## Johnsonian Polytopes

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

P.S. and of course, the 24-cell itself can be elongated with a cuboctahedral prism, in which case the square pyramids from each half will merge with the cubes in the prism, to form elongated square bipyramids. So the elongated 24-cell will have 18 octahedra, 6 elongated square bipyramids, and 6 triangular prisms.

Edit: I made a quick render of the elongated 24-cell:

Because of the large number of edges, I left visibility clipping on, so this is only half of the polytope. I colored the 3 elongated square bipyramids so they're clearly visible; also visible are 4 of the triangular prisms as well as 8 of the octahedra. Can you spot them all?
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Two new CRF rotundae!

Today, I suddenly realized that not only the 24-cell can be cut in half; various uniform 16-cell derivatives can, too. The 16-cell itself, of course, can be cut into two octahedral pyramids -- that is well-known and not so interesting (it's already included in Klitzing's list of segmentochora).

What is interesting, though, is that the truncated 16-cell can be cut in half as well. Six of the octahedral cells get cut into square pyramids, so you end up with a sort of octahedral rotunda, with an octahedron on the top, 8 truncated tetrahedra, 6 square pyramids, and a truncated octahedron base. This polychoron is obviously CRF, but it is not among Klitzing's segmentochora because its vertices lie in 3 parallel hyperplanes.

The runcitruncated 16-cell can also be cut, albeit not in half, but slightly above the halfway point, such that the equatorial rhombicuboctahedra get truncated into square cupola. The resulting polychoron is CRF, and consists of 6 square cupola, 8 truncated tetrahedra, 6 cubes, 12 hexagonal prisms, 1 small rhombicuboctahedron, and 1 great rhombicuboctahedron. It is a kind of rhombicuboctahedral rotunda. It's also not in Klitzing's list, because it has vertices in at least 3 parallel hyperplanes.

The other remaining part of the runcitruncated 16-cell after the cut is an elongated version of this rotunda: it has 6 elongated square cupolae, 8 truncated tetrahedra, 18 cubes, 20 hexagonal prisms, 1 small rhombicuboctahedron, and 1 great rhombicuboctahedron. It may be thought of as the rhombicuboctahedral rotunda augmented with a great rhombicuboctahedron prism.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

OK, here are some renders of the rhombicuboctahedral rotunda:

Centered on great rhombicuboctahedron:

Side-view:

These renders were made with visibility clipping turned off.

The coordinates of this rotunda are the coordinates of the runcitruncated 16-cell where the last coordinate is positive.

And here's the .off file for Marek & other Stella4D users.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

And here are the renders of the octahedral rotunda:

Side view:

View centered on truncated octahedron base:

Coordinates: same as truncated 16-cell, where last coordinate is ≥0.

Stella4D .off file: here.

Enjoy!
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Does anybody still read this thread? Well, I hope somebody's reading, 'cos I've discovered a new non-trivial CRF! Here is a projection of it:

(This is a cross-eyed stereo pair; see here if you're new to cross-eyed stereo viewing.)

Its cells are 1 cuboctahedron, 2 hexagonal prisms, 2 triangular cupolae, 4 square pyramids, 4 triangular prisms.

Here are its coordinates:
Code: Select all
`<4/sqrt(10), 0, 3/sqrt(3), 1><4/sqrt(10), 0, 3/sqrt(3), -1><4/sqrt(10), 0, 0, 2><4/sqrt(10), 0, 0, -2><4/sqrt(10), 0, -3/sqrt(3), 1><4/sqrt(10), 0, -3/sqrt(3), -1><4/sqrt(10), -4/sqrt(6), 2/sqrt(3), 0><4/sqrt(10), -4/sqrt(6), -1/sqrt(3), 1><4/sqrt(10), -4/sqrt(6), -1/sqrt(3), -1><-1/sqrt(10), 3/sqrt(6), 3/sqrt(3), 1><-1/sqrt(10), 3/sqrt(6), 3/sqrt(3), -1><-1/sqrt(10), 3/sqrt(6), 0, 2><-1/sqrt(10), 3/sqrt(6), 0, -2><-1/sqrt(10), 3/sqrt(6), -3/sqrt(3), 1><-1/sqrt(10), 3/sqrt(6), -3/sqrt(3), -1><-1/sqrt(10), -1/sqrt(6), 2/sqrt(3), -2><-1/sqrt(10), -1/sqrt(6), -4/sqrt(3), 0><-1/sqrt(10), -5/sqrt(6), 1/sqrt(3), -1><-1/sqrt(10), -5/sqrt(6), -2/sqrt(3), 0><-6/sqrt(10), 2/sqrt(6), 2/sqrt(3), 0><-6/sqrt(10), 2/sqrt(6), -1/sqrt(3), 1><-6/sqrt(10), 2/sqrt(6), -1/sqrt(3), -1><-6/sqrt(10), -2/sqrt(6), 1/sqrt(3), -1><-6/sqrt(10), -2/sqrt(6), -2/sqrt(3), 0>`

Here's the Stella4D file (hi Marek! ).

I discovered this CRF while searching for maximally-diminished uniform polychora (defined to be a CRF polychoron obtained by some diminishing of a uniform polychoron, such that no more vertices can be removed from it without making it non-CRF). The cantellated 5-cell can have 3 vertices on one triangular face removed from it, producing a CRF with 1 hexagonal prism, 2 triangular cupola, 2 octahedra, 3 cuboctahedra, 7 triangular prisms, and 3 square pyramids. That polychoron I tentatively call the "triangle-diminished cantellated 5-cell"; the hexagonal prism is opposite a triangular prism. Another triangle of vertices can be removed from it, to produce the polychoron shown above. This triangle has 1 vertex touching the apex of a square pyramid and shares an edge with the triangular prism opposite the hexagonal prism. (It's important exactly which triangle to remove, since removing the wrong one produces a non-CRF polychoron.)

I think the result is maximally-diminished, but I'm not 100% sure. (Another maximal diminishing of the cantellated 5-cell is the bisected cantellated 5-cell, which is the same as the segmentochoron cuboctahedron||truncated_tetrahedron (4.48 in Klitzing's list).)

