Bilbirothawroids (D4.3 to D4.9)

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 06, 2014 7:32 pm

Marek14 wrote:No, I meant in a journal. After all, we have quite a lot of unique results on the forum, don't we?

I dunno, it seems to me that some sort of closure should be attained before attempting to publish in a journal. If we have the complete 4D CRF list, then that would definitely be journal-worthy, on par with Johnson's list of 3D CRFs. Of course, that's still a long way away given the huge number of 4D CRFs, but still, closure in at least some subcategory should be reached, like Klitzing's list of segmentochora (which is surmised to be complete, given the orbiform requirement). Just because we have a handful of crown jewels isn't good enough, I think, since after all, it would seem a bit excessive if we submitted a journal paper for every new CRF we find!
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Re: Johnsonian Polytopes

Postby student91 » Thu Feb 06, 2014 7:40 pm

quickfur wrote:Wait, you mean Klitzing is claiming that those vertices will give a CRF result just by themselves? Because that is false, the images my program outputs is based on a convex hull calculation from the vertices; I did not insert the edges by hand!

If that's the case, then this is not a true CRF after all. :(

I'm not sure what Klitzing exactly claimed, but as far as I got, he hasn't suggested what vertices should be inserted to make it CRF. (the o3f and f3o-vertices are just the vertices filling the trid.ike's with vertices). I still hope it's CRF, or at least CRF-able. it would be very dissapointing if it wasn't :cry:
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Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 06, 2014 7:51 pm

quickfur wrote:
Marek14 wrote:No, I meant in a journal. After all, we have quite a lot of unique results on the forum, don't we?

I dunno, it seems to me that some sort of closure should be attained before attempting to publish in a journal. If we have the complete 4D CRF list, then that would definitely be journal-worthy, on par with Johnson's list of 3D CRFs. Of course, that's still a long way away given the huge number of 4D CRFs, but still, closure in at least some subcategory should be reached, like Klitzing's list of segmentochora (which is surmised to be complete, given the orbiform requirement). Just because we have a handful of crown jewels isn't good enough, I think, since after all, it would seem a bit excessive if we submitted a journal paper for every new CRF we find!


But this sort of results might take years more. We might have enough for at least one article right now...
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Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 06, 2014 8:11 pm

Marek14 wrote:
quickfur wrote:
Marek14 wrote:No, I meant in a journal. After all, we have quite a lot of unique results on the forum, don't we?

I dunno, it seems to me that some sort of closure should be attained before attempting to publish in a journal. If we have the complete 4D CRF list, then that would definitely be journal-worthy, on par with Johnson's list of 3D CRFs. Of course, that's still a long way away given the huge number of 4D CRFs, but still, closure in at least some subcategory should be reached, like Klitzing's list of segmentochora (which is surmised to be complete, given the orbiform requirement). Just because we have a handful of crown jewels isn't good enough, I think, since after all, it would seem a bit excessive if we submitted a journal paper for every new CRF we find!


But this sort of results might take years more. We might have enough for at least one article right now...

What do you think is worth inclusion? Klitzing's cube||icosahedron is already published along with the segmentochora; then there's the ursachora, the J91 castellated x5o3x prism, and now this one (if it turns out to be valid). What else? Maybe the 600-cell lunae and the bi-penta-cyclodiminished polychora with 5,5-duoprism symmetry? The partial Stott expansion products like my tesseract/octagon convex hull, and Klitzing's other discoveries along those lines?

Hmm, maybe the enumeration of duoprism augmentations? That one is still incomplete, actually, because each pyramid augmentation for an m,n-duoprism also has a Stott-expanded equivalent in a 2m,n-duoprism with cupolaic pyramid augments, which introduces orientation into the augments (as well as having many more slots to augment in the other ring due to the doubling of the size of the m-polygon) and so adds many more possibilities than the pyramid augments. I've been meaning to get to those, but just never had the time to sit down and do it. :\

I guess you're right, we do have enough material here to publish. What would be the underlying thrust of the paper, though? Dumping in everything we found seems a bit too incoherent, not to mention incomplete. But there has been so many discoveries in so many diverse areas, that I can't really think of a good unifying motif for them. Besides being CRF, that is, which is a bit too broad since we haven't really systematically gone through the possibilties. In this sense, the duoprism augmentations are probably the best candidates, since both of us have computed all possibilities with pyramid augments. If we complete the enumeration of 2n-prism||n-gon augments to it, then it should totally be journal material.
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Re: Johnsonian Polytopes

Postby Marek14 » Thu Feb 06, 2014 8:36 pm

quickfur wrote:What do you think is worth inclusion? Klitzing's cube||icosahedron is already published along with the segmentochora; then there's the ursachora, the J91 castellated x5o3x prism, and now this one (if it turns out to be valid). What else? Maybe the 600-cell lunae and the bi-penta-cyclodiminished polychora with 5,5-duoprism symmetry? The partial Stott expansion products like my tesseract/octagon convex hull, and Klitzing's other discoveries along those lines?

Hmm, maybe the enumeration of duoprism augmentations? That one is still incomplete, actually, because each pyramid augmentation for an m,n-duoprism also has a Stott-expanded equivalent in a 2m,n-duoprism with cupolaic pyramid augments, which introduces orientation into the augments (as well as having many more slots to augment in the other ring due to the doubling of the size of the m-polygon) and so adds many more possibilities than the pyramid augments. I've been meaning to get to those, but just never had the time to sit down and do it. :\

I guess you're right, we do have enough material here to publish. What would be the underlying thrust of the paper, though? Dumping in everything we found seems a bit too incoherent, not to mention incomplete. But there has been so many discoveries in so many diverse areas, that I can't really think of a good unifying motif for them. Besides being CRF, that is, which is a bit too broad since we haven't really systematically gone through the possibilties. In this sense, the duoprism augmentations are probably the best candidates, since both of us have computed all possibilities with pyramid augments. If we complete the enumeration of 2n-prism||n-gon augments to it, then it should totally be journal material.


