## More 4D compounds now available

- Jabe
**Posts:**49**Joined:**Sat Jan 12, 2008 6:30 am**Location:**Somewhere between Texas and the Fourth Dimension-
**Contact:**

Here's an idea that might help Stella render these "unStellable" scaliforms and compounds, could we not only define the vertex figure, but also give it a cover figure to help Stella find the polychoron - here's an example, for sistakix - we could take Hi's verf, and find the embedded 10-tet verf within the dodecahedron verf sides - this'll reveal the 10-ex compound - then dig in futher to locate sistakix - then send the sistakix verf with the hi verf cover - this would be a great feature to add to Stella to help it render these - maybe we could have a "dig into verf" feature to help find all sorts of scaliforms and semi-uniform figures in 4D

May the Fourth (dimension) be with you.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Five hecatonicosachora

Here's a picture made from another of the cell-regular 4D compounds, the less complicated conjugate of the compound of five great grand stellated hecatonicosachora posted in a previous reply. Its five components are identical ordinary convex regular hecatonicosachora (120-cells), so the compound has nowhere near as many cellets as its pointier conjugate. It does, however, have the same set of 2520 corners: 2400 have only a single vertex of one of the five components, whereas the other 120 have five coincident vertices, one from each component. At those corners the tetrahedral vertex figures combine into familiar compounds of five tetrahedra. Having two kinds of corners, the compound is thus not uniform. Its 600 cell realms, however, lie in the cell realms of a regular 600-cell (hexacosichoron), hence the description "cell-regular."

The 3D cross section is at level 0.555, as with a previous section, a bit away from the center pf the figure. The sectioning direction is fivefold-corner-first, which gives the whole cross section icosahedral rotational symmetry. The points of fivefold, threefold, and twofold rotational symmetry are pretty easy to pick out n the picture. A patient model maker could probably construct a physical model of this particular section in a week or so; Stella4D will easily draw the 300 nets of the individual pieces that need to be cut out and pasted together to build it. As a 3D model, the cross section is itself a compound (note the usual five different colors), each component being a cross section of one of the hecatonicosachora. Choosing the sectioning direction through a fivefold vertex inclines the components at the same angle to the sectioning realm, so the cross-section components are congruent polyhedra.

The 3D cross section is at level 0.555, as with a previous section, a bit away from the center pf the figure. The sectioning direction is fivefold-corner-first, which gives the whole cross section icosahedral rotational symmetry. The points of fivefold, threefold, and twofold rotational symmetry are pretty easy to pick out n the picture. A patient model maker could probably construct a physical model of this particular section in a week or so; Stella4D will easily draw the 300 nets of the individual pieces that need to be cut out and pasted together to build it. As a 3D model, the cross section is itself a compound (note the usual five different colors), each component being a cross section of one of the hecatonicosachora. Choosing the sectioning direction through a fivefold vertex inclines the components at the same angle to the sectioning realm, so the cross-section components are congruent polyhedra.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Five stellated hecatonicosachora

After typing this long post

Working out from the center, the very first of the nonillions of “stellations” (in four-space, I call them aggrandizements, from the word “grand,” as in “grand 120-cell”) of the regular hecatonicosachoron is the

The Schlaefli symbol of the stellated hecatonicosachoron is {5/2,5,3}. As with the hexacosichoron and eight of the nine other regular star-polychora, its 120 vertices occur in five sets among the 600 vertices of a regular hecatonicosachoron, in two mirror-image ways. This leads to the well-known compounds of five and ten star-polychora described by Coxeter in his book

This picture displays a single 3D cross section of the chiral compound of five stellated hecatonicosachora. It is at the 0.555 level (like the section in a preceding post), and lies in a sectioning realm orthogonal to an icosahedral symmetry axis. This gives the figure as a whole icosahedral rotational symmetry. Centers of fivefold, threefold, and twofold symmetry are easily found; a fivefold center appears near the middle of the picture, for example. It would not be excruciatingly difficult for a model maker (using Stella4D) to print, cut out, and paste together the 200 nets needed to build a physical model, although some of the edgelets are pretty short in this particular section. The figure is a compound of five polyhedra (note the usual five-color scheme), each a 3D cross section of one of the component stellated hecatoniicosachora, made congruent by the direction of the section. What fun, eh?

**twice**and completely destroying it**twice**, just because I accidentally happened to hit a wrong key at the website, I’ve learned my lesson and recomposed it*using my word processor*. That way I’ll at least have a copy to work on instead of losing the whole thing outright into bulletin-board cyberspace. Once you inadvertently leave the “post reply” page, everything you’ve laboriously typed and edited into a coherent 600+-word post is**GONE**. Why is there no way to recover I wasted a lot of precious time because of this So harken to my plight:*Use a word processor for long posts to this site.*Learn from my mistakes, avoid aggravation.*OK, then, for the third and I really, really hope final time:*Working out from the center, the very first of the nonillions of “stellations” (in four-space, I call them aggrandizements, from the word “grand,” as in “grand 120-cell”) of the regular hecatonicosachoron is the