Anybody up for naming this little pretty? I'm thinking bi-triangle-diminished cantellated 5-cell, but I don't like the name; it sounds ugly. Any suggestions?
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Apparently nobody is reading this thread anymore. Oh well. I have nowhere else to post this, so might as well put it here. Maybe someday, somebody will take notice.

I have been continuing my search for CRFs by diminishing uniform polychora, and I discovered that the cantellated tesseract is a veritable gold mine of CRFs.

For one thing, you can cut it into 3 segmentotopes: rhombicuboctahedron || truncated cube, truncated cube prism, truncated cube || rhombicuboctahedron. But this simplicity belies the richness of the cantellated tesseract...

... because you can also cut off a square orthobicupolic ring (hence, SOBR) to obtain a diminished cantellated tesseract, where two of the rhombicuboctahedra get truncated into elongated square cupolae, two octahedra get truncated into square pyramids, and the gap is filled in by a new octagonal prism. You can actually cut off 8 SOBRs this way, and you end up with an 8,8-duoprism. IOW, the cantellated tesseract is an augmented 8,8-duoprism!

Furthermore, once you cut off an SOBR, you can glue it back the "wrong way" (i.e., the octagonal prism base is rotated 45°). This causes the square pyramids to be "misaligned" so they don't join with other square pyramids to form octahedra, but instead remain as distinct square pyramids. The square cupola cells then join with the elongated square cupola in the bulk of the polychoron to form elongated square gyrobicupola (i.e., pseudo-rhombicuboctahedra, J37). We can cut off 4 SOBRs in two orthogonal rings and glue them back the "wrong way" and we end up with a CRF polychoron with 8 pseudo-rhombicuboctahedral cells, 24 square pyramids, 32 triangular prisms, and 4 octahedra. For lack of a better name, I will name this the "pseudo cantellated tesseract" (by analogy with the pseudo-rhombicuboctahedron).

But the fun doesn't stop here. Instead of gluing back an SOBR the "wrong way" (i.e., rotating the base of the octagonal prism), we can cut off 4 SOBRs all lying along a single great circle -- this makes a ring of 8 octagonal prisms -- then glue them back on their neighbouring octagonal prisms. This again will produce a polychoron with 8 rhombicuboctahedral cells, but now 4 of these cells are "misaligned" with the other 4, so the result is CRF but not uniform. So in effect, we have "gyrated" 4 of the rhombicuboctahedra so that the result is another kind of analogue of the pseudo-rhombicuboctahedron. I really don't know how to name this polychoron now, because we have two different gyrations going on: gyrating the SOBRs while keeping them on their base, and shifting the SOBRs to their neighbouring positions on a great circle.

Tentatively, I will use the term "spirallated" for gluing the SOBRs back the "wrong" way (spiral, because it's like you're wrapping a spiral around the ring of octagonal prisms in the 8,8-duoprism), and "gyrated" for shifting the SOBRs to their neighbouring positions on the great circle. So the polychoron with 8 pseudo-rhombicuboctahedra will be the tetraspirallated cantellated tesseract, and the polychoron with 4 "misaligned" rhombicuboctahedra will be the gyrated cantellated tesseract.

I haven't verified it yet, but I think it's possible to make a spiro-gyrated cantellated tesseract, which is both gyrated and spirallated.

These are all augmentations of the 8,8-duoprism by SOBRs in various orientations. Other possible 8,8-duoprism augmentations with SOBRs also give CRFs.

But wait, there's more! The cantellated tesseract isn't just a glorified 8,8-duoprism; it has symmetries that the 8,8-duoprism doesn't have. So it's possible to cut off SOBRs in a way that has no correspondence with the 8,8-duoprism at all. The number of possible such diminishings (including the 8,8-duoprism derivatives) is the number of subsets of the 24-cell's vertices such that no two vertices are adjacent. (Anybody care to calculate this number? I suck at combinatorics.) Each of these diminishings can be augmented with SOBRs glued on the "wrong" way, thus yielding a very large number of CRFs indeed.

The cantellated tesseract is proving to be an unexpectedly (at least to me) rich source of CRFs. And we haven't even gotten to the 600-cell family yet!
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

P.S. here are the coordinates of the spirallated cantellated tesseract paratetragyrated cantellated tesseract (8 pseudo-rhombicuboctahedra, 24 square pyramids, 32 trigonal prisms, 4 octahedra):

<0, -(1+√2), -(1+√2), ±√2>
<0, 1+√2, 1+√2, ±√2>
<±√2, -(1+√2), -(1+√2), 0>
<±√2, 1+√2, 1+√2, 0>
<-(1+√2), 0, ±√2, -(1+√2)>
<1+√2, 0, ±√2, 1+√2>
<-(1+√2), ±√2, 0, -(1+√2)>
<1+√2, ±√2, 0, 1+√2>

<±1, ±1, ±(1+√2), ±(1+√2)>
<±1, ±(1+√2), ±1, ±(1+√2)>
<±1, -(1+√2), 1+√2, ±1>
<±1, 1+√2, -(1+√2), ±1>
<-(1+√2), ±1, ±1, 1+√2>
<1+√2, ±1, ±1, -(1+√2)>
<±(1+√2), ±1, ±(1+√2), ±1>
<±(1+√2), ±(1+√2), ±1, ±1>

EDIT: I had some trouble finding a 4D viewpoint that would give a nice projection, but I think this one should show a good idea of what this funny little polychoron looks like:

(This is a cross-eyed stereo pair; just ignore the second one if you don't know/don't want to do cross-eyed viewing.)

This projection is centered on one of the square faces between the two square cupola in the SOBR that got rotated 45°. Above and below are two pseudo-rhombicuboctahedra. Wrapped around the sides of the projection are the images of a ring of 4 other pseudo-rhombicuboctahedra (or rather, half of each of them -- because of the complexity of the polychoron, I turned on visibility clipping so that the image isn't just a big messy tangle of edges).

On the left side of the "equator", you can see a square pyramid joined to a triangular prism; on the right, there are two triangular prisms touching each other. Due to the dichoral angle of these cells in the SOBR, these cells aren't coplanar, so they aren't augmented triangular prisms or gyrobifastigiums. There are cases in this polychoron (not shown) where two square pyramids are joined to each other: in those cases, the dichoral angle is just right and the cells are coplanar, so they merge into regular octahedra.