I can't really help with that. I am unfortunately pathologically incapable of systematic work on publication level :) I suspect something in my toratope threads is also new, but I have no real idea where to look :D
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Re: Johnsonian Polytopes

Postby quickfur » Thu Feb 06, 2014 8:58 pm

Marek14 wrote:
quickfur wrote:What do you think is worth inclusion?
[...]
I guess you're right, we do have enough material here to publish. What would be the underlying thrust of the paper, though?[...]


I can't really help with that. I am unfortunately pathologically incapable of systematic work on publication level :) I suspect something in my toratope threads is also new, but I have no real idea where to look :D

I'd do it, except that I have a full-time job and have other activities that take up my free time. (And polishing up a publication-worthy paper takes a lot of time.) Klitzing would be the best candidate? ;) *nudges Klitzing* :nod: We could just be the co-authors, and he will do the real work. :P
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 1:42 am

Good news!!! I have found a genuine CRF with J92 cells!!

It is based on what I had before; it turns out that student91's idea to add two more points per phi-scaled edge was correct. I calculated their coordinates by taking advantage of the fact that we know 3 points of the pentagon already, which gives us two edge vectors, from which we can easily calculate the chords to find the other two points. This produced pentagonal pyramids where the phi-scaled edges had been, and makes the near side of the polytope CRF. But the rest of the polytope was still not CRF: it turns out that these new points lie behind the original tridiminished icosahedra cells, so all of that is no more, leaving a bunch of new phi-scaled edges on the far side of the polytope where a bunch of irregular cells appear. Furthermore, more square pyramids appeared, adding new square faces that didn't have any obvious solution how to interface them with the pentagonal parts of the surface.

However, I then noticed that the last coordinate of the new points is exactly the same as the last coordinate of the top/bottom triangular faces of the J92's -- meaning that the big irregular cell that had appeared previously with the new phi-scaled edges lies exactly on the bisecting hyperplane of the circumscribing 3-sphere. So the solution was very simple: make a mirror image of this polychoron and paste the two together at this hyperplane. I tried this, and voila! All phi-scaled edges disappeared, leaving a CRF polychoron with 4 J92 cells, 3 bidiminished icosahedra, and three absolutely peculiar clusters of tetrahedra and square pyramids surrounding a triangular prism, that interface the square faces introduced earlier with the rest pentagonal parts of the polytope, and a bunch of pentagonal pyramids, tetrahedra, etc..

I haven't fully counted all the cells yet, but it shouldn't be too hard, basically you have 4 J92's with dichoral angle 60° at the hexagons and 120° at the opposite triangles, and 12 other vertices in the same hyperplane as the top/bottom triangles of the J92's, that complete the pentagons spanning 1 hexagon vertex and two o3F vertices, one from each J92 sharing a hexagonal face. These define all the vertices of this CRF. You should be able to make the lace city from this quite easily, and then it shouldn't be too hard to read off the cells.

Here is a look at this crazy new CRF, seen from the top view (looking down on the top triangle face of a pair of J92's):

Image

I have to run now, so I'll post more renders later.

P.S. The pentagonal pyramids you can see here are the ones that appeared after the 12 vertices suggested by student91 were added.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 3:45 am

Wow, it's quite tricky to get good projection images of this strange tetra-J92 polytope! Here's a sideview from the 4D viewpoint, but I tweaked the 3D viewpoint to look from the top because otherwise the projection is just a narrow spindle:

Image

This is a parallel projection, looking at the side cells flanking the J92's. The green and yellow cells are J92's seen at a 90° angle. The short edge where they join is their top triangular face, and the long edge where they join is the hexagonal face. The middle of this projection, where the closest cell to the 4D viewpoint is, is where you see a triangular prism flanked by square pyramids and tetrahedra in a configuration that links the squarish part of the J92's with the pentagonal parts of the bidiminished icosahedra.

The top and bottom points are 4 of the vertices that student91 suggested to add to "complete the pentagon" where the phi-scaled edges previously were. The next image shows the same projection (same 4D viewpoint) rotated in 3D so that you can see where the edges from the top vertex links to the J92's:

Image

You can see some distorted pentagons linking those vertices to the J92. Notice that the bottom tip of the pentagon is pointing inward? That's where the phi-scaled edge was; the two tetrahedra that Klitzing pointed out actually had a concave dichoral angle, lying on either side of this tip! After adding student91's vertices (two of which are at the top in this image), that concave gap is now filled by pentagonal pyramids.

Now to show you just how narrow this CRF is, here's a view of the same projection (again, same 4D viewpoint) looking in 3D from the top of the spindle:

Image

Here, you can see the cycle of 4 J92's with dichoral angles 60° and 120°.

Alright, time for more interesting 4D viewpoints:

Image

Here's the original viewpoint of my first attempt with the phi-scaled edges. Notice here that now, the phi-scaled edge has been replaced by a pair of pentagonal pyramids. But those vertices overhang the original square pyramid lacing the two J92's; so here you can see that a new square pyramid has been added. The square bottom of this new square pyramid and the square face of the adjacent J92 are interfaced by a triangular prism (not shown here, it lies on the far side of the polytope from this 4D viewpoint). The upper right yellow pentagonal face of the yellow J92 is part of an outline that shows one of the bidiminished icosahedra (it lies on the limb of this projection). Notice how the bidiminished icosahedra and pentagonal pyramids fit just nicely in with the square pyramids in a totally unexpected fashion.

I thought it'd be interesting to look at this CRF from an oblique angle, to see more clearly how all those cells lacing the J92's around the hexagonal face are fitted together, so here's one:

Image

Here, I show one of the square pyramids that link a J92's square face to its neighbouring J92's triangular face. Notice that this pyramid is next to another square pyramid, which links to the triangular prism outlined in red. If you look carefully, you can see that this pair of pyramids touch one of a pair of pentagonal pyramids between the green and yellow J92's. A most fascinating structure!
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 3:53 am

And here are the cell counts: 4 J92's, 12 tetrahedra, 30 square pyramids, 6 triangular prisms, 24 pentagonal pyramids, 6 metabidiminished icosahedra (J62); total: 82 cells, 262 polygons (206 triangles, 30 squares, 24 pentagons, 2 hexagons), 246 edges, 66 vertices.