**stellated hecatonicosachoron**. It is constructed by drawing each of the 120 dodecahedral cells of the hecatonicosachoron out into a small stellated dodecahedron: extending the dodecahedral edges until they meet in new vertices (an operation often called “edge-stellation” but for which I reserve the term "stellation" itself, thereby justifying the name “stellated hecatonicosachoron”). This decreases the number of vertices from 600 to 120---one above each of the underlying hecatonicosachoric cells---and increases the number of cells at each vertex from four to twelve. The 600 original vertices do remain externally visible in the star-polychoron, but only as false “re-entrant vertices”: corners where four cell-realms intersect. This is analogous to the situation in three-space with the small stellated dodecahedron, which may be modeled by affixing pentagonal pyramids to the faces of an underlying dodecahedron. There, three face-planes intersect in each false vertex of the underlying dodecahedron. In the 3D case, the pentagons’ interiors are hidden from external view beneath the pyramids (only the vertices and edges are visible, as “re-entrant” elements), but in 4D, even the pentagons of the underlying hecatonicosachoron (central parts of the pentagrammatic faces) remain visible (as “re-entrant” elements) on the exterior. It is just the interiors of the underlying dodecahedra that become hidden in the star-polychoron.The Schlaefli symbol of the stellated hecatonicosachoron is {5/2,5,3}. As with the hexacosichoron and eight of the nine other regular star-polychora, its 120 vertices occur in five sets among the 600 vertices of a regular hecatonicosachoron, in two mirror-image ways. This leads to the well-known compounds of five and ten star-polychora described by Coxeter in his book

*Regular Polytopes*. Coxeter’s symbol for the compound of five is {5,3,3}[*5*{5/2,5,3}]{3,3,5}, which tells us that there are five stellated hecatonicosachora inside (*i.e.*, together having the 600 corners of) a regular hecatonicosachoron {5,3,3}, surrounding (having the 600 face-realms of) a regular hexacosichoron {3,3,5}. Each corner of the case hecatonicosachoron is home to just one stellated-hecatonicosachoric vertex.This picture displays a single 3D cross section of the chiral compound of five stellated hecatonicosachora. It is at the 0.555 level (like the section in a preceding post), and lies in a sectioning realm orthogonal to an icosahedral symmetry axis. This gives the figure as a whole icosahedral rotational symmetry. Centers of fivefold, threefold, and twofold symmetry are easily found; a fivefold center appears near the middle of the picture, for example. It would not be excruciatingly difficult for a model maker (using Stella4D) to print, cut out, and paste together the 200 nets needed to build a physical model, although some of the edgelets are pretty short in this particular section. The figure is a compound of five polyhedra (note the usual five-color scheme), each a 3D cross section of one of the component stellated hecatoniicosachora, made congruent by the direction of the section. What fun, eh?

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Five great hecatonicosachora

The

N.B.: A

I call these kinds of little bits and pieces surrounding a core polytope “stellachunks,” and an entire set of them that just covers an underlying figure a “stellalayer.” If the hecatonicosachoron is labeled A, then the 120 dodecahedral pyramids are the “B stellachunks,” all together forming the “B stellalayer.” The 720 pentagonal double pyramids are the “C stellachunks,” forming the “C stellalayer.” And so on. Bruce Chilton has recently found all 57 stellalayers (A through BL) of the hecatonicosachoron.

Here is a picture of a net of one of those 720 double-pyramid stellachunks that model the great hecatonicosachoron:

This stellachunk net has two maroon pentagonal pyramids and five teal tetrahedral wedges (“dyadic double pyramids”) as cells. It folds up in four-space, along the face-planes between the joined cells, to produce a pentagonal double pyramid of the correct shape to make the great hecatonicosachoron: an irregular heptachoron (seven cells total). One maroon pyramid in the picture is hidden by the five attached tetrahedra. The triangles are the familiar acute and obtuse isosceles golden triangles. Unfortunately, Stella4D is not yet equipped to display such irregular polychora except as nets.

Adding the 720 double pyramids to a stellated hecatonicosachoron changes it in the following ways: First, the small-stellated-dodecahedral cells expand into great dodecahedra. Their pentagrammatic faces expand, or as I call it, “greaten” (hence the name “great hecatonicosachoron”; greatening is the same operation as face-stellation) into the regular pentagons that have the same vertices. These meet other such pentagons and become the faces of the great-dodecahedral cells of the great hecatonicosachoron. When the double pyramids are added, their pentagonal-pyramidal cells blend with and cover up the pentagonal pyramids of the points of the stellated hecatonicosachoron, while the five wedge-cells combine to create the new exterior (or “surcell”) of the figure. This works because the shapes of the lateral cells are exactly the shapes of the 30 tetrahedra added to a small stellated dodecahedron to make it a great dodecahedron. All the edges of the stellated hecatonicosachoron become false “re-entrant edges” of the great hecatonicosachoron, whose true edges turn out to be the 720 wedge edges of the double pyramids. Five tetrahedra surround the wedge edge of the double pyramid, which tells us that five great dodecahedra of the great hecatonicosachoron share each edge---accounting for the second 5 in the Schlaefli symbol, {5,5/2,5}. It should also be apparent that the true edges of the great hecatonicosachoron connect neighboring points of the underlying stellated hecatonicosachoron, so they are also edges of the hexacosichoron (600-cell) that is the convex hull of both star-polychora. (So the great hecatonicosachoron is in the same

Having thus become acquainted with the great hecatonicosachoron, let’s have “humanity’s first look” at the chiral regular compound of five:

This 3D cross section is at the usual level of 0.555, along the usual icosahedral symmetry axis, colored in the usual five colors. So it has centers of fivefold, threefold, and twofold symmetry and overall icosahedral rotational symmetry. One of the fivefold symmetry centers is parked close to the center of the picture. This particular section comprises 770 nets, so it is quite a bit more complicated than the two preceding five-compound sections. The components are congruent and transitive on the icosahedral symmetry group, as with those previous two compounds. The sectioning movie, of which this is one frame, is quite attractive!

