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Compounds of 10 regular polychora

 
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Dinogeorge



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PostPosted: Sat Feb 16, 2008 11:14 pm    Post subject: Ten hecatonicosachora Reply with quote

Well, here’s “humanity’s first look” at the cell-regular 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. Smile
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Dinogeorge



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PostPosted: Tue Feb 19, 2008 11:30 pm    Post subject: Ten stellated hecatonicosachora Reply with quote

And here is “humanity’s first look” at the compound of ten stellated hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (lavender pink and teal), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions (in this case, each such “polyhedron” is itself a further compound of three). They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Stella4D will calculate and print the 1742 nets required to build this figure, should one desire to make a model, which would be like building a lumpy beach ball from little tiny pieces. A few nets are complicated but most involve just one or two snivs.

The stellated hecatonicosachoron has 120 cells, 720 faces, 1200 edges, and 120 vertices, so unlike the underlying ten hecatonicosachora, this compound of ten has only 600 vertices rather than 2520, and is both vertex-regular and cell-regular. Coxeter’s notation for this compound is 2{5,3,3}[10{5/2,5,3}]2{3,3,5}, which shows that the vertices coincide by twos and the cells “pair up” in their realms. Indeed, the vertex figure is just the pair of dodecahedra displayed as the cells of the preceding compound, as one might expect from how the edge-stellation operation works.



Here is how the small-stellated-dodecahedral cells pair up in their realms. This is a uniform compound of two small stellated dodecahedra with octahedral symmetry, although only its tetrahedral subgroup is used in the 4D compound:



As we progress through the ten-compounds of the hecatonicosachoric stellations the way we did with the five-compounds, the cross sections will become wilder and wilder. Stella4D can routinely make real-time sectioning movies of the 4D compounds, but the wildest ones get pretty jumpy as the computer strives to keep up. To see the movies, however, you'll just have to get Stella4D. Wink
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Last edited by Dinogeorge on Mon Feb 25, 2008 8:15 pm; edited 2 times in total
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PostPosted: Wed Feb 20, 2008 8:08 am    Post subject: Ten great hecatonicosachora Reply with quote

This, as far as I know, is “humanity’s first look” at the compound of ten great hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (light yellow and red), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of orange. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions. They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Stella4D will calculate and print the 3044 nets required to build this figure, should one desire to make a model. A few nets are complicated but most involve just one or two snivs.

The great hecatonicosachoron has 120 cells, 720 faces, 720 edges, and 120 vertices, so its compound of ten has the 600 vertices of a hecatonicosachoron and is both vertex-regular and cell-regular. The vertex figure is the pair of small stellated dodecahedra displayed as the cell-pair of the preceding compound, which is what Stella4D uses to construct the figure. Interestingly, and quite oddly, not every such compound of two small stellated dodecahedra will work with Stella4D’s uniform-figure-generation routine. One vertex figure that I use will work with pentagons but not with equits (to produce this compound), another that I use will work with equits but not with pentagons (to produce the compond of ten great icosahedral hecatonicosachira), and a third I have will not work with either! It must be something about the order in which the faces are stored, but beyond that I know nothing. I am simply thankful I was lucky enough to stumble onto vertex figures that worked at all. It makes me wonder whether I may someday find the right forms of the vertex figures of the two ten-compounds I have as yet been unable to create with Stella4D.

Coxeter’s notation for this compound is 2{5,3,3}[10{5,5/2,5}]2{3,3,5}, which shows that both vertices and cells “pair up.” The palindromic Schlaefli symbol indicates that the compound is self-dual. In fact, it is completely self-dual, in the sense that the Stella4D dual function will return exactly the same figure in exactly the same orientation (although the colors will change; try it sometime!).



Here is how the great-dodecahedral cells pair up in their realms, as a uniform compound of two great dodecahedra with octahedral symmetry, although only its tetrahedral subgroup is used in the 4D compound:



These ten-compounds are all systematically similar enough that I can reuse most of the text from the preceding compound and just change a few names and numbers. That should account for any feelings of déjà vu the reader may have experienced reading this! (Sorry about that, but it does speed things up.) Twisted Evil
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Last edited by Dinogeorge on Wed Feb 20, 2008 7:21 pm; edited 1 time in total
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PostPosted: Wed Feb 20, 2008 5:53 pm    Post subject: Ten icosahedral hecatonicosachora Reply with quote

This, as far as I know, is “humanity’s first look” at the compound of ten icosahedral hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (light blue and maroon), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of taupe. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions. They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Stella4D will calculate and print the 5042 nets required to build this figure, should one desire to make a model. A few nets are themselves quite complicated but most involve just one or two snivs. The cells of the compound’s components never get too far below the surchoron, which accounts for the large regions of overlapping facelets in its sections.