And here are the vertices, adjusted from what I originally derived so that they are centered on the origin:
Code: Select all
#
# Tetra-J92: a CRF crown jewel with 4 triangular hebesphenorotundae (J92's).
#

#
# Top two J92's:
#

# x3o:
<0, 2/√3, phi^2/√3, 0>
<±1, -1/√3, phi^2/√3, 0>

# f3x:
<±1, phi^3/√3, phi/√3, ±1>
<±phi^2, -1/(phi*√3), phi/√3, ±1>
<±phi, -(phi+2)/√3, phi/√3, ±1>

# o3F:
<±phi^2, phi^2/√3, 1/√3, ±phi>
<0, -2*phi^2/√3, 1/√3, ±phi>

# Hexagonal base:
<±1, ±√3, 0, ±phi^2>
<±2, 0, 0, ±phi^2>

#
# Bottom two J92's, in gyro orientation.
#

# x3o:
<0, -2/√3, -phi^2/√3, 0>
<±1, 1/√3, -phi^2/√3, 0>

# f3x:
<±1, -phi^3/√3, -phi/√3, ±1>
<±phi^2, 1/(phi*√3), -phi/√3, ±1>
<±phi, (phi+2)/√3, -phi/√3, ±1>

# o3F:
<±phi^2, -phi^2/√3, -1/√3, ±phi>
<0, 2*phi^2/√3, -1/√3, ±phi>


#
# 12 equatorial vertices to "complete the pentagons" and make the result CRF.
#

# Note: the following cannot be further combined with ± because they involve an
# even number of sign changes in the second the third coordinate.
<±phi^2,  (phi+3)/√3,  1/(phi*√3), 0>
<±phi^2, -(phi+3)/√3, -1/(phi*√3), 0>
<±1,  (3 + 2*phi)/√3, -1/(phi*√3), 0>
<±1, -(3 + 2*phi)/√3,  1/(phi*√3), 0>
<±(phi+2),  phi/√3,  1/(phi*√3), 0>
<±(phi+2), -phi/√3, -1/(phi*√3), 0>


And here is the lace city:
Code: Select all
               x3o
          f3x       f3x
     o3F                 o3F
               x3F
x3x                           x3x
               F3x
     F3o                 F3o
          x3f       x3f
               o3x


And finally, the all-important question: what shall we name this beauty? I'm at a loss for names, frankly. The structure is just so unique. It seems to incorporate much of the 600-cell's structure, but also adds its own twists to it. Keiji? ;)

P.S. And here's the Stella4D file for Marek. :]
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Re: Johnsonian Polytopes

Postby student91 » Fri Feb 07, 2014 6:38 am

*mind boggled, dont know what to say* :o_o:
It's the most awesome thing we've found so far, and probably the most awesome thing we'll ever find :D
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Re: Johnsonian Polytopes

Postby Keiji » Fri Feb 07, 2014 7:30 am

quickfur wrote:I'm at a loss for names, frankly. The structure is just so unique. It seems to incorporate much of the 600-cell's structure, but also adds its own twists to it. Keiji? ;)


Indeed, I'm having a hard time understanding the structure. It was tempting to treat it like a duoprism, with the ring of J92s and the ring of J62s orthogonal, but then I noticed there were only three J62s visible and you said there were six in total. Would I then be right in saying that the 6 J62s are split into two "rings" of three, one "ring" around each J92-J92 triangle?
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 7:41 am

Hmm, if the previous J91-based polychoron was an exotic prism, this looks like an exotic duoprism since the 4 J92's form a ring.

A very interesting thing about this polychoron is its bisecting cuts. The bisecting cut stemming from the J62's is especially interesting since it looks like a "gyrated dodecahedron" if there's such a thing. Image is attached.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 4:05 pm

Keiji wrote:
quickfur wrote:I'm at a loss for names, frankly. The structure is just so unique. It seems to incorporate much of the 600-cell's structure, but also adds its own twists to it. Keiji? ;)


Indeed, I'm having a hard time understanding the structure. It was tempting to treat it like a duoprism, with the ring of J92s and the ring of J62s orthogonal, but then I noticed there were only three J62s visible and you said there were six in total. Would I then be right in saying that the 6 J62s are split into two "rings" of three, one "ring" around each J92-J92 triangle?

The J62's are not in a ring; they are just flanking cells around the J92-J92 triangles to fill in the gaps there. Take a look at this:

Image

This is a 4D viewpoint looking at one of the J92-J92 triangles. The J62's are outlined in red. For clarity, I elided the edges that aren't on the J92's and J62's. So you can see that between these two J92's, there's an alternating pattern of square pyramids and J62's flanking the triangle. For a better understanding of how these J62's relate to the J62's around the other triangle, here's projection centered on one of the J92's:

Image

I found it difficult to make the J62's clear, so I decided to render them in blue rather than just outlining in red. You can see that the J62's around one triangle are linked to the J62's around the other triangle by a triangular prism and pairs of pentagonal pyramids, and they are in gyro arrangement.

EDIT: I realize the pentagonal pyramids are a bit hard to see, so here's one highlighted:

Image

EDIT#2: And here is the other pentagonal pyramid next to this one:

Image
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 5:00 pm

I was thinking about the current results a bit.

Basically, we're trying to build new CRF polychora via "scaffolding" -- you start with rough structure of polychoron and then try to fill it in CRF manner.

In that case, I think there is one very potent group of scaffoldings we haven't thought about yet. Let's look at grand antiprism. It's based on two rings of pentagonal antiprisms.
The question is as follows: what other CRF things could be built based on the general "two rings" method?

The J92 polychoron is based on a single ring of J92's, but I think there are some elements of second ring, it's just thin enough to not actually have full rows of cells -- if I see it correctly, the second ring (which can be put in various positions is a circle of form J62-square pyramid-tetrahedron-tetrahedron-square pyramid-J62-square pyramid-triangular prism-triangular prism-square pyramid where tetrahedra are joined to their neighbours through edges.