**great hecatonicosachoron**, Schlaefli symbol {5,5/2,5}, is only the second of the nonillions of aggrandizements of the regular hecatonicosachoron. Just as the stellated hecatonicosachoron may be modeled by sticking 120 dodecahedral pyramids onto the hecatonicosachoron’s dodecahedral cells, so the great hecatonicosachoron may be modeled by sticking 720 pentagonal double pyramids into the pentagonal “valleys” of an underlying stellated hecatonicosachoron. The double pyramids surmount the pentagonal centers of the pentagrammatic faces, the two pyramidal cells of each double pyramid blending with two adjacent lateral cells across the valleys between the stellated hecatonicosachoron’s dodecahedral pyramids. This situation is similar to modeling a great dodecahedron in three-space by sticking 30 wedge-shaped tetrahedra onto a small stellated dodecahedron, into the valleys formed by pairs of its points.N.B.: A

**double pyramid**is a polytope that is a pyramid whose base is itself a pyramid. Double pyramids exist in spaces of any number of dimensions from one up, but they’re pretty trivial in dimensions 1--3. In four-space, the base of the base is a plane polygon I call the “base face” of the double pyramid; the edge that joins the two apices I call the “wedge edge.” If the base face has*n*corners, the double pyramid has*n*+2 cells: two*n*-gonal pyramids (hence the name “double pyramid”) and*n*tetrahedra (the “lateral cells”). The simplest double pyramid is the triangular double pyramid, which has a triangular base face and five cells altogether. It is better known as a pentachoron, of course. In a pentachoron, any face can serve as the base face, with the lone edge that has no end in the base face serving as the wedge edge.I call these kinds of little bits and pieces surrounding a core polytope “stellachunks,” and an entire set of them that just covers an underlying figure a “stellalayer.” If the hecatonicosachoron is labeled A, then the 120 dodecahedral pyramids are the “B stellachunks,” all together forming the “B stellalayer.” The 720 pentagonal double pyramids are the “C stellachunks,” forming the “C stellalayer.” And so on. Bruce Chilton has recently found all 57 stellalayers (A through BL) of the hecatonicosachoron.

Here is a picture of a net of one of those 720 double-pyramid stellachunks that model the great hecatonicosachoron:

This stellachunk net has two maroon pentagonal pyramids and five teal tetrahedral wedges (“dyadic double pyramids”) as cells. It folds up in four-space, along the face-planes between the joined cells, to produce a pentagonal double pyramid of the correct shape to make the great hecatonicosachoron: an irregular heptachoron (seven cells total). One maroon pyramid in the picture is hidden by the five attached tetrahedra. The triangles are the familiar acute and obtuse isosceles golden triangles. Unfortunately, Stella4D is not yet equipped to display such irregular polychora except as nets.

Adding the 720 double pyramids to a stellated hecatonicosachoron changes it in the following ways: First, the small-stellated-dodecahedral cells expand into great dodecahedra. Their pentagrammatic faces expand, or as I call it, “greaten” (hence the name “great hecatonicosachoron”; greatening is the same operation as face-stellation) into the regular pentagons that have the same vertices. These meet other such pentagons and become the faces of the great-dodecahedral cells of the great hecatonicosachoron. When the double pyramids are added, their pentagonal-pyramidal cells blend with and cover up the pentagonal pyramids of the points of the stellated hecatonicosachoron, while the five wedge-cells combine to create the new exterior (or “surcell”) of the figure. This works because the shapes of the lateral cells are exactly the shapes of the 30 tetrahedra added to a small stellated dodecahedron to make it a great dodecahedron. All the edges of the stellated hecatonicosachoron become false “re-entrant edges” of the great hecatonicosachoron, whose true edges turn out to be the 720 wedge edges of the double pyramids. Five tetrahedra surround the wedge edge of the double pyramid, which tells us that five great dodecahedra of the great hecatonicosachoron share each edge---accounting for the second 5 in the Schlaefli symbol, {5,5/2,5}. It should also be apparent that the true edges of the great hecatonicosachoron connect neighboring points of the underlying stellated hecatonicosachoron, so they are also edges of the hexacosichoron (600-cell) that is the convex hull of both star-polychora. (So the great hecatonicosachoron is in the same

**regiment**as the hexacosichoron, which is the regiment’s**colonel**.) Incidentally, the palindromic Schlaefli symbol indicates that this hectonicosachoron is self-dual.Having thus become acquainted with the great hecatonicosachoron, let’s have “humanity’s first look” at the chiral regular compound of five:

This 3D cross section is at the usual level of 0.555, along the usual icosahedral symmetry axis, colored in the usual five colors. So it has centers of fivefold, threefold, and twofold symmetry and overall icosahedral rotational symmetry. One of the fivefold symmetry centers is parked close to the center of the picture. This particular section comprises 770 nets, so it is quite a bit more complicated than the two preceding five-compound sections. The components are congruent and transitive on the icosahedral symmetry group, as with those previous two compounds. The sectioning movie, of which this is one frame, is quite attractive!

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Re: Five great hecatonicosachora

Make that A through BE. Sorry about the typo!Dinogeorge wrote:And so on. Bruce Chilton has recently found all 57 stellalayers (A through BL) of the hecatonicosachoron.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Five icosahedral hecatonicosachora

Over the years, a kind of unofficial terminology has evolved among people who play with polytopes. For example, in increasing order of dimensionality, there are four venerable names for the different kinds of polytope elements: vertex (plural: vertices), edge, face, and cell. This series has recently been indefinitely extended with teron, peton, exon, etc. There are now also names for polytope elements going the

Star-polytopes in general, and star-polychora in particular, need some more terms. I’ve already used the term “surcell” in previous posts. The surcell was named as a pun on the word “surface”: the “-face” part is two-dimensional, so to obtain the analogous three-dimensional term, change “-face” to “-cell.” In

The surtope of a simple polytope is the polytope itself, of course. But the surtope of a star-polytope is not the same figure as the star-polytope itself. It’s just the simple “shell” of exterior topelets that exactly encloses the star-polytope. A

The surhedron of a star-polyhedron, the surchoron of a star-polychoron, and in general the surtope of a star-polytope are usually polytopes with a lumpy, bumpy topography loaded with several kinds of peaks and valleys. The term “peak” itself is already in use (see above), so I use the term