The icosahedral hecatonicosachoron has 120 cells, 1200 faces, 720 edges, and 120 vertices, so its compound of ten has the 600 vertices of a hecatonicosachoron and is both vertex-regular and cell-regular. The vertex figure is the pair of great dodecahedra displayed as the cell-pair of the preceding compound, which is what Stella4D uses to construct the figure. Coxeter’s notation for this compound is 2{5,3,3}[10{3,5,5/2}]2{3,3,5}, which shows that both vertices and cells “pair up”:



Here is how the icosahedral cells pair up in their realms, as a uniform compound of two icosahedra with octahedral symmetry, although only its tetrahedral subgroup is used in the 4D compound:



Note the large taupe areas where two equits overlap. These are the external regions that bound the volumes common to both cells, whose sections appear likewise colored taupe in the 3D cross section.

These compounds are all systematically similar enough that I can reuse most of the text from the preceding compound and just change a few names and numbers. That should account for any feelings of déjà vu the reader may have experienced reading this! (Sorry about that, but it does speed things up.) Laughing
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PostPosted: Thu Feb 21, 2008 6:16 pm    Post subject: Ten hexacosichora Reply with quote

This, as far as I know, is “humanity’s first look” at the compound of ten hexacosichora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (dark blue and red), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of purple. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions. They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Alas, Stella4D cannot calculate and print the nets required to build this figure, should one desire to make a model. There are too many face planes and the stellation cell diagrams become way too huge, even though the number of actual nets does not seem too unwieldy. (My computer grinds to a halt. It would be nice if Stella4D could decouple the operation of net making from the operation of stellation.)

The hexacosichoron is the regular polychoron with 600 cells, 1200 faces, 720 edges, and 120 vertices. This compound of ten is vertex-regular and has the 600 vertices of a hecatonicosachoron. It is not, however, cell-regular, since the cells are of two kinds: those that come together by tens as compounds of ten tetrahedra (120 compounds, each inscribed in a different cell of the circumscribing hecatonicosachoron), and the remainder, which come together in 2400 pairs. The vertex figure of the compound is the pair of icosahedra displayed as the cell-pair of the preceding compound, which is what Stella4D uses to construct the figure. Coxeter’s notation for this compound is 2{5,3,3}[10{3,3,5}], which shows that the vertices “pair up” but only indicates that the cell realms are not those of a regular polychoron, not what that polychoron might be:



Here is how the tetrahedral cells group together in their 2520 realms, as compounds of two and ten. Twenty-four hundred realms have two tetrahedra arranged this way:



And 120 realms have ten tetrahedra arranged this way. This well-known regular compound is pictured in numerous books and other publications on polyhedra and polyhedron model making:



The two-tetrahedra compound is just two of the ten. The compounds of ten lie entirely in the compound's surchoron, and cross sections of some of them occur as coplanar face-groups in the cross section pictured above. The tetrahedral pairs "fill in the gaps" between the compounds of ten.

The convex core of the compound is that 2520-cell polychoron I alluded to in a previous post. Its cells are 120 icosahedra and 2400 unequal equit bipyramids truncated at their taller apices. The ten-compound is quite "shallow," in that the realms of the cells do not pass very deeply into the figure. The icosahedra of the core are the centers of the compounds of ten tetrahedra and are thus, in fact, part of the compound's surchoron. Here is a picture of one of the icosahedra (red) and how it joins to twelve of its 20 neighboring monotruncated equit bipyramids (MEBs; dark blue; the colors bear no relationship to the colors I used for the compound except that they're the same):



Note the teeny gap between the MEBs, which is all the room there is in which to fold up the net.