EDIT 2: the Stella4D file is here.
Last edited by quickfur on Tue Aug 07, 2012 4:30 am, edited 3 times in total.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

quickfur wrote:[...] Tentatively, I will use the term "spirallated" for gluing the SOBRs back the "wrong" way (spiral, because it's like you're wrapping a spiral around the ring of octagonal prisms in the 8,8-duoprism), and "gyrated" for shifting the SOBRs to their neighbouring positions on the great circle. So the polychoron with 8 pseudo-rhombicuboctahedra will be the tetraspirallated cantellated tesseract, and the polychoron with 4 "misaligned" rhombicuboctahedra will be the gyrated cantellated tesseract.
[...]

Hmm, I just realized that "gyration" is equivalent to 4 "spirallations", so maybe we should just stick with "gyro" to avoid coining unnecessary terms. So the polychoron with 8 pseudo-rhombicuboctahedra is a paratetragyrated cantellated tesseract (the 4 SOBRs being gyrated lie in two orthogonal rings, and are opposite each other within the same ring). The polychoron with 8 rhombicuboctahedra, 4 of which are "misaligned", is just the orthotetragyrated cantellated tesseract (all 4 SOBRs in the same great circle are gyrated; this is equivalent to shifting the SOBRs in the other ring by 1 position).
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Just looked at the thread after some time Thanks for the off files
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:Just looked at the thread after some time Thanks for the off files

Hey Marek, good to hear from you again. Any new CRF-related ideas recently?

A few days ago I started thinking about CRFs that have elongated square bipyramids for cells, and came up with two that have 24 of them. It turns out that they are augmented runcinated tesseract and augmented cantellated 16-cell, respectively. The respective augments are cubical pyramids and octahedron||rhombicuboctahedron. Here's a projection of the augmented cantellated 16-cell:

Again, these are cross-eyed stereo pairs. The cells are: 8 octahedra, 24 elongated square bipyramids, 16 truncated tetrahedra, 32 hexagonal prisms, and 160 triangular prisms. Its coordinates are:

apacs<0, 0, √2, 2+2√2>
apacs<1, 1, 1+√2, 1+2√2>

Here's the .off file.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Hmm, no new ideas recently, just the realization that any cutting should produce shapes whose vertex figures can be traced on the vertex figure of the original shape (for example rectified 600-cell has an uniform pentagonal prism as vf, and shapes that can be traced there include vfs of pentagonal prism, icosidodecahedron, rhombicosidodecahedron and even a truncated dodecahedron, though this is unlikely to be a clean cut).

EDIT: Toying with your new shape. Seems that there are 2 different types of triangular prisms, 64 of them stand on octahedra and their second base leads into truncated tetrahedra, while 96 has both of their bases on elongated square bipyramids. Understanding the augmentations now, seems this shape has a complex augmentation/diminishings possible as after removing cube prism you can further remove cube || octagon from the main shape.
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:Hmm, no new ideas recently, just the realization that any cutting should produce shapes whose vertex figures can be traced on the vertex figure of the original shape (for example rectified 600-cell has an uniform pentagonal prism as vf, and shapes that can be traced there include vfs of pentagonal prism, icosidodecahedron, rhombicosidodecahedron and even a truncated dodecahedron, though this is unlikely to be a clean cut).

I've been toying with the idea of a brute force CRF algo, or perhaps an interactive CRF builder. The fundamental idea is this:

(1) We all know that the definition of CRF requires that the cells be either uniform or Johnson. Barring the prisms and antiprisms, this means a finite number of cell types are possible. Consequently, there is only a finite number of cell pairs possible (basically n2 where n is the number of uniform + Johnson polyhedra).

(2) Since there are a finite number of cell types, and cell faces are always regular polygons, we can make a table of all possible dihedral angles between two given polygons. We can index this table by polygon degrees -- for simplicity, we can restrict our attention to 3, 4, 5, 6, 8, 10 (since we're not considering prisms and antiprisms). So given two polygons P1 and P2, we can look up all possible dihedral angles between them.

(3) Now, let's say we have some CRF where cell A joins to cell B via a shared face F. F must be a regular polygon. Look around each edge of the polygon. Each edge will be shared between F, and at least a face from A and a face from B. Suppose, for simplicity, that there are exactly 3 faces around the edge; so it will be F, a face from A, and a face from B. Then each pair of faces must have a dihedral angle that's listed in the table we made in step (2). Furthermore, given a particular dichoral angle between A and B, all dihedral angles around the edges of F must be listed in the table.

This gives us a way of constructing CRFs manually: given any two cells A and B that share a face F, we can compute the possible dichoral angles between A and B that are permissible in a CRF construction, as follows: loop around the edges of F, and check the other faces attached to each edge. For simplicity, we assume the edge has 3 faces meeting at it (we can handle the more complicated case later -- but even then, there is a maximum of 5 faces per edge, because the tetrahedron has the smallest dihedral angle and only 5 tetrahedra can fit around an edge). Then the edge will be surrounded by faces F, G, and H, where G is a face from A, and H is a face from B. We then use the table in step (2) to lookup all possible dihedral angles between G and H.

Now here's the key to this idea: given a list of possible dihedral angles between faces for each edge in F, we now compute what the dichoral angle between A and B must be in order for the dihedral angle between G and H to be among the known dihedral angles. Fixing a specific dichoral angle Z will also fix all dihedral angles around F, and furthermore, all dihedral angles around F produced by dichoral angle Z must be listed in the table. This means that if even one edge ends up with a wrong dihedral angle (i.e. not listed in the table), we will reject that value of Z. So this quickly narrows down the possible dichoral angles between A and B -- presumably to a relatively small finite number.

Then for each viable dichoral angle Z, we can also compute which cells can be fitted around F by looking it up in the table. So that also reduces the number of possible of cells that can be fitted between A and B.

Once we select a particular cell to fit into the edge around F, then we can check the cell types permissible for the neighbouring edges to see if it will intersect the cell we chose. If so, then the combination is rejected as invalid, since CRFs must be convex. So this again narrows down the possibilities.

Also, for each new cell we add to our CRF candidate, it will share faces with the existing cells, so we can repeat the same algorithm to find other cell candidates to fill in the remaining gaps. In this case, the dichoral angle will already be fixed, so actually the number of possibilities will be much smaller -- if the new cell C shares faces F' and F'' with existing cells, then if the dichoral angle between C and the existing cells causes one or more edges of F' and F'' to have an invalid dihedral angle, then C must be rejected as invalid, since it will not lead to a CRF result.