The opposite of a point is a

Polyhedra:

Points (convex vertex) and dimples (concave vertex)

Aretes (convex edge) and valleys (concave edge)

Two facelets (

Three or more facelets meet about a point or a dimple

Polychora:

Points (convex vertex) and dimples (concave vertex)

Crests (convex edge) and depressions (concave edge)

Aretes (convex face) and valleys (concave face)

Two cellets (

Three or more cellets meet in a crest or depression

Four or more cellets meet about a point or dimple

To get specific, the stellated hecatonicosachoron’s surchoron has 120 dodecahedral points separated by 720 pentagonal valleys. The great hecatonicosachoron’s surchoron is bult by sticking 720 stellachunks into the valleys, which raises the 720 valleys into crests that surround 1200 equit depressions (three congruent cellets around an edge). And to build the icosahedral hecatonicosachoron’s surchoron we stick 1200 stellachunks (of a shape to be described below) into the equit depressions. This models the D aggrandizement of the hecatonicosachoron, which is a regular polychoron whose cells are 120 icosahedra. Its Schlaefli symbol is {5,3,5/2}.

Restating the situation, the B aggrandizement of the hecatonicosachoron was created by adding 120 dodecahedral pyramids (the points) onto the underlying A aggrandizement (the hecatonicosachoron itself). This covered the hecatonicosachoron completely and left a surcell composed of 1440 pentagonal pyramids (two per valley), which models the exterior of the stellated hecatonicosachoron. The C aggrandizement was modeled by adding 720 pentagonal double pyramids to this surchoron, into the valleys between the dodecahedral points. This covered the B aggrandizement completely and left a surchoron composed of 3600 tetrahedral wedges (3D "triangular double pyramids"), which models the exterior of the great hecatonicosachoron.

The D aggrandizement is created by aggrandizing the 120 great dodecahedral cells of the great hecatonicosachoron into the regular icosahedra that have the same vertices and edges. We model this by adding 1200 flattened pentachoric stellachunks into the triangular depressions of the C aggrandizement surchoron. Here is a picture of the net of one such stellachunk.

The three tetrahedral-wedge-shaped cells are colored teal; these are the same shape (and color) as the cellets of the underlying great hecatonicosachoron in my earlier post. They blend out when the D stellalayer is added, leaving a surchoron composed of 2400 (two per stellachunk) light yellow flattened tetrahedra. (In the picture of the net, one of them is completely covered by the other cells.) These have the shape of the flattened tetrahedra that may be placed into the dimples of a great dodecahedron to change it into a regular icosahedron. The D stellalayer models the

All this is fairly well known to us polytopers. But now, here is “humanity’s first look” at the compound of five icosahedral hecatonicosachora:

The picture shows the usual level 0.555 3D cross section of the compound, taken by a sectioning realm perpendicular to an axis of icosahedral symmetry. The figure is colored in the usual five colors, one for each component. The closeup

zooms in on a center of fivefold symmetry. Note that a rotation of 2pi/5 of the cross section about this or any fivefold axis will send each component into a component of a different color: All five components are congruent. I was surprised at how complicated this section turned out to be as a physical polyhedron model: Stella4D calculated that it requires a total of 3860 nets to build, although many of them are one- and two-piece “sniv nets.” I cannot be certain from the account in

*other*way, in decreasing order of dimensionality: facet, ridge, and peak. A**facet**of an*n*-dimensional polytope is any of its (*n*-1)-dimensional elements, its “flat sides.” In four-space (and only in four-space), facets are three-dimensional polytopes (polyhedra) more usually called “cells.” A**ridge**is an (*n*-2)-dimensional element (if there is one) of an*n*-dimensional polytope, and a**peak**is an (*n*-3)-dimensional element (if there is one). In four-space, ridges are the faces (not to be confused with “facets”!), while 4D peaks are the edges. In any polytope of dimension greater than 2 (in which facets, ridges, and peaks all exist), exactly two facets adjoin at each ridge, whereas three or more facets must always be incident around a peak.Star-polytopes in general, and star-polychora in particular, need some more terms. I’ve already used the term “surcell” in previous posts. The surcell was named as a pun on the word “surface”: the “-face” part is two-dimensional, so to obtain the analogous three-dimensional term, change “-face” to “-cell.” In

*n*dimensions, the corresponding term I use is “surtope.” This then begets the infinitude of names of the other kinds of surtopes. In the present 4D context, "surcell" denotes the simple polychoron (no intersecting cells, no self-intersecting faces, etc.; but coincident elements may be OK in certain exceptional circumstances) formed by just the*external cellets*of a star-polychoron. In the plane, on the other hand, we already have a term that means the same as a “suredge” (of a polygom), namely,**periphery**. Thus we have, in order of increasing dimension starting with 2, the following sequence of surtopes: periphery, surhedron (or surface), surchoron (or surcell), surteron, surpeton, surexon, etc. Inasmuch as the term “surface” may apply to any kind of two-dimensional manifold, flat or curved, such as the surface of a sphere, I created the term “surhedron” especially for the external surface of a polyhedron. Likewise, a “surcell” can be any three-dimensional manifold, such as the surcell of a glome (sphere in four dimensions), so I created the term “surchoron” specifically for the external surcell of a polychoron. I hope this doesn’t come across as too nitpicky or confusing. I should have been using "surchoron" instead of "surcell" all along.The surtope of a simple polytope is the polytope itself, of course. But the surtope of a star-polytope is not the same figure as the star-polytope itself. It’s just the simple “shell” of exterior topelets that exactly encloses the star-polytope. A