And here is a picture of a tower net that folds up in four-space to build a model of the convex core. All the icosahedra are hidden by their 20 attached MEBs, so the whole net is dark blue:



I tried to make a net in which the MEBs were colored in five different colors, four of each color “tetrahedrally” adjoining each icosahedron, but it drove me nuts trying also to make sure that no two adjoining MEBs (after the net is folded up in four-space) ever got the same color. I think it’s possible to do it, but I cannot waste my time finding it. It’s like solving the Rubik’s cube. Confused
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PostPosted: Sun Feb 24, 2008 6:52 pm    Post subject: Missing ten-compounds Reply with quote

Here is where I would have continued with the compounds of ten great stellated hecatonicosachora and grand hecatonicosachora in a hecatonicosachoron, but so far my efforts to coax Stella4D into displaying them for me have come to naught.

Up until now, the compounds have been rather more like "lumpy" glomes than “pointy,” like stars should be, but the compound of ten great stellated hecatonicosachora crosses the “pointy” threshold. Its points are still pretty fat, but it’s the best we can do until we reach the compound of ten great grand stellated hecatonicosachora. The reason for this is, of course, that the lateral cellets of the points are the sharper equit pyramids with lateral acute golden triangles, rather than the blunter pentagonal pyramids with lateral acute golden triangles. Most of the upcoming compounds have points that are inscribable in the points of the missing compound of ten great stellated hecatonicosachoron, so those pictures will indeed be “pointier” than the pictures displayed thus far.

The missing compound of ten grand hecatonicosachora is the last of the “lumpy” figures. Its lumps are highly sculptured because of the great-icosahedral vertex figures. For some reason, Stella4D doesn’t like to use the pair of great icosahedra that is the vertex figure of the compound of ten, although she will use it all right to generate the compound of ten grand hexacosichora.

Visitors to the ten-compound thread of the forum may have noticed the sequence in which the compounds appear, namely, that the cell-pairs of one compound become the vertex figures of the next. It is easy to list ten of the twelve regular pentagonal polychora in this kind of sequence, which is actually a cycle, since the first follows likewise from the last:

    {5,3,3}
    {5/2,5,3}
    {5,5/2,5}
    {3,5,5/2}
    {3,3,5}
    {5/2,3,3}
    {5,5/2,3}
    {5/2,5,5/2}
    {3,5/2,5}
    {3,3,5/2}

The first two numbers of the Schlaefli symbol of an entry become the last two numbers of the Schalefli symbol of the next entry.

Which two are omitted? Well, {5/2,3,5} and {5,3,5/2}, the very star-polychora whose compounds Stella4D won’t construct. Coincidence (heh, heh) Twisted Evil ? Naturally, I’ll continue the presentation of the ten-compounds in the order of the above sequence.

Of course, this sequence thing is phony, right? But, as an exercise, try to put all twelve pentagonal polychora into just that kind of sequence. Rolling Eyes
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PostPosted: Sun Feb 24, 2008 8:54 pm    Post subject: Ten great grand stellated hecatonicosachora Reply with quote

This, as far as I know, is “humanity’s first look” at the spectacular compound of ten great grand stellated hecatonicosachora about a hexacosichoron. As with the preceding compound, I painted it with just two colors (white and red), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of pink. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions (in this case, each such “polyhedron” is itself a further compound of no less than 53 true polyhedra, mostly cross sections of individual points that are partially merged around the center). They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Despite the “extreme” complexity of the figure, Stella4D will, after a few minutes of waiting, calculate and print the 1954 nets required to build this figure, should one desire to make a model. Some are pretty complicated, but most include just one or two snivs.

The great grand stellated hecatonicosachoron is a regular star-polychoron with 120 cells, 1200 faces, 720 edges, and 600 vertices. Its compound of ten is cell-regular but not vertex-regular (it is biform), having the same 2520 vertices as its conjugate, the compound of ten hecatonicosachora. The cells come together in 600 pairs, in the cell realms of a core hexacosichoron, as compounds of great stellated dodecahedra. The two vertex figures of the compound are the pair of tetrahedra displayed as the cell-pair of the preceding compound of ten hexacosichora, and the familiar regular compound of ten tetrahedra, likewise displayed. Coxeter’s notation for this compound is [10{5/2,3,3}]2{3,3,5}, which shows that the cells “pair up” but says only that the vertices are not those of a regular polychoron:



This, of course, is merely one frame of the sectioning movie of this compound, which Stella4D will play in real time. The movie gets a bit jumpy in the central regions as the computer tries to keep up. To see the movie, one will of course need to have a copy of Stella4D handy.