Instead of current lace towers that have planar layers, we'd have lace towers with toroidal layers. Grand antiprism has two, but CRF polychora could have more.

We could use, say, square antiprisms or even snub square antiprisms (J85). And any sort of cupolas and bicupolas, maybe even some augmented prisms, diminished or parabidiminished rhombicosidodecahedron -- any uniform or Johnson solid with two parallel faces. If necessary, you are not even required to keep identical dichoral angles between members of a ring, or you can mix various shapes in one ring.

Start from a ring like that and add more CRF layers until you close it completely or until you can fill the remaining space with another ring.

The benefit of this construction? There are LOTS of possible ring configurations, meaning we'd have lots of chances for CRFs, and some of them might pay off. Actually, I'm now starting to wonder if there might be some infinite families hiding in there.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 6:14 pm

Marek14 wrote:I was thinking about the current results a bit.

Basically, we're trying to build new CRF polychora via "scaffolding" -- you start with rough structure of polychoron and then try to fill it in CRF manner.

Well, I don't know of any other way to find crown jewels, since the usual methods of augmenting/diminishing uniform polychora or cutting and pasting CRF polychora no longer applies here. In fact, in the process of discovering the J91-based castellated prism and this J92-based twin-wedge (in a sense, it's just joining two copies of the initial non-CRF wedges together), I've had many ideas of new features to add to my polytope programs to help with this kind of piecemeal assembly of CRFs. Maybe a full-fledged "4D lego" program might be in the works. :P

In that case, I think there is one very potent group of scaffoldings we haven't thought about yet. Let's look at grand antiprism. It's based on two rings of pentagonal antiprisms.
The question is as follows: what other CRF things could be built based on the general "two rings" method?

Duoprisms and their augmentations? :P

Keep in mind, that the two-rings structure is but the most basic Hopf fibration structure of the 3-sphere. In Jonathan Bowers' term, it's just the "Hopf function" of the digon, producing in the continuous case the duocylinder, and in the discrete case the duoprisms. Of course, based on that we can build all sorts of interesting things, and generally small polychora like the J92 one will exhibit some features of this two-ring structure, because that's the most likely structure to happen in a relatively-small shape.

But it's far from the only possible Hopf fibration based structure, though. Take a look at the swirldiminished rectified 600-cell that I rendered recently (Bowers' "spidrox"):

Image

It has a 12-ring structure of alternating prisms and antiprisms, in a 1+5+5+1 arrangement, corresponding with the "Hopf function" of the dodecahedron. It also contains a 20-ring series of twisting square pyramids, 5 of which are shown here:

Image

This one has a 5+(5+5)+5 structure, and corresponds with the "Hopf function" of the icosahedron.

Going back a little farther, the bi-24-diminished 600-cell (bidex) was shown to have a 1+3+3+1 ring structure, corresponding with the "Hopf function" of an octahedron (seen as a triangular antiprism).

Now remember that these symmetries are all already present in the uniform polychora, it's just that they also happen to have a much higher degree of symmetry, so usually we don't pay attention to (or notice) the inherent spiralling symmetry hidden in their higher symmetry group. So it seems reasonable to assume that spidrox and bidex are but two of potentially numerous examples of CRF polychora that exhibit that kind of structure. (And indeed, if you relax the convexity requirement, Bowers has entire families of swirlprisms to show you.) And since the Hopf fibration is continuous, and these are only the most symmetric discrete partitionings thereof, it seems reasonable to assume that other less symmetric partitionings are possible, which may be expressed in some as-yet unknown CRFs. So if we're going to be exploring rings, we shouldn't limit ourselves to just the two-ring structure, 'cos there are many more out there! And probably, many of them will lead to CRFs, perhaps many more crown jewels!

The J92 polychoron is based on a single ring of J92's, but I think there are some elements of second ring, it's just thin enough to not actually have full rows of cells -- if I see it correctly, the second ring (which can be put in various positions is a circle of form J62-square pyramid-tetrahedron-tetrahedron-square pyramid-J62-square pyramid-triangular prism-triangular prism-square pyramid where tetrahedra are joined to their neighbours through edges.

Instead of current lace towers that have planar layers, we'd have lace towers with toroidal layers. Grand antiprism has two, but CRF polychora could have more.

We could use, say, square antiprisms or even snub square antiprisms (J85). And any sort of cupolas and bicupolas, maybe even some augmented prisms, diminished or parabidiminished rhombicosidodecahedron -- any uniform or Johnson solid with two parallel faces. If necessary, you are not even required to keep identical dichoral angles between members of a ring, or you can mix various shapes in one ring.

There is no need to restrict ring members to have parallel faces. For example, spidrox sports 20 rings of square pyramids that run along great circles, filling in the gaps between the rings of alternating prisms/antiprisms. These square pyramids are connected with a 3-fold twist, where the square faces rotate among 3 orientations as you go down the ring. They are almost protein-like in structure; you have the internal 3-fold twist of the square pyramids, and then the secondary 5-fold swirling of the rings around each ring of prisms/antiprisms.

Now, I haven't gotten around to it yet, but I'm planning to some day construct a x5o3x3o swirldiminishing with the same structure as spidrox, which will have a 20-ring structure with triangular prisms inserted between the twisting square pyramids (alongside a ring of alternating decagonal prisms and parabidiminished x5o3x's). So the twisting rings of square pyramids is probably not just a coincidence with spidrox; it's probably a general 4D phenomenon! One can imagine that if we construct a ring structure that doesn't need to correspond with any uniform polychoron, the possibilities will be far greater!

Start from a ring like that and add more CRF layers until you close it completely or until you can fill the remaining space with another ring.

The benefit of this construction? There are LOTS of possible ring configurations, meaning we'd have lots of chances for CRFs, and some of them might pay off. Actually, I'm now starting to wonder if there might be some infinite families hiding in there.

Never forget the Hopf fibration of the 3-sphere (that is, its discrete equivalents thereof), which features prominently in 4D convex polytopes. :D The two-ring structure is but one of almost limitless possibilities. It just takes imagination (and lots of persistence!! -- constructing that J92 crown jewel was a great exercise in persistence; I wouldn't have found it if student91 hadn't pointed out the other way of "completing the pentagon" that I overlooked).