**cellet**, by the way, is a piece of a cell bounded by other cells of the polytope that intersect or adjoin the cell. We add the suffix -let to the name of a polytope element to get the name of a piece of the element, as in “edgelet, “facelet,” “facetlet,” “ridgelet,” “peaklet,” “teronlet,” “petonlet,” and so forth. “Topelet” would be the general term.The surhedron of a star-polyhedron, the surchoron of a star-polychoron, and in general the surtope of a star-polytope are usually polytopes with a lumpy, bumpy topography loaded with several kinds of peaks and valleys. The term “peak” itself is already in use (see above), so I use the term

**point**, as in “a five-pointed star has five points” (not to be confused with the usual concept of a mathematical/geometrical point, or with a vertex) for a configuration of facetlets about a vertex.. This configuration must be, loosely speaking, “lumplike” (locally convex) so that the point “sticks out” from within its neighborhood. In four-space, points are polyhedral, defined by the surhedron of the vertex figure at that vertex. So, in particular, there must be at least four cells/cellets at a point, and we speak of tetrahedral points, dodecahedral points, and so forth. Depending on the vertex-figure polyhedron, points may have lateral grooves and even holes in them.The opposite of a point is a

**dimple**. Whereas a point “sticks out,” a dimple “sticks in.” Otherwise, a dimple is just like a point. In four-space, dimples, like points, are polyhedral. A ridge that “sticks out” I call an**arete**, which is a kind of sharply defined ridge in mountaineering. And a ridge that “sticks in” I call, simply, a**valley**. In 4D and higher spaces, a peak that “sticks out” I call a**crest**, and a peak that “sticks in” I call, again simply, a**depression**. Here’s a little chart summarizing these terms for 3D and 4D:Polyhedra:

Points (convex vertex) and dimples (concave vertex)

Aretes (convex edge) and valleys (concave edge)

Two facelets (

*i.e.*, faces of a surhedron) meet at an arete or a valleyThree or more facelets meet about a point or a dimple

Polychora:

Points (convex vertex) and dimples (concave vertex)

Crests (convex edge) and depressions (concave edge)

Aretes (convex face) and valleys (concave face)

Two cellets (

*i.e.*, cells of a surchoron) meet at an arete or valleyThree or more cellets meet in a crest or depression

Four or more cellets meet about a point or dimple

To get specific, the stellated hecatonicosachoron’s surchoron has 120 dodecahedral points separated by 720 pentagonal valleys. The great hecatonicosachoron’s surchoron is bult by sticking 720 stellachunks into the valleys, which raises the 720 valleys into crests that surround 1200 equit depressions (three congruent cellets around an edge). And to build the icosahedral hecatonicosachoron’s surchoron we stick 1200 stellachunks (of a shape to be described below) into the equit depressions. This models the D aggrandizement of the hecatonicosachoron, which is a regular polychoron whose cells are 120 icosahedra. Its Schlaefli symbol is {5,3,5/2}.

Restating the situation, the B aggrandizement of the hecatonicosachoron was created by adding 120 dodecahedral pyramids (the points) onto the underlying A aggrandizement (the hecatonicosachoron itself). This covered the hecatonicosachoron completely and left a surcell composed of 1440 pentagonal pyramids (two per valley), which models the exterior of the stellated hecatonicosachoron. The C aggrandizement was modeled by adding 720 pentagonal double pyramids to this surchoron, into the valleys between the dodecahedral points. This covered the B aggrandizement completely and left a surchoron composed of 3600 tetrahedral wedges (3D "triangular double pyramids"), which models the exterior of the great hecatonicosachoron.

The D aggrandizement is created by aggrandizing the 120 great dodecahedral cells of the great hecatonicosachoron into the regular icosahedra that have the same vertices and edges. We model this by adding 1200 flattened pentachoric stellachunks into the triangular depressions of the C aggrandizement surchoron. Here is a picture of the net of one such stellachunk.

The three tetrahedral-wedge-shaped cells are colored teal; these are the same shape (and color) as the cellets of the underlying great hecatonicosachoron in my earlier post. They blend out when the D stellalayer is added, leaving a surchoron composed of 2400 (two per stellachunk) light yellow flattened tetrahedra. (In the picture of the net, one of them is completely covered by the other cells.) These have the shape of the flattened tetrahedra that may be placed into the dimples of a great dodecahedron to change it into a regular icosahedron. The D stellalayer models the

**icosahedral hecatonicosachoron**, so called because it has 120 interpenetrating icosahedra as its cells (of which just those flattened tetrahedra would be visible on the exterior to a 4D model maker).All this is fairly well known to us polytopers. But now, here is “humanity’s first look” at the compound of five icosahedral hecatonicosachora:

The picture shows the usual level 0.555 3D cross section of the compound, taken by a sectioning realm perpendicular to an axis of icosahedral symmetry. The figure is colored in the usual five colors, one for each component. The closeup

zooms in on a center of fivefold symmetry. Note that a rotation of 2pi/5 of the cross section about this or any fivefold axis will send each component into a component of a different color: All five components are congruent. I was surprised at how complicated this section turned out to be as a physical polyhedron model: Stella4D calculated that it requires a total of 3860 nets to build, although many of them are one- and two-piece “sniv nets.” I cannot be certain from the account in

*Regular Polytopes*, but I’m pretty sure it was my old mentor, H. S. M. Coxeter himself, who discovered the vertex-regular and cell-regular compounds of five and ten regular star-polychora. Sadly, he died some four years ago,*just before*desktop computer technology had advanced to the point where sectioning movies, in full color, could easily be made. He would certainly have been thrilled to see his figures “in the flesh.- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Five hexacosichora

In three-space, to obtain an icosahedron from a great dodecahedron’s surhedron, we simply fill up its 20 equit dimples with 20 flattened equit pyramids. These are

The

In four-space, we can likewise stuff the 600 tetrahedral dimples of the surchoron of the icosahedral hecatonicosachoron with short irregular pentachora that are low regular-tetrahedral pyramids. The short apex sticks into a dimple, the opposite regular tetrahedron becomes one of the 600 cells of the regular hexacosichoron that has the same vertices, edges, and faces (but not the same cells) as the icosahedral hecatonicosachoron.