Here is how the great stellated dodecahedral cells pair up in their 600 realms, as biform compounds:



Constructing the ten-gogishi compound (as Jonathan Bowers might call it) required first building its dual, the ten-compound of grand hexacosichora in a hecatonicosachoron, which Stella4D did using the vertex figure of two great icosahedra, and then dualizing the result. I had been waiting a long time to see this figure; didn't think I'd see it in my lifetime Exclamation Very Happy
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Last edited by Dinogeorge on Tue Feb 26, 2008 6:08 am; edited 1 time in total
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PostPosted: Mon Feb 25, 2008 1:59 am    Post subject: Ten great grand hecatonicosachora Reply with quote

This, as far as I know, is “humanity’s first look” at the compound of ten great grand hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (turquoise and red), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a light shade of taupe. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions (in this case, each such “polyhedron” is itself a further compound of 16 true polyhedra). They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Despite the “high” complexity of the figure, Stella4D will, after a few minutes of waiting, heroically calculate and print the 4382 nets required to build this figure, should one ever desire to make a model. Some of the nets are pretty complicated, but most comprise just one or two snivs. The last net in particular is a real “microsniv.”

The great grand hecatonicosachoron is a regular star-polychoron with 120 cells, 720 faces, 1200 edges, and 120 vertices. Its compound of ten is both cell-regular and vertex-regular, having the 600 corners of a hecatonicosachoron and the 600 cell-realms of a hexacosichoron. As per earlier discussions at this forum, the compound edge-stellates into the preceding compound of ten great grand stellated hecatonicosachora. Its cells come together as 600 uniform compounds of two great dodecahedra. The vertex figure of the compound is the pair of great stellated dodecahedra displayed as the cell-pair of the preceding compound. Coxeter’s notation for this compound is 2{5,3,3}[10{5,5/2,3}]2{3,3,5}, which shows that the cells “pair up” in their realms and that the vertices of the components also pair up at the corners of a regular hecatonicosachoron.



This, of course, is one frame of the sectioning movie of this compound, which Stella4D will play in real time. The movie gets a bit jumpy in the central regions as the computer tries to keep up.

Here is a closeup of the cross section, looking down a fivefold symmetry axis:



This looks like a closeup of Saturn's satellite Hyperion Exclamation Each of the 3920 points (by Stella4D's count) is the cross section of a crest in the surchoron of the compound.

And here is how the great dodecahedral cells pair up in their 600 realms, as uniform compounds. We've seen this compound before, but here it is again in different colors and from a somewhat different viewpoint:



In order to construct the ten-compound of great grand hecatonicosachora, I had to ask Stella4D to create the dual compound of ten great icosahedral hecatonicosachora from its small-stellated-dodecahedral-pair vertex figure and then dualize that. Stella4D really doesn't like to build polychoric compounds from vertex figures that are these biform pairs of dodecahedra or great stellated dodecahedra. Crying or Very sad
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Last edited by Dinogeorge on Tue Feb 26, 2008 6:09 am; edited 1 time in total
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PostPosted: Mon Feb 25, 2008 7:58 am    Post subject: Ten grand stellated hecatonicosachora Reply with quote

This, as far as I know, is “humanity’s first look” at the compound of ten grand stellated hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (orange and maroon), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a light tan (I can find no such regions in this particular section, but they do appear in others). The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra in left- and right-handed versions (in this case, each such “polyhedron” is itself a further compound of two). They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Despite the “high” complexity of the figure, Stella4D will, after a few minutes of waiting, superheroically calculate and print the 4562 nets required to build it, should one ever desire to make a model. Some of the nets are pretty complicated, but most comprise just one or two snivs. The last two nets in particular are real “microsnivs.”

The grand stellated hecatonicosachoron is a regular star-polychoron with 120 cells, 720 faces, 720 edges, and 120 vertices. Its compound of ten is both cell-regular and vertex-regular, having the 600 corners of a hecatonicosachoron and the 600 cell-realms of a hexacosichoron. As per earlier discussions at this forum, the compound greatens into the preceding compound of ten great grand hecatonicosachora. Its cells come together as 600 uniform compounds of two small stellated dodecahedra. The vertex figure of the compound is the pair of great dodecahedra displayed as the cell-pair of the preceding compound. Coxeter’s notation for this compound is 2{5,3,3}[10{5/2,5,5/2}]2{3,3,5}, which shows that the cells “pair up” in their 600 hexacosichoric realms and that the vertices of the components also pair up at the corners of a regular hecatonicosachoron. The palindromic Schlaefli symbol indicates that the compound is completely self-dual, like its conjugate, the compound of ten great hecatonicosachora. If you ask Stella4D to build the dual, all she will do is change the colors (if anything) of the figure. This is unlike an incompletely self-dual figure such as the tetrahedron, dualization of which causes it to turn upside-down.