But speaking of two-ring structures... one thing I've always wanted to find was a CRF that sports square antiprisms instead of pentagonal antiprisms in the two rings. I tried constructing a 4-membered ring with square antiprisms before but it turned out to be non-CRF. But maybe there is some CRF solution out there with a larger square antiprism ring?
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 6:20 pm

Hmm... Still looks like there might be a good number of CRFs waiting for discovery.

I guess that what we need is a "polychoron builder" application where you could add polychora to faces of existing polychora and "snap" two faces together, foldint the result into 4D, all interactive and graphical :)

At this point, I was thinking mainly about rings of snub square antiprisms...
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 7:18 pm

Marek14 wrote:Hmm... Still looks like there might be a good number of CRFs waiting for discovery.

Indeed! Before the discovery of the castellated prism and the J92 crown jewel (anybody has a good name for it yet? it's kinda getting tedious to refer to it as "that thing with 4 J92 cells" :P), I was vacillating between thinking there should be a handful of 4D crown jewels, and thinking that maybe there are none because of the increased linking requirements in 4D (in 3D you only had to worry about joining up the edges of regular polygons; in 4D, you have faces in all directions that all must match up to close the shape). But after discovering them, and esp. this last one with a completely unexpected structure, I'm also tempted to think there should be many more out there.

I guess that what we need is a "polychoron builder" application where you could add polychora to faces of existing polychora and "snap" two faces together, foldint the result into 4D, all interactive and graphical :)

Ah, but the problem with that is that I suck at designing GUIs. Well, actually, I suck at using GUIs. I find them a great hindrance to what I need to do, so my own polytope programs are all CLI-based. My latest partial rewrite of the main polytope viewer, which is still not completely working, at least has enough functionality now to understand Wendy-style permutation operators and Unicode characters like √, so you can literally input
Code: Select all
epacs<0, 1, (1+√5)/2>
and it will produce all the vertices of an icosahedron. You can even construct the coordinates piecemeal with the concatenation operator '~':
Code: Select all
epacs<0, 1, phi> ~ <±1>
produces the icosahedron prism: the epacs applies to the first vector segment, generating the coordinates of an icosahedron, then the concatenation appends <+1> and <-1> to each result. This allows for very convenient algebraic representations of coordinates in my polytope source files, so I can compute convex hulls and render them, and at the same time retain their algebraic formulation for formal mathematical verification of their CRF-ness, etc..

The current (old) version of my polytope viewer has a crude face-lattice querying system, so you can ask for cells that contain faces with 5 vertices, for example, and it will pick up pentagonal pyramids, dodecahedra, etc., but exclude cubes and tetrahedra. It can also filter by visibility (by checking whether the hyperplane normal of a facet faces the current viewpoint). In fact, this query-language is what I use to do almost all my renders when I want to highlight certain cells, or color a group of cells a certain way. Having an explicit scripting language for this means that I can save my renders as script files, so that I can re-create them at any time without needing to click through endless layers of menus, and I can edit them to use precise values (e.g., 1+sqrt(5) instead of 3.23608, which will throw povray off because of roundoff errors and will produce many rendering artifacts.

However, the current query language is rather handicapped, and often I find myself having to manually copy-n-paste intermediate results so that I can later filter them to get what I want (this task is no joke when you're dealing with things like the omnitruncated 120-cell which has 14400 vertices, 28800 edges, and who knows how many more other surtopes). I've been thinking about implementing an improved query language that will run more efficiently (the current implementation is a very inefficient O(n^2) algorithm).

Along with these improvements, I'm thinking of supporting partial polytopes, that is, polytopes that aren't closed up, so they are like cell complexes not necessarily enclosing space. These can be used as "CRFs-in-progress" where you can add/delete cells, snap cells together, etc.. Since I suck at designing/using GUIs, this will probably be script-based, and probably the input would be like a connectivity graph specification, so you can specify graph nodes to be specific polyhedra, then specify how they are to be connected to each other, like "connect cell 0 to cell 1 via ridge 16 with dichoral angle 60°". Probably, some dichoral angles can be replaced with "snap", meaning the program should automatically determine the angle based on adjacent cells. Things like mismatching ridges will generate errors, and there can be commands to check CRF-ness, self-intersection, etc.. Having it in script form also means I can go back and edit an earlier step without needing to redo every subsequent step after making the change, as you'd be forced to with a GUI-based undo/redo function.

Armed with something like this, I'd be able to explore CRFs much more easily, and perhaps even do partially-automated searches (initial part of script specifies a CRF fragment as seed, then give a command for the program to explore all possible CRF extensions up to depth 5 or something, then look at the output for something that looks promising).

At this point, I was thinking mainly about rings of snub square antiprisms...

Hmm... I wonder if you could link those up in a ring! Am I thinking what you're thinking? ;)

Although, I'm a bit worried that perhaps things might not work out, because we got lucky that both J91 and J92 had strong connections to the pentagonal polyhedra, so they fit more easily into the 600-cell framework which is known to work. The snub square antiprism may be much harder to close up in 4D... although if we do find a CRF with it, that would indicate that maybe all Johnson solids can occur as cells in CRFs other than just their own prisms! In any case, these may be considered as the "hard crown jewels", as opposed to "soft crown jewels" that are partially derived from existing uniforms/regulars like the 600-cell. They would have structures that are completely unrelated to any known uniforms.
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 7:31 pm

The question is this:

Imagine a ring of snub square antiprisms. Is there an upper limit of size of such ring? I.e. Can it be proven that there can be no CRF based on it above certain size of ring?
If not, then we have potentially unbounded playground, just with this one type of ring.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 8:11 pm

Marek14 wrote:The question is this:

Imagine a ring of snub square antiprisms. Is there an upper limit of size of such ring? I.e. Can it be proven that there can be no CRF based on it above certain size of ring?
If not, then we have potentially unbounded playground, just with this one type of ring.