As in three-space, the next (E) aggrandizement of the hecatonicosachoron (in my ongoing series of posts) is constructed by stuffing 600 unequal tetrahedral bipyramids (instead of the tetrahedral pyramids) into the tetrahedral dimples of the icosahedral hecatonicosachoron. The short apices fit into the dimples as before, the 600 slightly taller opposite apices become the 600 points of the E aggrandizement. Here is a 3D net of the stellachunk:

The four light yellow equit pyramids (inside) form the bottom (short) apex, the red equit pyramids (outside) form the top (slightly taller) apex. Unlike the previous aggrandizements, the E aggrandizement is no longer a regular polychoron. Its cells are the 120 B greatenings of the icosahedral cells of the icosahedral hecatonicosachoron. There are many, many irregular aggrandizements between the inner ones already covered and the ones farther out that form regular polychora. Here is a picture of the B greatening of an icosahedron:

The faces are 20 irregular hexagons, and the polyhedron itself is the dual of the

The dual is a uniform polyhedron in which faces of two different kinds, three equits and three pentagrams, meet alternately, six around each of the 20 vertices. It is also the vertex figure of the dual of the E aggrandizement of the hecatonicosachoron, which is a uniform star-polychoron whose cells are 600 tetrahedra and 120 great icosahedra. Stella4D will draw sections of either of these polychora, but so far Stella4D gets lost trying to construct the uniform compounds of five and ten of them.

Here is the 0.555 section of the E aggrandizement of the hecatonicosachoron:

and here is the 0.555 section of the uniform polychoron dual of the E aggrandizement:

The uniform polychoron is color-coded to the vertex figure, but the E aggrandizement I left monochromatic. It’s too tedious to color each of its 120 cells in, say, a five-color scheme.

Finally, here is a 0.555 section of the compound of five hexacosichora in a hecatonicosachoron, perpendicular to an axis of icosahedral symmetry. Coxeter’s notation for this compound, which is vertex-regular but not cell-regular, is {5,3,3}[

In the compound, which has a total of five times 600 = 3000 tetrahedral cells, 600 of the cells form compounds of five in the 120 dodecahedral cells of the circumscribing hecatonicosachoron. The other 2400 tetrahedra “fill up the gaps” between the five-tetrahedra compounds. In the section, some of the five-tetrahedra compounds show up as sets of five coplanar triangles and quadrilaterals, easily visible in the section picture above. Stella4D warns that this compound polyhedron has “extreme” complexity as far as making its nets goes, likely because it has lots of different face planes, and crashes (runs out of memory) if one proceeds further.

*not*stellachunks of the great dodecahedron, because the “top” faces, opposite the low apex that sinks into the dimples, do not lie in the face planes of the underlying great dodecahedron. They lie, rather, in 20 face planes new to the figure: the planes of the faces of the icosahedron that has the same vertices and edges as the great dodecahedron.The

*true*D stellachunks of the dodecahedron are 20 unequal equit bipyramids, in which one apex is short and the opposite apex is tall. The short apices fit exactly into and blend out the great dodecahedron’s surhedron’s dimples, while the tall apices, separated by 30 rather deep valleys, become the 20 points of the great stellated dodecahedron.In four-space, we can likewise stuff the 600 tetrahedral dimples of the surchoron of the icosahedral hecatonicosachoron with short irregular pentachora that are low regular-tetrahedral pyramids. The short apex sticks into a dimple, the opposite regular tetrahedron becomes one of the 600 cells of the regular hexacosichoron that has the same vertices, edges, and faces (but not the same cells) as the icosahedral hecatonicosachoron.

As in three-space, the next (E) aggrandizement of the hecatonicosachoron (in my ongoing series of posts) is constructed by stuffing 600 unequal tetrahedral bipyramids (instead of the tetrahedral pyramids) into the tetrahedral dimples of the icosahedral hecatonicosachoron. The short apices fit into the dimples as before, the 600 slightly taller opposite apices become the 600 points of the E aggrandizement. Here is a 3D net of the stellachunk:

The four light yellow equit pyramids (inside) form the bottom (short) apex, the red equit pyramids (outside) form the top (slightly taller) apex. Unlike the previous aggrandizements, the E aggrandizement is no longer a regular polychoron. Its cells are the 120 B greatenings of the icosahedral cells of the icosahedral hecatonicosachoron. There are many, many irregular aggrandizements between the inner ones already covered and the ones farther out that form regular polychora. Here is a picture of the B greatening of an icosahedron:

The faces are 20 irregular hexagons, and the polyhedron itself is the dual of the

**small ditrigonary (or ditrigonal) icosidodecahedron**. It has vertices of two different kinds: those surrounded by three hexagons, and those surrounded by five in a pentagrammatic pattern. Here is a picture of its dual:The dual is a uniform polyhedron in which faces of two different kinds, three equits and three pentagrams, meet alternately, six around each of the 20 vertices. It is also the vertex figure of the dual of the E aggrandizement of the hecatonicosachoron, which is a uniform star-polychoron whose cells are 600 tetrahedra and 120 great icosahedra. Stella4D will draw sections of either of these polychora, but so far Stella4D gets lost trying to construct the uniform compounds of five and ten of them.

Here is the 0.555 section of the E aggrandizement of the hecatonicosachoron:

and here is the 0.555 section of the uniform polychoron dual of the E aggrandizement:

The uniform polychoron is color-coded to the vertex figure, but the E aggrandizement I left monochromatic. It’s too tedious to color each of its 120 cells in, say, a five-color scheme.