This, of course, is one frame of the sectioning movie of this compound, which Stella4D will play in real time. The movie gets a bit jumpy in the central regions as the computer tries to keep up.

Here is a closeup of the cross section, looking down a fivefold symmetry axis:



It would tax a model maker's patience to the ultimate to piece together those pesky little pentagrammatic rosettes right at the fivefold axes. Arggh Exclamation Shocked

And here is how the small-stellated-dodecahedral cells pair up in their 600 realms, as uniform compounds. We’ve seen this compound before, but here it is in different colors and from a somewhat different viewpoint:



Visitors should note that these pictures are merely cross sections of the actual four-dimensional star-compounds. The four-dimensional aspect of these figures is quite hidden from even the best of us geometers, although we may over time become very familiar with the figures in our mind's eye. Four-dimensionally, they're a lot more complicated than the little hints that the cross-section pictures provide. Compare, say, a great stellated dodecahedron with a single one of its two-dimensional cross sections, which might typically be a few interpenetrating polygons. Now scale the analogy up one dimension, using one of the preceding cross-section pictures in place of the interpenetrating polygons to try to infer the true appearance of the 4D star-compound itself. The mind boggles Exclamation Smile
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Last edited by Dinogeorge on Tue Feb 26, 2008 6:10 am; edited 1 time in total
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robertw
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Joined: 10 Jan 2008
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Location: Melbourne, Australia

PostPosted: Mon Feb 25, 2008 2:02 pm    Post subject: Re: Ten grand stellated hecatonicosachora Reply with quote

Dinogeorge wrote:
Compare, say, a great stellated dodecahedron with a single one of its two-dimensional cross sections, which might typically be a few interpenetrating polygons.


And just for fun, here's a cross-section of the great stellated dodecahedron, taken at the 0.555 depth that George has grown fond of:



As you said, this section is just two overlapping pentagrams. By the way, to get nicely coloured 2D cross-sections, use Stella's
"Color->Basic Color Scheme->Color Along Cross-Section Direction".

If you scroll down a bit on the Stella Screenshots page you can see several animated 2D cross-sections of 3D polyhedra too (http://www.software3d.com/ScreenShots.php), and scroll down further to see some animated 3D cross-sections of 4D polytopes.

Personally, I like the cross-sections of the great icosahedron. A small animated one can be seen on the page above, but here's a larger single frame, this time taken at 0.310 for aesthetic appeal:



Rob.
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Dinogeorge



Joined: 13 Jan 2008
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Location: San Diego, California

PostPosted: Mon Feb 25, 2008 7:55 pm    Post subject: Ten great icosahedral hecatonicosachora Reply with quote

robertw wrote:
Personally, I like the cross-sections of the great icosahedron. A small animated one can be seen on the page above, but here's a larger single frame, this time taken at 0.310 for aesthetic appeal:



Rob.


Very nice Exclamation Speaking of the great icosahedron, the present ten-compound features 1200 of them, and the 3200 faces of its 3D cross section below are cross sections of 700 of those cells (many are compound faces, hence the discrepancy in the numbers). Bruce Chilton made animated sectioning “movies” of the regular polyhedra, including the star-polyhedra, back in the 1960s by hand as a set of flipbooks. These might have been the first such animations ever made, since few people then thought of polyhedra in terms of their sectioning series. We were both impressed by the simple symmetry and beauty of the great icosahedron’s sections, as are you today. His flipbooks inspired my 1970 M.Sc. thesis on displaying 4D objects as computer-animated sectioning movies.

As far as I know, this is “humanity’s first look” at the compound of ten great icosahedral hecatonicosachora in a hecatonicosachoron. As with the preceding compound, I painted it with just two colors (orange and blue, my old high-school colors), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of taupe (just a few occasional locations in this section, best seen in the closeup). The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra (in this case, each such “polyhedron” is itself a further compound of five polyhedra) in left- and right-handed versions. They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Despite the “high” complexity of the figure, Stella4D will, after a few minutes of waiting, heroically calculate and print the 2862 nets required to build it, should one ever desire to make a model. Some of the nets are pretty complicated, but most comprise just one or two snivs.