Of course, things get a bit more complicated if we allow 4 or 5 faces around each edge; in that case, we have to allow the dihedral angle to be a sum of up to 4 dihedral angles from the table. This will greatly increase the number of possibilities -- although it will also narrow the number of cell types to be considered, since most CRF polyhedra have too large a dihedral angle to be able to fit so many around a single edge. As a last resort, if there's no easier way, we can precompute a table of all possible sums of permissible dihedral angles, so we can outright reject angles that can't possibly be a sum of permissible angles.

In any case, this algo allows us to construct CRFs piecemeal, either in a brute-force exhaustive search algo, or in an interactive application where the user can explore different possible options and the program will automatically compute the next possible cell candidates to put into the CRF. I think the interactive approach may be best for discovering interesting CRFs, as a brute-force approach may just get stuck for a long time enumerating 600-cell diminishings, for example, instead of finding the interesting CRFs like the crown jewels. (OTOH, if we limit the number of cell types to be considered, the program can be made to brute-force search all 600-cell diminishings and give us an accurate count, hopefully.)

EDIT: Toying with your new shape. Seems that there are 2 different types of triangular prisms, 64 of them stand on octahedra and their second base leads into truncated tetrahedra, while 96 has both of their bases on elongated square bipyramids.

Yeah, this is quite apparent when you look at the projections I have above. Each octahedron is surrounded by 8 triangular prisms that connect to truncated tetrahedra, and 12 triangular prisms that bridge two elongated square bipyramids.

Understanding the augmentations now, seems this shape has a complex augmentation/diminishings possible as after removing cube prism you can further remove cube || octagon from the main shape.

Removing a cube prism? Is that a typo? 'cos I don't see any cube prisms (==tesseract?).
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Sorry, I meant cube pyramids.

As for prisms and antiprisms, I suspect we could just add some of them later, and search only for shapes that contain at least one of them.
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Yeah I was thinking of restricting the prisms/antiprisms to degree 3, 4, 5, 6, 8, and 10, so they don't add any new polygon degrees, but existing degrees will be covered. I suspect this will cover most valid CRFs anyway (except for the infinite family of antiprism wedges enumerated by Klitzing and rediscovered by Mrrl).
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

I'm trying to compile the list of dihedral angle, but I got this idea: what if you tried to build the polychora primarily from a net of faces instead of cells? Any such net that has only regular faces could be filled in by convex cells, couldn't it?
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

So, let's look at dihedral angles. Stella is a big help here since its net mode computes dihedral angles automatically.
I include all prisms and antiprisms up to 10, for completeness.

BTW, you shouldn't automatically assume that there can be only 5 faces to an edge -- dihedral angles of some Johnson solids can get quite small.

Now, let's talk "possibilities". Basically, in some Johnson solids, the same dihedral angle exists at several nonequivalent places. I note this. Some of those places might be also assymetrical (i.e. joining two identical polygons, but without an axis of symmetry passing through the edge, so you have to try to fit it in both orientations).
The exact number of possiblities for CRF purposes will have to be checked, some possibilities might be chiral.
For the weird solids (sphenocorona and up) I've given up on describing the edges and counting possibilities...

31.7175 (dpc4)
4-10 in pentagonal cupola

37.3774 (dpp)
3-5 in pentagonal pyramid
3-10 in pentagonal cupola

45
4-8 in square cupola

54.7356 (90 - d4/2)
3-4 in square pyramid
4-6 in triangular cupola
3-8 in square cupola

60
4-4 in triangular prism
4-4 in elongated triangular pyramid (asymmetrical)
4-4 in elongated triangular dipyramid
4-4 in augmented triangular prism

63.4349 (2*dpc4)
5-10 in pentagonal rotunda
4-4 in pentagonal orthobicupola
5-5 in metabidiminished icosahedron
5-5 in tridiminished icosahedron (asymmetrical)
5-5 in augmented tridiminished icosahedron (asymmetrical)
5-5 in bilunabirotunda

69.0948 (dpc4 + dpp = d20/2)
3-4 in pentagonal gyrobicupola (equatorial)

70.5288 (let's call it d4 because it's an important number)
3-3 in tetrahedron
6-6 in truncated tetrahedron
3-6 in triangular cupola
3-3 in elongated triangular pyramid (asymmetrical)
3-3 in triangular dipyramid (apex) (asymmetrical)
3-3 in elongated triangular dipyramid (asymmetrical)
3-3 in augmented tridiminished icosahedron (augment) (asymmetrical)
6-6 in augmented truncated tetrahedron

72.973
4-4 in sphenomegacorona

74.7547 (2*dpp)
3-3 in pentagonal dipyramid (equatorial)
3-3 in pentagonal orthobicupola

79.1877 (dpr3; dpr3 + 2*dpc4 = did)
3-10 in pentagonal rotunda

86.7268
3-3 in sphenomegacorona

90
4-4 in cube
8-8 in truncated cube
3-4 in triangular prism
4-5 in pentagonal prism
4-6 in hexagonal prism
4-7 in heptagonal prism
4-8 in octagonal prism
4-9 in enneagonal prism
4-10 in decagonal prism
3-4 in elongated triangular pyramid (base)
4-4 in elongated square pyramid (2 possibilities) (both asymmetrical)
4-5 in elongated pentagonal pyramid
4-4 in elongated square dipyramid
4-6 in elongated triangular cupola (2 possibilities)
4-8 in elongated square cupola (2 possibilities)
4-10 in elongated pentagonal cupola (2 possibilities)
4-10 in elongated pentagonal rotunda (2 possibilities)
3-4 in gyrobifastigium (individual prisms)
4-4 in square orthobicupola (equatorial)
3-4 in augmented triangular prism (base/side)
3-4 in biaugmented triangular prism (base/side)
4-5 in augmented pentagonal prism (2 possibilities)
4-5 in biaugmented pentagonal prism (2 possibilities)
4-6 in augmented hexagonal prism (3 possibilities)
4-6 in parabiaugmented hexagonal prism
4-6 in metabiaugmented hexagonal prism (3 possibilities)
4-6 in triaugmented hexagonal prism
8-8 in augmented truncated cube (2 possibilities) (both asymmetrical)
8-8 in biaugmented truncated cube