Finally, here is a 0.555 section of the compound of five hexacosichora in a hecatonicosachoron, perpendicular to an axis of icosahedral symmetry. Coxeter’s notation for this compound, which is vertex-regular but not cell-regular, is {5,3,3}[

*5*{3,3,5}]. As with the previous sections, this makes the sections of the individual components congruent. They’re colored in my usual scheme of five colors (red, white, teal, light yellow, and maroon), and the section is displayed more or less looking down a fivefold symmetry axis:In the compound, which has a total of five times 600 = 3000 tetrahedral cells, 600 of the cells form compounds of five in the 120 dodecahedral cells of the circumscribing hecatonicosachoron. The other 2400 tetrahedra “fill up the gaps” between the five-tetrahedra compounds. In the section, some of the five-tetrahedra compounds show up as sets of five coplanar triangles and quadrilaterals, easily visible in the section picture above. Stella4D warns that this compound polyhedron has “extreme” complexity as far as making its nets goes, likely because it has lots of different face planes, and crashes (runs out of memory) if one proceeds further.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Re: Five hexacosichora

Oops! The color-coding was backward. I have put theDinogeorge wrote:here is the 0.555 section of the uniform polychoron dual of the E aggrandizement:

The uniform polychoron is color-coded to the vertex figure, but the E aggrandizement I left monochromatic.

**corrected color-coded figure**into the quote above. Sorry about that.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Pictures

Incidentally, forum visitors may download for their own use any of my pictures by simply right-clicking on them and then left-clicking on the "save picture as" menu item. They're pretty snazzy when printed on glossy photoprint stock with an inkjet printer.

- robertw
- Site Admin
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**Contact:**

### Re: Five hexacosichora

Just for your info, you can edit your old posts to correct errors.Dinogeorge wrote:Oops! The color-coding was backward. I have put thecorrected color-coded figureinto the quote above. Sorry about that.

Also, images are not uploaded to the forum, but referenced indirectly using the link you used. If you changed the image at that link, it would show up differently here too.

Rob.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Re: Five hexacosichora

Cool, but how? I don't see any "edit your post" buttons.robertw wrote:Just for your info, you can edit your old posts to correct errors.

- robertw
- Site Admin
**Posts:**560**Joined:**Thu Jan 10, 2008 6:47 am**Location:**Melbourne, Australia-
**Contact:**

### Re: Five hexacosichora

Dinogeorge wrote:For me, there are "edit" buttons next to the "quote" button in the top right corner of each post. As admin I see these on every post, but I thought other users would still be able to see these on their own posts, provided they are logged in. Is this not the case?robertw wrote:I don't see any "edit your post" buttons.

Rob.

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### Re: Five hexacosichora

Ah, yes. I see them. Didn't notice them before. They appear just with my posts, meaning of course that I can edit only my own posts and not someone else's.robertw wrote:For me, there are "edit" buttons next to the "quote" button in the top right corner of each post. As admin I see these on every post, but I thought other users would still be able to see these on their own posts, provided they are logged in. Is this not the case?.

Seeing as the correction is already in for my earlier post, I'll not bother editing it. But I'll keep this feature in mind if I need to correct a post in the future.

Thanks!

- Dinogeorge
**Posts:**71**Joined:**Sun Jan 13, 2008 7:09 am**Location:**San Diego, California

### More on hecatonicosachoric aggrandizements

Unfortunately, Stella4D gets lost when I try to construct the vertex/cell-regular compounds of ten great stellated hecatonicosachora {5/2,3,5} and ten grand hecatonicosachora {5,3,5/2} from their vertex figures. Perhaps not coincidentally, these two polychora are each other's duals. These would have been the next two figures in my ongoing romp through “humanity’s first looks” at the regular star-polychoron compounds. The star-polychora themselves are, of course, readily constructed by Stella4D with just a few mouse clicks. But for some reason those particular compounds’ vertex figures just don’t work, even though the same vertex figures

and now the

As far as I know, sectioning movies of these and the other eight regular star-polychora were first created, with considerable labor, by Bruce Chilton in the mid-1990s. Nowadays, a mere ten years later, Stella4D will section these and many other interesting polychora with just a few mouse clicks, and the virtual polychora may be moved around in four-space to be sectioned from any direction, not just the four axes of higher-order polyhedral symmetries.

Anyway, I

The great stellated hecatonicosachoron is developed from the great hecatonicosachoron by stellation (that is, edge-stellation: so it is also the "stellated great hecatonicosachoron"), which expands each of the 120 great-dodecahedral cells into great stellated dodecahedra. It has icosahedral points. The grand hecatonicosachoron is then developed from the great stellated hecatonicosachoron by replacing each great stellated dodecahedron with the regular dodecahedron that has the same vertices. It has highly sculptured great-icosahedral points (actually, they're more corners than points). As might be imagined, the dodecahedra of the grand hecatonicosachoron divide one another up into lots and lots of small cellets, all of which occur among the enormous set of hecatonicosachoron stellachunks.