That's easy, what's the dihedral angle A of the triangle sharing an edge with a square to the square? In the maximal case, we have a flat 3D stacking of snub square antiprisms, so the angle between the triangles sharing the same square edge will be 360°-2A. This is the upper limit of the dihedral angle of anything we attach to this edge, since we must have positive angle defect in order for it to form a closed 4D shape. So anything with dihedral angle ≥360°-2A can be ruled out from fitting here. Since from what I can tell the snub square antiprism has quite a low height, 360°-2A should be relatively small, meaning that only a limited number of CRF cell types can fit into this edge. So in the maximal case, find the combination of fitting cells whose sum of dihedral angles is as close to 360°-2A as possible, while still being strictly less than 360°-2A. This will represent the upper bound on the ring size. Of course, the actual maximum ring size may be a lot smaller, because not every combination of size cells is going to have the ring close up.

In fact, this latter requirement might be more fruitful to investigate first: what combination of cells fitting into that triangle-square edge between two snub square antiprisms (J85's) will lead to a dichoral angle between J85's that evenly divides the circle? This will tell us which cell combinations will allow ring closure assuming the ring spans a regular polygon. Of course, there may be ways of closing the ring by inserting different cells, but regular rings will be easier to construct, and might be worth investigating first. For irregular rings, you still have the requirement that some combination of J85-J85 dichoral angles Ai must have (180° - Ai) sum up to 360°; if not, then ring closure is impossible.

So it looks like the first task is to enumerate all possible side cells fitting into that edge, and the corresponding J85-J85 dichoral angles. Then compute which combinations of angles will have (180°-A) sum up to 360°. That will tell us which cells can be fitted into that ring if we want the ring to close. Assuming, of course, all ring members are J85. But let's not overstretch ourselves with other possibilities at the moment. In fact, if it turns out there are too many combinations, the first thing to try is to find which dichoral angles evenly divides the circle, so that at least we can find all the regular rings first before we go on to more exotic combinations.
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Re: Johnsonian Polytopes

Postby Keiji » Fri Feb 07, 2014 8:18 pm

quickfur wrote:Ah, but the problem with that is that I suck at designing GUIs. Well, actually, I suck at using GUIs. I find them a great hindrance to what I need to do, so my own polytope programs are all CLI-based.


Thank you. I'm glad I'm not the only one.

And no, I haven't thought up a name for the hebesphenochoron yet.
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 8:24 pm

Well, the first result (easy) is that any ring of snub square antiprisms must have even number of cells since otherwise the squares on both ends wouldn't match.

Now, the dihedral angle triangle/square is 145.441. Double that is 290.882, meaning that maximum fit is 69.118.

But the smallest dihedral angle between two triangles (according to dihedral list I compiled some time ago) is 70.5288 in tetrahedron, and so we can't actually fit anything in there and J85 won't work as ring element, at least not by itself. Good to know.

Now, if we take square antiprism, the 3-4 dihedral angle is 103.836, so 360-2A = 152.328, which is much more generous. You could potentially fit there quite a lot of shapes and even some combinations of shapes. Largest thing that could fit would be augmented sphenocorona, and next largest would be gyroelongated square cupola/bicupola.

It would be also interesting to see what happens if we'd alternate square antiprisms and square snub antiprisms. In that case the remaining angle would be 110.723 which would allow you to fit an octahedron there, with just over 1 degree of free space. Since octahedron is highly symmetrical (and since the same angle appears in various variants, like square pyramids), this might be a shape worth considering.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 9:15 pm

Marek14 wrote:Well, the first result (easy) is that any ring of snub square antiprisms must have even number of cells since otherwise the squares on both ends wouldn't match.

Now, the dihedral angle triangle/square is 145.441. Double that is 290.882, meaning that maximum fit is 69.118.

But the smallest dihedral angle between two triangles (according to dihedral list I compiled some time ago) is 70.5288 in tetrahedron, and so we can't actually fit anything in there and J85 won't work as ring element, at least not by itself. Good to know.

Well, you could always try fitting them together by their triangular faces... you'd have to do a twisting orientation type thing, like the square pyramids in spidrox, in order to get the ring to close, though. And angles will be much more difficult to calculate.

Now, if we take square antiprism, the 3-4 dihedral angle is 103.836, so 360-2A = 152.328, which is much more generous. You could potentially fit there quite a lot of shapes and even some combinations of shapes. Largest thing that could fit would be augmented sphenocorona, and next largest would be gyroelongated square cupola/bicupola.

I wonder if we can get something out of a ring of alternating square antiprisms and square prisms. If you look at spidrox carefully, you'll see how the triangular side faces of the antiprisms allow connecting to the prisms in an adjacent ring via a square pyramid. Without this, the thing wouldn't fit together. So maybe there's a way to do an analogous thing with alternating square prisms/antiprisms so that you have rings of them that swirl around each other.

It would be also interesting to see what happens if we'd alternate square antiprisms and square snub antiprisms. In that case the remaining angle would be 110.723 which would allow you to fit an octahedron there, with just over 1 degree of free space. Since octahedron is highly symmetrical (and since the same angle appears in various variants, like square pyramids), this might be a shape worth considering.

And don't forget, the octahedron can be cut into a square pyramid while still having the same dihedral angle at the ring joints. This flexibility allows it to interface with a greater variety of cells as you move outwards from the ring.
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 9:22 pm

quickfur wrote:I wonder if we can get something out of a ring of alternating square antiprisms and square prisms. If you look at spidrox carefully, you'll see how the triangular side faces of the antiprisms allow connecting to the prisms in an adjacent ring via a square pyramid. Without this, the thing wouldn't fit together. So maybe there's a way to do an analogous thing with alternating square prisms/antiprisms so that you have rings of them that swirl around each other.

It would be also interesting to see what happens if we'd alternate square antiprisms and square snub antiprisms. In that case the remaining angle would be 110.723 which would allow you to fit an octahedron there, with just over 1 degree of free space. Since octahedron is highly symmetrical (and since the same angle appears in various variants, like square pyramids), this might be a shape worth considering.