95.1524 (3*dpc4)
4-5 in pentagonal gyrocupolarotunda (equatorial)

95.2466 (da10)
3-10 in decagonal antiprism
3-10 in gyroelongated pentagonal cupola
3-10 in gyroelongated pentagonal rotunda

95.843
3-9 in enneagonal antiprism

96.1983
3-3 in snub disphenoid (type 1) (2 possibilities) (1 asymmetrical)

96.5945 (da8)
3-8 in octagonal antiprism
3-8 in gyroelongated square cupola

97.4555
3-4 in sphenocorona
3-4 in augmented sphenocorona

97.5723
3-7 in heptagonal antiprism

98.8994 (da6)
3-6 in hexagonal antiprism
3-6 in gyroelongated triangular cupola

99.7356 (135 - d4/2)
3-4 in square gyrobicupola (equatorial)

100.194
4-4 in disphenocingulum

100.812 (d20 - dpp = dpp + 2*dpc4)
3-5 in pentagonal antiprism
3-5 in gyroelongated pentagonal pyramid
3-5 in pentagonal orthocupolarotunda (equatorial)
3-5 in metabidiminished icosahedron (2 possibilities)
3-5 in tridiminished icosahedron (2 possibilities)
3-5 in augmented tridiminished icosahedron (main body)
3-5 in bilunabirotunda
3-5 in triangular hebesphenorotunda

102.524
4-4 in hebesphenomegacorona

103.836 (da4)
3-4 in square antiprism
3-4 in gyroelongated square pyramid

108
4-4 in pentagonal prism
4-4 in elongated pentagonal pyramid (asymmetrical)
4-4 in elongated pentagonal dipyramid
4-4 in augmented pentagonal prism (2 possibilities) (1 asymmetrical)
4-4 in biaugmented pentagonal prism

109.471 (180 - d4)
3-3 in octahedron
3-6 in truncated tetrahedron
6-6 in truncated octahedron
3-3 in square pyramid
3-3 in elongated square pyramid (asymmetrical)
3-3 in gyroelongated square pyramid (apex) (asymmetrical)
3-3 in elongated square dipyramid (asymmetrical)
3-3 in gyroelongated square dipyramid (apex) (asymmetrical)
4-4 in triangular orthobicupola
3-3 in square orthobicupola
3-3 in augmented triangular prism (augment) (asymmetrical)
3-3 in biaugmented triangular prism (augment) (2 possibilities) (both asymmetrical)
3-3 in triaugmented triangular prism (augment) (asymmetrical)
3-3 in augmented pentagonal prism (asymmetrical)
3-3 in biaugmented pentagonal prism (2 possibilities) (both asymmetrical)
3-3 in augmented hexagonal prism (asymmetrical)
3-3 in parabiaugmented hexagonal prism (asymmetrical)
3-3 in metabiaugmented hexagonal prism (2 possibilities) (both asymmetrical)
3-3 in triaugmented hexagonal prism (asymmetrical)
3-6 in augmented truncated tetrahedron (main body) (2 possibilities)
3-3 in augmented sphenocorona

109.524
3-4 in sphenocorona
3-4 in augmented sphenocorona

110.905 (dpc4 + dpr3)
3-4 in pentagonal orthocupolarotunda (equatorial)
3-4 in bilunabirotunda
4-6 in triangular hebesphenorotunda
3-4 in triangular hebesphenorotunda

111.735
3-3 in hebesphenomegacorona

114.645
3-3 in snub square antiprism (middle edges) (2 possibilities)

114.736 (150 - d4/2)
3-4 in augmented triangular prism (augment/side)
3-4 in biaugmented triangular prism (augment/side)

116.565 (d12 = dpp + dpr3)
5-5 in dodecahedron
10-10 in truncated dodecahedron
3-3 in pentagonal gyrocupolarotunda
5-5 in augmented dodecahedron (4 possibilities) (all asymmetrical)
5-5 in parabiaugmented dodecahedron (2 possibilities) (1 asymmetrical)
5-5 in metabiaugmented dodecahedron (7 possibilities) (5 asymmetrical)
5-5 in triaugmented dodecahedron (4 possibilities) (all asymmetrical)
10-10 in augmented truncated dodecahedron (4 possibilities) (all asymmetrical)
10-10 in parabiaugmented truncated dodecahedron (2 possibilities) (1 asymmetrical)
10-10 in metabiaugmented truncated dodecahedron (7 possibilities) (5 asymmetrical)
10-10 in triaugmented truncated dodecahedron (4 possibilities) (all asymmetrical)
5-10 in diminished rhombicosidodecahedron
5-10 in paragyrate diminished rhombicosidodecahedron
5-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
5-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)
5-10 in parabidiminished rhombicosidodecahedron
5-10 in metabidiminished rhombicosidodecahedron (3 possibilities)
5-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)
5-10 in tridiminished rhombicosidodecahedron (3 possibilities)

117.019 (dsp)
4-4 in sphenocorona

117.356
3-3 in sphenomegacorona

118.892
3-3 in sphenocorona
3-3 in augmented sphenocorona

120
4-4 in hexagonal prism
4-4 in elongated triangular cupola (prism) (asymmetrical)
4-4 in elongated triangular orthobicupola (prism) (asymmetrical)
4-4 in elongated triangular gyrobicupola (prism)
4-4 in augmented hexagonal prism (2 possibilities) (both asymmetrical)
4-4 in parabiaugmented hexagonal prism
4-4 in metabiaugmented hexagonal prism (asymmetrical)

121.717 (90 + dpc4)
4-4 in elongated pentagonal cupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal orthobicupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal gyrobicupola (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal orthocupolarotunda (cupola/prism) (asymmetrical)
4-4 in elongated pentagonal gyrocupolarotunda (cupola/prism) (asymmetrical)
4-10 in diminished rhombicosidodecahedron
4-10 in paragyrate diminished rhombicosidodecahedron
4-10 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
4-10 in bigyrate diminished rhombicosidodecahedron (3 possibilities)
4-10 in parabidiminished rhombicosidodecahedron
4-10 in metabidiminished rhombicosidodecahedron (3 possibilities)
4-10 in gyrate bidiminished rhombicosidodecahedron (5 possibilities)
4-10 in tridiminished rhombicosidodecahedron (3 possibilities)