The great icosahedral hecatonicosachoron is a regular star-polychoron with 120 cells, 1200 faces, 720 edges, and 120 vertices. Its compound of ten is both cell-regular and vertex-regular, having the 600 corners of a hecatonicosachoron and the 600 cell-realms of a hexacosichoron. Its cells come together as 600 uniform compounds of two great icosahedra. The vertex figure of the compound is the pair of small stellated dodecahedra displayed as the cell-pair of the preceding compound. Coxeter’s notation for this compound is 2{5,3,3}[10{3,5/2,5}]2{3,3,5}, which shows that the cells “pair up” in their 600 hexacosichoric realms and that the vertices of the components also pair up at the corners of a regular hecatonicosachoron.



This, of course, is one frame of the sectioning movie of this compound, which Stella4D will play in real time. The movie gets a bit jumpy in the central regions as the computer tries to keep up.

Here is a closeup of the cross section, looking down a fivefold symmetry axis:



And here is how the great-icosahedral cells pair up in their 600 realms, as octahedrally symmetric uniform compounds (but only the tetrahedral symmtery subgroup is used in the compound):



One more compound of ten to go, as this series enters its home stretch. Very Happy
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Dinogeorge



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PostPosted: Tue Feb 26, 2008 12:33 am    Post subject: Ten grand hexacosichora Reply with quote

The compound of ten great icosahedral hecatonicosachora in a hecatonicosachoron is one of the first I built with Stella4D once I discovered it was making compounds it had failed to create before. I posted pictures of a section shortly after this forum opened. So here I present it again, this time painted with just two colors (red and teal), one for each chiral subset of five. Where the external facelets are coplanar, the colors blend into a shade of light maroon. The picture shows the usual 3D cross section at depth 0.555 orthogonal to an axis of icosahedral symmetry. The ten components appear sectioned as ten congruent chiral polyhedra (in this case, each such “polyhedron” is itself a further compound of five) in left- and right-handed versions. They may be interchanged by suitable reflections and rotations about the figure’s axes of symmetry. Alas, the “extreme” complexity of the figure does prevent Stella4D from calculating and printing the nets for building a model of this cross section. It has 5980 faces, each a triangular or quadrilateral section of one of the 6000 tetrahedral cells (all but 20 of them in this case Exclamation ).

The grand hexacosichoron is a regular star-polychoron with 600 cells, 1200 faces, 720 edges, and 120 vertices. Its compound of ten is vertex-regular, having the 600 corners of a hecatonicosachoron, but not cell-regular, the convex core being the figure exhibited with the conjugate compound, of ten "ordinary" hexacosichora, above. In the ten-compound, there are 120 realms in which the tetrahedral cells make compounds of ten (circumscribed about the great-stellated-dodecahedral cells of the great grand stellated hecatonicosachoron that has the ten-compound's 600 vertices), and 2400 more realms in which the tetrahedral cells pair up: exactly as in the conjugate compound. The vertex figure of the compound is the pair of great icosahedra displayed as the cell-pair of the preceding compound. Coxeter’s notation for this compound is 2{5,3,3}[10{3,5/2,5}], which shows that the vertices “pair up” at the 600 corners of a regular hecatonicosachoron but that the cell realms are not those of a regular polychoron.



This, of course, is one frame of the sectioning movie of this compound, which Stella4D will play in real time. The movie gets a bit jumpy in the central regions as the computer tries to keep up.

Here is a closeup of the cross section, looking down a fivefold symmetry axis:



This finishes my series on ten of the twelve ten-compounds of regular pentagonal polychora, which followed from my series on the five-compounds. There are still plenty of symmetric compound polychora that Stella4D will generate. It'll be some time before I've exhausted her repertoire Exclamation Laughing
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PostPosted: Wed Mar 19, 2008 11:28 pm    Post subject: Another view Reply with quote

Here's a view of the compound of ten grand 600-cells 0.555 3D cross section a bit off center, so that it looks like a surrealistic planetoid with tall peaks and deep valleys that you're passing over in a spaceship.



This one is in ten colors rather than two. Smile
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