And don't forget, the octahedron can be cut into a square pyramid while still having the same dihedral angle at the ring joints. This flexibility allows it to interface with a greater variety of cells as you move outwards from the ring.[/quote]

Well, square antiprism/square prism ring clearly works (it's part of square antiduoprism), so there might be some possibilities in extending it...
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 9:24 pm

Another thing to keep in mind is that in a 2-ring structure, the size of the first ring profoundly influences what you can put into the second ring, and vice versa. Since the two rings lie in orthogonal planes, their cells also lie in hyperplanes 90° to each other. This means that an n-membered regular ring will induce an n-fold polygonal symmetry in the cells of the orthogonal m-membered ring. Of course, this symmetry can be altered by inserting various flanking cells, but it does constrain the possibilities somewhat, because if you end up with n=7, for example, you will have rather limited choices for the members of the orthogonal ring. This in turn influences what can be put into the first ring, because limited choices for the orthogonal ring means limited available ring sizes m, which means the members of the first ring will have a symmetry constrained by the value of m. It just so happens that for the duoprisms, this limitation turns out to be completely flexible, because you can bend adjacent members of a ring arbitrarily, so there's effectively no limitation, but when your ring contains multiple cell types with only a finite number of possible dichoral angles, this constraint will kick in and limit what kind of ring pairs you can make.
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 9:25 pm

Marek14 wrote:[...] Well, square antiprism/square prism ring clearly works (it's part of square antiduoprism), so there might be some possibilities in extending it...

Yeah, that's what I was thinking too.
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Re: Johnsonian Polytopes

Postby student91 » Fri Feb 07, 2014 10:12 pm

quickfur wrote:[...]
Never forget the Hopf fibration of the 3-sphere (that is, its discrete equivalents thereof), which features prominently in 4D convex polytopes. :D The two-ring structure is but one of almost limitless possibilities. It just takes imagination (and lots of persistence!! -- constructing that J92 crown jewel was a great exercise in persistence; I wouldn't have found it if student91 hadn't pointed out the other way of "completing the pentagon" that I overlooked).
[...]

Glad I could help.

quickfur wrote:Although, I'm a bit worried that perhaps things might not work out, because we got lucky that both J91 and J92 had strong connections to the pentagonal polyhedra, so they fit more easily into the 600-cell framework which is known to work. The snub square antiprism may be much harder to close up in 4D... although if we do find a CRF with it, that would indicate that maybe all Johnson solids can occur as cells in CRFs other than just their own prisms! In any case, these may be considered as the "hard crown jewels", as opposed to "soft crown jewels" that are partially derived from existing uniforms/regulars like the 600-cell. They would have structures that are completely unrelated to any known uniforms.

I don't think something without a basic framework will work very well. think of it this way: you start with a cell, with coordinates {(..,..,..,..),(..,..,..,..),...}. if those coordinates aren't related to any known thing, it will get unwieldy coordinates. you can add more vertices, (building a lego building), and the coordinates get even more complex. eventually, you're at the back of the polytope, and you're trying to close it up (if you'd get that far). you need edges of length 1 to close it up nicely. because the coordinates are that horrible, it's unlikely that they will make a nice distance of 1. If it is related to something known, it would ofc. work out beautifully. This is just a feeling though, you're free to surprise me with awesome things :D .

(anybody has a good name for it yet? it's kinda getting tedious to refer to it as "that thing with 4 J92 cells" )
If we don't know a describing name for it, we could just use a non-describing one, couldn't we? I mean, the corona don't have a very describing name either (how are these shapes related to our sun's corona?). I suggest, because those things are crown jewels, we give it a jewel name, e.g. diamond or The Arkenstone ;) . (I've been watching the hobbit lately).
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Re: Johnsonian Polytopes

Postby quickfur » Fri Feb 07, 2014 10:48 pm

student91 wrote:[...]
quickfur wrote:Although, I'm a bit worried that perhaps things might not work out, because we got lucky that both J91 and J92 had strong connections to the pentagonal polyhedra, so they fit more easily into the 600-cell framework which is known to work. The snub square antiprism may be much harder to close up in 4D... although if we do find a CRF with it, that would indicate that maybe all Johnson solids can occur as cells in CRFs other than just their own prisms! In any case, these may be considered as the "hard crown jewels", as opposed to "soft crown jewels" that are partially derived from existing uniforms/regulars like the 600-cell. They would have structures that are completely unrelated to any known uniforms.

I don't think something without a basic framework will work very well. think of it this way: you start with a cell, with coordinates {(..,..,..,..),(..,..,..,..),...}. if those coordinates aren't related to any known thing, it will get unwieldy coordinates. you can add more vertices, (building a lego building), and the coordinates get even more complex. eventually, you're at the back of the polytope, and you're trying to close it up (if you'd get that far). you need edges of length 1 to close it up nicely. because the coordinates are that horrible, it's unlikely that they will make a nice distance of 1. If it is related to something known, it would ofc. work out beautifully. This is just a feeling though, you're free to surprise me with awesome things :D .

I know what you're getting at, but sometimes even "nice" things have ugly coordinates. Take the snub cube, for example. It requires solving a bunch of degree-3 polynomials, and the resulting coordinates have nested cube roots. Yet the thing is uniform! So we cannot say that just because those crazy nested cube roots are coming out, the polyhedron isn't going to close up. Similarly, the snub disphenoid has some pretty crazy coordinates that I haven't even been able to derive myself, I just copied somebody else's work. :P

In fact, I've been thinking a little more about the 3D crown jewels. The two that we've managed to construct CRFs from so far are both related to the pentagonal polyhedra, or, more generically speaking, their coordinates involves solving at most quadratic equations (the golden ratio is the root of the quadratic x^2-x-1=0). Geometrically speaking, this means that the position of the vertices are constrained enough that to solve them, you only need at most two simultaneous constraints. The pentagons basically dictate that certain layers are scaled by phi, so the relative scale of things can be fixed first, and then you use that to solve the other parameters.