121.743
3-3 in snub disphenoid (type 2) (2 possibilities) (both asymmetrical)

124.702
3-3 in disphenocingulum

125.264 (90 + d4/2)
3-4 in cuboctahedron
4-6 in truncated octahedron
3-8 in truncated cube
6-8 in truncated cuboctahedron
3-4 in triangular cupola (2 possibilities)
3-4 in elongated triangular cupola (cupola) (2 possibilities)
3-4 in gyroelongated triangular cupola (cupola) (2 possibilities)
3-4 in triangular orthobicupola (2 possibilities)
3-4 in elongated triangular orthobicupola (cupola) (2 possibilities)
3-4 in elongated triangular gyrobicupola (cupola) (2 possibilities)
3-4 in gyroelongated triangular bicupola (cupola) (2 possibilities)
3-4 in augmented truncated tetrahedron (augment) (2 possibilities)
3-8 in augmented truncated cube (3 possibilities)
3-8 in biaugmented truncated cube

126.87 (4*dpc4)
5-5 in pentagonal orthobirotunda

126.964 (da10 + dpc)
3-4 in gyroelongated pentagonal cupola (cupola/antiprism)
3-4 in gyroelongated pentagonal bicupola (cupola/antiprism)
3-4 in gyroelongated pentagonal cupolarotunda (cupola/antiprism)

127.377 (90 + dpp)
3-4 in elongated pentagonal pyramid
3-4 in elongated pentagonal dipyramid
3-4 in elongated pentagonal cupola (cupola/prism)
3-4 in elongated pentagonal orthobicupola (cupola/prism)
3-4 in elongated pentagonal gyrobicupola (cupola/prism)
3-4 in elongated pentagonal orthocupolarotunda (cupola/prism)
3-4 in elongated pentagonal gyrocupolarotunda (cupola/prism)

127.552
3-3 in square antiprism
3-3 in gyroelongated square pyramid (antiprism band) (asymmetrical)
3-3 in gyroelongated square dipyramid (antiprism band)

128.496
3-3 in hebesphenomegacorona

128.571
4-4 in heptagonal prism

129.445
3-3 in sphenomegacorona

131.442
3-3 in augmented sphenocorona

132.624 (da10 + dpp)
3-3 in gyroelongated pentagonal cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal bicupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (cupola/antiprism) (asymmetrical)

133.591
3-3 in disphenocingulum

133.973
3-4 in hebesphenomegacorona

135
4-4 in rhombicuboctahedron
4-8 in truncated cuboctahedron
4-4 in octagonal prism
4-4 in square cupola
4-4 in elongated square cupola (3 possibilities) (all asymmetrical)
4-4 in gyroelongated square cupola (asymmetrical)
4-4 in square orthobicupola (cupola) (asymmetrical)
4-4 in square gyrobicupola (asymmetrical)
4-4 in elongated square gyrobicupola (3 possibilities) (2 of them asymmetrical)
4-4 in gyroelongated square bicupola (asymmetrical)
4-4 in augmented truncated cube (asymmetrical)
4-4 in biaugmented truncated cube (asymmetrical)

135.992
3-3 in sphenocorona
3-3 in augmented sphenocorona

136.336
3-4 in disphenocingulum

137.24
3-4 in sphenomegacorona

138.19 (d20 = 2*(dpc4 + dpp)
3-3 in icosahedron
6-6 in truncated icosahedron
3-3 in pentagonal antiprism
3-3 in pentagonal pyramid
3-3 in elongated pentagonal pyramid (asymmetrical)
3-3 in gyroelongated pentagonal pyramid (3 possibilities) (all asymmetrical)
3-3 in pentagonal dipyramid (apex) (asymmetrical)
3-3 in elongated pentagonal dipyramid (asymmetrical)
3-3 in augmented dodecahedron (asymmetrical)
3-3 in parabiaugmented dodecahedron (asymmetrical)
3-3 in metabiaugmented dodecahedron (3 possibilities) (all asymmetrical)
3-3 in triaugmented dodecahedron (3 possibilities) (all asymmetrical)
3-3 in metabidiminished icosahedron (4 possibilities) (3 asymmetrical)
3-3 in tridiminished icosahedron (asymmetrical)
3-3 in augmented tridiminished icosahedron (main body) (asymmetrical)
3-6 in triangular hebesphenorotunda
3-3 in triangular hebesphenorotunda

140
4-4 in enneagonal prism

141.058 (2*d4)
3-3 in triangular dipyramid (equatorial)
3-3 in triangular orthobicupola
3-6 in augmented truncated tetrahedron (augment/main body)

141.31
3-3 in hebesphenomegacorona

141.595 (45 + da8)
3-4 in gyroelongated square cupola (cupola/antiprism)
3-4 in gyroelongated square bicupola (cupola/antiprism)

142.623 (did)
3-5 in icosidodecahedron
5-6 in truncated icosahedron
3-10 in truncated dodecahedron
6-10 in truncated icosidodecahedron
3-5 in pentagonal rotunda (3 possibilities)
3-5 in elongated pentagonal rotunda (3 possibilities)
3-5 in gyroelongated pentagonal cupola (rotunda) (3 possibilities)
3-5 in pentagonal orthocupolarotunda (rotunda) (3 possibilities)
3-5 in pentagonal gyrocupolarotunda (3 possibilities)
3-5 in pentagonal orthobirotunda (3 possibilities)
3-5 in elongated pentagonal orthocupolarotunda (3 possibilities)
3-5 in elongated pentagonal gyrocupolarotunda (3 possibilities)
3-5 in elongated pentagonal orthobirotunda (3 possibilities)
3-5 in elongated pentagonal gyrobirotunda (3 possibilities)
3-5 in gyroelongated pentagonal cupolarotunda (3 possibilities)
3-5 in gyroelongated pentagonal birotunda (3 possibilities)
3-10 in augmented truncated dodecahedron (main body) (7 possibilities)
3-10 in parabiaugmented truncated dodecahedron (main body) (3 possibilities)
3-10 in metabiaugmented truncated dodecahedron (main body) (14 possibilities)
3-10 in triaugmented truncated dodecahedron (main body) (9 possibilities)
3-5 in bilunabirotunda
3-5 in triangular hebesphenorotunda

