope I’m not being too pedantic here :!:

Eight of the regular polychora with hexacosichoric symmetry are constructible as aggrandizements of the hecatonicosachoron (counting the hecatonicosachoron itself as one), and humanity has now had its “first look” at the chiral five-compounds of six of them, right in this Stella4D forum. Humanity has also had its “first look” at three of the remaining four chiral five-compounds here as well: the compound of five hexacosichora, the compound of five icosahedral hecatonicosachora, and the compound of five great icosahedral hecatonicosachora. The latter, however, has not yet been displayed in the “canonical” way that the others have been, namely, as the 0.555 section orthogonal to an icosahedral symmetry axis. The picture below remedies this inadequacy:


The compound of five congruent polyhedra seen here (colored in the usual five colors; each polyhedron is a 3D cross section of one of the five great icosahedral hecatonicosachora) requires 720 moderately intricate nets, none of which is just a sniv or two. It has a “high” complexity, despite which Stella4D calculates the nets if you ask her to. In case you’re interested, she also finds a remarkable 459,398 kinds of stellachunks in the stellation cell diagram for this figure. The vertex figure of the great icosahedral hecatonicosachoron, which is a small stellated dodecahedron, is quite evident in the picture in the cross sections of the numerous points.

The great icosahedral hecatonicosachoron is the dual of the great grand hecatonicosachoron and the conjugate of the icosahedral hecatonicosachoron. Stella4D won’t build the compound of ten great grand hecatonicosachora directly, but, fortunately, she will create it as the dual of the compound of ten great icosahedral hecatonicosachora. From there it is a short exercise to remove the correct five components to make the compound of five. This makes me very happy! :D

Five grand hexacosichora

Posted: Tue Feb 12, 2008 12:58 am
by Dinogeorge
The great icosahedron is the most complicated of the regular polyhedra in terms of the number of its external facelets, with a surhedron of 180 faces. Its 4D analogue, the grand hexacosichoron (600-cell) likewise is the most complicated of the regular polychora, with a surchoron of 36000 cells (according to Jonathan Bowers’s counts on the Wikipedia; Jonathan calls this figure a gax). Back in the 1990s, Bruce Chilton created the first sectioning movies of it, along its four main symmetry axes, but now it can routinely be sectioned in any direction by Stella4D with just a few mouse clicks. Such is the advance of technology. Pretty soon there will be nothing left for geometers to do except click mouse buttons and look at the resulting pretty pictures and movies :!:

Naturally, the compound of ten (“ten-gax”) was one of the first I displayed in this forum, because of its astounding complexity. I was surprised that the current version of Stella4D generated the compound of ten directly from its vertex figure, which is a uniform compound of two great icosahedra. Previous Stella4D versions that I attempted this compound with became “lost.” Once I had the compound of ten, it was pretty straightforward to delete five of them to make the chiral vertex-regular compound of five. Here is “humanity’s first look” at it:


and here is a closeup, approximately down an axis of fivefold symmetry:


The great-icosahedral vertex figures are very evident as cross sections of its points. This cross section is at the usual level of 0.555, in a realm orthogonal to an icosahedral symmetry axis. The cross section is a compound of five congruent polyhedra (colored in the usual five colors), each a section of one of the component grand hexacosichora. The 3000 tetrahedra of the compound are distributed as 120 compounds of five, plus 2400 more "loners." I'm pretty sure (because of conjugacy) the compounds of five tetrahedra lie in the 120 realms of the great-stellated-dodecahedral cells of the great grand stellated dodecahedron that has the 600 corners of the compound. Since the grand hexacosichoron has the same vertices, edges, and faces (but not the same cells; that is, belongs to the same company) as the great icosahedral hecatonicosachoron, the section of the five-compound above has the same vertices and edges (but not the same faces) as the preceding section of the compound of five great icosahedral hecatonicosachora.

Coxeter’s symbol for this compound is {5,3,3}[5{3,3,5/2}], and it is the dual of the compound of five great grand stellated hecatonicosachora. Whereas the latter has 2520 vertices (that is, 2400 + 120), the compound of five grand hexacosichora has 2520 cell realms (and only 600 vertices), which surround a tiny polychoron at its center that has 2520 cells: 2400 unequal equit bipyramids with their long apices truncated, and 120 regular icosahedra, the stumps of that truncation. The stump icosahedra correspond to the previously described 120 hexachoric vertices of the compound of five great grand stellated hecatonicosachora. If you don’t truncate the 120 vertices, the underlying 2400-cell with the congruent unequal-equit-bipyramidal cells is exactly the Catalan dual of the sidpixhi. The colored compound of five grand hexacosichora produces a nicely five-colored version of the truncated sidpixhi dual by using the “create convex hull” command with Stella4D.

This finishes my series on the compounds of five regular polychora listed in Coxeter’s Regular Polytopes. I eagerly look forward to someday constructing the compounds of {5/2,3,5} and its dual/conjugate {5,3,5/2}, the two that we have not yet looked at.

Should I post a similar series on the compounds of ten :?: :wink:

Re: Five grand hexacosichora

Posted: Thu Feb 21, 2008 12:49 pm
by robertw
Dinogeorge wrote:Should I post a similar series on the compounds of ten :?: :wink:
George has indeed continued his series for the compounds of ten regular polychora, but I have split this into a new topic. You can find that new topic here:

If you were watching this topic and wish to also watch the new one, you will need to click on the "Watch this topic for replies" link at the bottom of the new topic.


Posted: Tue Aug 26, 2008 8:05 pm
by 3katie3
fantastic pictures. just beautiful. i love looking at them.