Now look at something like the snub square antiprism. There are 4 layers of vertices, the top and bottom are o4x and x4o. To fix the points in the middle two layers, you need to simultaneously solve for the heights of the x4o (resp. o4x's) and the height of the middle layers, with unit edge lengths constraining the height between the middle layers and between the middle layer to the outer layer. There is no direct constraint that lets you solve for one height first, because the circumradius of the middle layers is dependent on the height. Almost all of the parts are interdependent, so I'm expecting that you need to solve at least a cubic equation in order to get everything fixed. So in a sense, the snub square antiprism is somehow "inherently" more complex than the bilunabirotunda or the triangular hebesphenorotunda.

So I'm thinking we can classify polytopes by the minimum degree of the polynomials you must solve in order to construct them. Roughly speaking, this measures the number of degrees of freedom you have to simultaneously reconcile. Let's call it the "complexity" of the polytope. Most of the symmetric polytopes would have complexity 2, because the most you need to solve is a quadratic equation. However, there are some notable polytopes of complexity 3, such as the snub cube, the snub dodecahedron, the snub disphenoid, and probably several others of the 3D crown jewels. Interestingly enough, we have not yet discovered 4D analogues of any complexity 3 polyhedron: we don't know of any snub cube equivalent, for example, and attempts to construct a 4D snub disphenoid haven't produced anything yet, AFAIK. Neither have we produced any 4D CRF that contains a 3D complexity 3 polyhedron. Now, this is just my gut feeling, I have no proof of this, but could it be because these are inherently harder to construct?

If so, we could say that 4D CRFs of complexity 3 are the "real" crown jewels (or "hard" crown jewels, as opposed to "soft" crown jewels, or "greater" crown jewels vs. "lesser" crown jewels), because they require solving a non-trivial degree ≥3 polynomial and as a result will probably have a very unique construction. Probably, they will not have any analogues in other dimensions because of their inherent complexity.

(anybody has a good name for it yet? it's kinda getting tedious to refer to it as "that thing with 4 J92 cells" )
If we don't know a describing name for it, we could just use a non-describing one, couldn't we? I mean, the corona don't have a very describing name either (how are these shapes related to our sun's corona?). I suggest, because those things are crown jewels, we give it a jewel name, e.g. diamond or The Arkenstone ;) . (I've been watching the hobbit lately).

"Corona" actually doesn't refer to the sun; it comes from the Latin word for "crown". A fitting name for crown jewels. :P But jokes aside, it refers to a net of triangles that serve as a kind of "crown" placed on top of the base part of the polyhedron (at least, that's how Johnson defined it).

Maybe Wendy would be a good person to ask, if we want to go the Greek/Latin route for naming, since her Polygloss is basically an experiment in constructing a consistent higher-dimensional terminology using Greek/Latin roots.

Or we can use Google Translate. :P The problem is, I'm at a loss as to how to even begin describing this crown jewel, much less how to name it.
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Re: Johnsonian Polytopes

Postby Marek14 » Fri Feb 07, 2014 11:39 pm

So far you could just call it "hebesphenochoron" as we don't have any others yet :)

Your complexity argument is very interesting. In the end, we might replicate Mohs hardness scale in geometry :)
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Re: Johnsonian Polytopes

Postby quickfur » Sat Feb 08, 2014 1:13 am

Marek14 wrote:So far you could just call it "hebesphenochoron" as we don't have any others yet :)

Hmm. That's quite ambiguous, actually, because in 3D there are two "hebespheno"'s: hebesphenomegacorona (J89) and triangular hebesphenorotunda (J92). Now, assuming it is the only 4D CRF with four J92's, I suppose we could call it the "tetrahebesphenorotunda" or "tetrahebesphenorotundachoron", perhaps? That's quite a mouthful, though. :|

Alternatively, if we consult Johnson's definitions of these names, "spheno" means "wedge", consisting of a pair of "lunes" (a lune is defined to be a triangle-square-triangle patch), and "hebespheno" means a "blunter complex of three lunes". If we strip the 3D-centric definitions and go to the root meaning of these words, then we may allude to the "siamese wedge" construction of this crown jewel by naming it something like "siamese disphenochoron". :mrgreen: Unfortunately, that obscures its relation to J92.

Or maybe we should just assign a numerical ID to it and call it CR4 (for crown jewel #4, according to this list), and defer naming it until we find enough examples of other crown jewels, which will give us a better naming scheme -- I'm assuming that Johnson didn't just invent those names out of the blue, but studied those shapes carefully to find common features, like the triangle-square-triangle patch that recurs among many of the 3D crown jewels. Since we don't have enough 4D crown jewels to make any such generalizations yet, maybe it's better to defer naming them?

So then, CR1 would be cube||icosahedron, CR2 would be the ursachora (maybe we can denote members of a class like CR2-4 for the tetrahedral ursachoron, CR2-8 for the octahedral, and CR2-20 for the icosahedral). CR3 would be the castellated H101 prism, perhaps CR3e for the expanded variant, then CR4 for the 4-J92 polychoron, etc.. Alternatively, just arbitrarily assign a single index to every crown jewel, variant or not, in some arbitrary order, just like the Johnson polyhedra Jxx numbers. :)

Your complexity argument is very interesting. In the end, we might replicate Mohs hardness scale in geometry :)

Haha, wrong kind of hardness there, but sure. :P
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Re: Johnsonian Polytopes

Postby Keiji » Sat Feb 08, 2014 8:45 am

I would propose numbering them as such:

CJ4.1 snubdis antiprism
CJ4.2.1 tetrahedral ursachoron
CJ4.2.2 expanded tetrahedral ursachoron
CJ4.2.3 octahedral ursachoron
CJ4.2.4 expanded octahedral ursachoron
CJ4.2.5 icosahedral ursachoron
CJ4.2.6 expanded icosahedral ursachoron
CJ4.3.1 castellated rhodoperihedral prism (what is H101, by the way?)
CJ4.3.2 castellated rhodopantohedral prism (the expanded version of CJ4.3.1)
CJ4.4 hebesphenochoron (or whatever we are calling it)

I like the idea about the polynomial "complexity", although perhaps it should be called something else as it is an order rather than a complexity - perhaps coordinate vertex polynomial order, or CVP order for short? In any case it would be nice to have this property calculated for each polytope.
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