142.983
3-4 in snub cube

143.479
3-3 in sphenocorona
3-3 in augmented sphenocorona

143.738
3-3 in sphenomegacorona

144
4-4 in decagonal prism
4-4 in elongated pentagonal cupola (prism) (asymmetrical)
4-4 in elongated pentagonal rotunda (asymmetrical)
4-4 in elongated pentagonal orthobicupola (prism) (asymmetrical)
4-4 in elongated pentagonal gyrobicupola (prism)
4-4 in elongated pentagonal orthocupolarotunda (prism) (asymmetrical)
4-4 in elongated pentagonal gyrocupolarotunda (prism) (asymmetrical)
4-4 in elongated pentagonal orthobirotunda (asymmetrical)
4-4 in elongated pentagonal gyrobirotunda

144.144
3-3 in snub square antiprism (other edges of triangles adjacent to squares) (2 possibilities) (both asymmetrical)

144.736 (180 - d4/2)
3-4 in rhombicuboctahedron
4-6 truncated cuboctahedron
3-4 in square cupola
3-4 in elongated square pyramid
3-4 in elongated square dipyramid
4-4 in elongated triangular cupola (cupola/prism) (asymmetrical)
3-4 in elongated square cupola (2 possibilities)
3-4 in gyroelongated square cupola (cupola)
3-4 in square orthobicupola
3-4 in square gyrobicupola (cupola)
3-4 in elongated triangular orthobicupola (cupola/prism) (asymmetrical)
4-4 in elongated triangular gyrobicupola (cupola/prism) (asymmetrical)
3-4 in elongated square gyrobicupola (2 possibilities)
3-4 in gyroelongated square bicupola (cupola)
3-3 in augmented triangular prism (augment/base) (asymmetrical)
3-3 in biaugmented triangular prism (augment/base) (asymmetrical)
3-3 in triaugmented triangular prism (augment/base) (asymmetrical)
3-5 in augmented pentagonal prism
3-5 in biaugmented pentagonal prism
3-6 in augmented hexagonal prism
3-6 in parabiaugmented hexagonal prism
3-6 in metabiaugmented hexagonal prism
3-6 in triaugmented hexagonal prism
3-4 in augmented truncated cube (augment)
3-8 in augmented truncated cube (augment/main body)
3-4 in biaugmented truncated cube (augment)
3-8 in biaugmented truncated cube (augment/main body)

145.222
3-3 in hexagonal antiprism
3-3 in gyroelongated triangular cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in in gyroelongated triangular bicupola (antiprism) (3 possibilities) (1 asymmetrical)

145.441
3-4 in snub square antiprism

148.283
4-5 in rhombicosidodecahedron
4-10 in truncated icosidodecahedron
4-5 in pentagonal cupola
4-5 in elongated pentagonal cupola
4-5 in gyroelongated pentagonal cupola
4-5 in pentagonal orthobicupola
4-5 in pentagonal gyrobicupola
4-5 in pentagonal orthocupolarotunda
4-5 in pentagonal gyrocupolarotunda (cupola)
4-5 in elongated pentagonal orthobicupola
4-5 in elongated pentagonal gyrobicupola
4-5 in elongated pentagonal orthocupolarotunda (cupola)
4-5 in elongated pentagonal gyrocupolarotunda (cupola)
4-5 in gyroelongated pentagonal bicupola
4-5 in augmented truncated dodecahedron
4-5 in parabiaugmented truncated dodecahedron
4-5 in metabiaugmented truncated dodecahedron (3 possibilities)
4-5 in triaugmented truncated dodecahedron (3 possibilities)
4-5 in gyrate rhombicosidodecahedron (7 possibilities)
4-5 in parabigyrate rhombicosidodecahedron (3 possibilities)
4-5 in metabigyrate rhombicosidodecahedron (15 possibilities)
4-5 in trigyrate rhombicosidodecahedron (10 possibilities)
4-5 in diminished rhombicosidodecahedron (6 possibilities)
4-5 in paragyrate diminished rhombicosidodecahedron (5 possibilities)
4-5 in metagyrate diminished rhombicosidodecahedron (22 possibilities)
4-5 in bigyrate diminished rhombicosidodecahedron (21 possibilities)
4-5 in parabidiminished rhombicosidodecahedron (2 possibilities)
4-5 in metabidiminished rhombicosidodecahedron (11 possibilities)
4-5 in gyrate bidiminished rhombicosidodecahedron (20 possibilities)
4-5 in tridiminished rhombicosidodecahedron (6 possibilities)

148.434
3-3 in disphenocingulum

149.565
3-3 in hebesphenomegacorona

150
3-4 in gyrobifastigium (blend)

150.222
3-3 in heptagonal antiprism

151.33 (90 + da8 - d4/2)
3-3 in gyroelongated square cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated square bicupola (cupola/antiprism) (asymmetrical)

152.191
3-3 in augmented sphenocorona

152.93
3-5 in snub dodecahedron

152.976
3-4 in hebesphenomegacorona

153.235
3-3 in snub cube (2 possibilities) (1 asymmetrical)

153.435 (90 + 2*dpc4)
4-5 in elongated pentagonal rotunda
4-5 in elongated pentagonal orthocupolarotunda (rotunda/prism)
4-5 in elongated pentagonal gyrocupolarotunda (rotunda/prism)
4-5 in elongated pentagonal orthobirotunda
4-5 in elongated pentagonal gyrobirotunda
4-4 in gyrate rhombicosidodecahedron (asymmetrical)
4-4 in parabigyrate rhombicosidodecahedron (asymmetrical)
4-4 in metabigyrate rhombicosidodecahedron (3 possibilities) (all asymetrical)
4-4 in trigyrate rhombicosidodecahedron (3 possibilities) (all asymmetrical)
4-4 in paragyrate diminished rhombicosidodecahedron (asymmetrical)
4-4 in metagyrate diminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)
4-4 in bigyrate diminished rhombicosidodecahedron (5 possibilities) (all asymmetrical)
4-4 in gyrate bidiminished rhombicosidodecahedron (3 possibilities) (all asymmetrical)