Thus, above the E stellalayer, the hecatonicosachoron stellachunks rapidly become more varied and numerous and have less symmetric shapes. The farthest stellachunks that make up the great stellated hecatonicosachoron lie in the N stellalayer, and those that make up the grand hecatonicosachoron lie in the U stellalayer, according to Bruce Chilton’s diagram of the complete cell of the hecatonicosachoron (which he calls Heinz, from its 57 stellalayers). Here is a picture of the cell of the F aggrandizement of the hecatonicosachoron, based on Bruce’s Heinz diagram:

One hundred twenty of these pass through one another and fit together to make the F aggrandizement. Its faces are twelve red regular pentagons and 20 yellow irregular nine-pointed stars. In the latter kind of face, three of the vertices lie on three of its own edges (the three long ones). The F stellachunks turn out to be little flattened octachora: irregular tetrahedral bipyramids that have two of those short tetrahedral cells from the E stellalayer plus six more wedges, whose shapes are pretty obvious in the picture above. Here is a shot of the net:

The two short pyramids are red (one is entirely hidden) and the six wedges are orange. The whole surchoron comprises 1200 of these stellachunks, from which the red cells are blended out with valleys in the stellalayer below, leaving just the total of 7200 orange wedges (six per stellachunk), arranged in 1200 low hexahedral points, as the F hecatonicosachron’s surchoron. Alas, there’s not yet any way to construct the F aggrandizement with Stella4D short of supplying the coordinates of all the vertices and incidence tables for all the cellets, so I have no picture to show here. I wouldn’t mind someday seeing how the wedges come together at the 120 star-shaped vertices (where all those edges intersect in the F cell).

*do*work when called upon to generate regular compounds that have other faces (specifically, equits instead of pentagons or pentagrams). Oh, well, that’s life. I understand Robert Webb is working on a fix. Meanwhile, here are pictures of the 0.555 sections of those two star-polychora,*in lieu*of pictures of their compounds of five (which are the left- or right-handed chiral sets from the achiral compounds of ten). First the**great stellated hecatonicosachoron**:and now the

**grand hecatonicosachoron**:As far as I know, sectioning movies of these and the other eight regular star-polychora were first created, with considerable labor, by Bruce Chilton in the mid-1990s. Nowadays, a mere ten years later, Stella4D will section these and many other interesting polychora with just a few mouse clicks, and the virtual polychora may be moved around in four-space to be sectioned from any direction, not just the four axes of higher-order polyhedral symmetries.

Anyway, I

*would*someday really like to have a look at the compounds of five and ten of these. As one might infer from the above pictures, sections of the compounds of the great stellated hecatonicosachoron would be quite “pointy,” while the sections of the compounds of the grand hecatonicosachoron would be massively sculptured.The great stellated hecatonicosachoron is developed from the great hecatonicosachoron by stellation (that is, edge-stellation: so it is also the "stellated great hecatonicosachoron"), which expands each of the 120 great-dodecahedral cells into great stellated dodecahedra. It has icosahedral points. The grand hecatonicosachoron is then developed from the great stellated hecatonicosachoron by replacing each great stellated dodecahedron with the regular dodecahedron that has the same vertices. It has highly sculptured great-icosahedral points (actually, they're more corners than points). As might be imagined, the dodecahedra of the grand hecatonicosachoron divide one another up into lots and lots of small cellets, all of which occur among the enormous set of hecatonicosachoron stellachunks.

Thus, above the E stellalayer, the hecatonicosachoron stellachunks rapidly become more varied and numerous and have less symmetric shapes. The farthest stellachunks that make up the great stellated hecatonicosachoron lie in the N stellalayer, and those that make up the grand hecatonicosachoron lie in the U stellalayer, according to Bruce Chilton’s diagram of the complete cell of the hecatonicosachoron (which he calls Heinz, from its 57 stellalayers). Here is a picture of the cell of the F aggrandizement of the hecatonicosachoron, based on Bruce’s Heinz diagram:

One hundred twenty of these pass through one another and fit together to make the F aggrandizement. Its faces are twelve red regular pentagons and 20 yellow irregular nine-pointed stars. In the latter kind of face, three of the vertices lie on three of its own edges (the three long ones). The F stellachunks turn out to be little flattened octachora: irregular tetrahedral bipyramids that have two of those short tetrahedral cells from the E stellalayer plus six more wedges, whose shapes are pretty obvious in the picture above. Here is a shot of the net:

The two short pyramids are red (one is entirely hidden) and the six wedges are orange. The whole surchoron comprises 1200 of these stellachunks, from which the red cells are blended out with valleys in the stellalayer below, leaving just the total of 7200 orange wedges (six per stellachunk), arranged in 1200 low hexahedral points, as the F hecatonicosachron’s surchoron. Alas, there’s not yet any way to construct the F aggrandizement with Stella4D short of supplying the coordinates of all the vertices and incidence tables for all the cellets, so I have no picture to show here. I wouldn’t mind someday seeing how the wedges come together at the 120 star-shaped vertices (where all those edges intersect in the F cell).

Last edited by Dinogeorge on Fri Feb 08, 2008 9:12 pm, edited 1 time in total.

- robertw
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### Re: More on hecatonicosachoric aggrandizements

What are the vertex figures for those (and respective face choice)? I'll have a look at them.Dinogeorge wrote:Unfortunately, Stella4D gets lost when I try to construct the vertex/cell-regular compounds of ten great stellated hecatonicosachora {5/2,3,5} and ten grand hecatonicosachora {5,3,5/2} from their vertex figures.

I am? Well, I do look at it every now and then, but I don't know how to fix it. So if you have any ideas about how to decide which orientation the next verf should be matched with, I'd like to hear it.I understand Robert Webb is working on a fix.

I had the idea to give preference to an orientation if it also represented a reflection through the edge from the previous vertex. This holds for all uniform polyhedra except for snubs. However, turns out it isn't so reliable for compounds, eg the compound of 10 tetrahedra, as Jonathan pointed out (also 5 octahedra). And I suspect it doesn't hold for some of these 4D compounds either.

This sounds like the beginning of a good Stella4D review, hint hint!As far as I know, sectioning movies of these and the other eight regular star-polychora were first created, with considerable labor, by Bruce Chilton in the mid-1990s. Nowadays, a mere ten years later, Stella4D will section these and many other interesting polychora with just a few mouse clicks, and the virtual polychora may be moved around in four-space to be sectioned from any direction, not just the four axes of higher-order polyhedral symmetries.

Currently the Reviews Forum is looking a bit empty

Rob.