153.635 (180 + da6 - d4/2)
3-4 in gyroelongated triangular cupola (cupola/antiprism)
3-4 in gyroelongated triangular bicupola (cupola/antiprism)

153.942 (dpp + d12)
3-5 in augmented dodecahedron
3-5 in parabiaugmented dodecahedron
3-5 in metabiaugmented dodecahedron (3 possibilities)
3-5 in triaugmented dodecahedron (3 possibilities)
3-10 in augmented truncated dodecahedron (augment/main body)
3-10 in parabiaugmented truncated dodecahedron (augment/main body)
3-10 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-10 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-5 in gyrate rhombicosidodecahedron
3-5 in parabigyrate rhombicosidodecahedron
3-5 in metabigyrate rhombicosidodecahedron (3 possibilities)
3-5 in trigyrate rhombicosidodecahedron (3 possibilities)
3-5 in paragyrate diminished rhombicosidodecahedron
3-5 in metagyrate diminished rhombicosidodecahedron (3 possibilities)
3-5 in bigyrate diminished rhombicosidodecahedron (5 possibilities)
3-5 in gyrate bidiminished rhombicosidodecahedron (3 possibilities)

153.962
3-3 in octagonal antiprism
3-3 in gyroelongated square cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated square bicupola (antiprism) (3 possibilities) (1 asymmetrical)

154.419
3-4 in disphenocingulum

154.722
3-4 in sphenomegacorona

156.866
3-3 in enneagonal antiprism

157.148
3-3 in hebesphenomegacorona

158.375 (2*dpr3)
3-3 in pentagonal orthobirotunda

158.572 (90 + da4 - d4/2)
3-3 in gyroelongated square pyramid (apex/antiprism) (asymmetrical)
3-3 in in gyroelongated square dipyramid (apex/antiprism) (asymmetrical)

158.682 (da10 + 2*dpc4)
3-5 in gyroelongated pentagonal rotunda (rotunda/antiprism)
3-5 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)
3-5 in gyroelongated pentagonal birotunda (rotunda/antiprism)

159.095
3-4 in rhombicosidodecahedron
4-6 in truncated icosidodecahedron
3-4 in pentagonal cupola
3-4 in elongated pentagonal cupola (cupola)
3-4 in gyroelongated pentagonal cupola (cupola)
3-4 in pentagonal orthobicupola
3-4 in pentagonal gyrobicupola (cupola)
3-4 in pentagonal orthocupolarotunda (cupola)
3-4 in pentagonal gyrocupolarotunda
3-4 in elongated pentagonal orthobicupola (cupola)
3-4 in elongated pentagonal gyrobicupola (cupola)
3-4 in elongated pentagonal orthocupolarotunda (cupola)
3-4 in elongated pentagonal gyrocupolarotunda (cupola)
3-4 in gyroelongated pentagonal bicupola (cupola)
3-4 in augmented truncated dodecahedron (augment)
3-4 in parabiaugmented truncated dodecahedron (augment)
3-4 in metabiaugmented truncated dodecahedron (augment) (5 possibilities)
3-4 in triaugmented truncated dodecahedron (augment) (5 possibilities)
3-4 in gyrate rhombicosidodecahedron (7 possibilities)
3-4 in parabigyrate rhombicosidodecahedron (3 possibilities)
3-4 in metabigyrate rhombicosidodecahedron (14 possibilities)
3-4 in trigyrate rhombicosidodecahedron (9 possibilities)
3-4 in diminished rhombicosidodecahedron (6 possibilities)
3-4 in paragyrate diminished rhombicosidodecahedron (5 possibilities)
3-4 in metagyrate diminished rhombicosidodecahedron (21? possibilities -- I'm honestly not quite sure here)
3-4 in bigyrate diminished rhombicosidodecahedron (17? possibilities)
3-4 in parabidiminished rhombicosidodecahedron (2 possibilities)
3-4 in metabidiminished rhombicosidodecahedron (9 possibilities)
3-4 in gyrate bidiminished rhombicosidodecahedron (13 possibilities)
3-4 in tridiminished rhombicosidodecahedron (4 possibilities)
3-4 in bilunabirotunda
3-4 in triangular hebesphenorotunda

159.187
3-3 in decagonal antiprism
3-3 in gyroelongated pentagonal cupola (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated pentagonal rotunda (antiprism) (2 possibilities) (both asymmetrical)
3-3 in gyroelongated pentagonal bicupola (antiprism) (3 possibilities) (1 asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (antiprism) (4 possibilities) (all asymmetrical)
3-3 in gyroelongated pentagonal birotunda (3 possibilities) (1 asymmetrical)

159.892
3-3 in sphenocorona
3-3 in augmented sphenocorona

160.529 (90 + d4)
3-4 in elongated triangular pyramid (apex)
3-4 in elongated triangular dipyramid
3-4 in elongated triangular cupola (cupola/prism)
3-4 in elongated triangular orthobicupola (cupola/prism)
3-4 in elongated triangular gyrobicupola (cupola/prism)

161.483
3-3 in sphenomegacorona

162.736 (198 - d4/2)
3-4 in augmented pentagonal prism (augment/side)
3-4 in biaugmented pentagonal prism (2 possibilities)

164.172
3-3 in snub dodecahedron (2 possibilities) (1 asymmetrical)

164.207 (270 - 3*d4/2)
3-4 in augmented truncated tetrahedron (augment/main body)

164.257
3-3 in snub square antiprism (vertical edges) (asymmetrical)

166.441
3-3 in snub disphenoid (type 3) (2 possibilities)

166.811
3-3 in disphenocingulum

169.188 (90 + dpr3)
3-4 in elongated pentagonal rotunda
3-4 in elongated pentagonal orthocupolarotunda (rotunda/prism)
3-4 in elongated pentagonal gyrocupolarotunda (rotunda/prism)
3-4 in elongated pentagonal orthobirotunda
3-4 in elongated pentagonal gyrobirotunda

169.428 (d4 + da6)
3-3 in gyroelongated triangular cupola (cupola/antiprism) (asymmetrical)
3-3 in gyroelongated triangular bicupola (cupola/antiprism) (asymmetrical)

169.471 (240 - d4)
3-3 in biaugmented triangular prism (augment/augment)
3-3 in triaugmented triangular prism (augment/augment)

170.264 (135 + d4/2)
3-4 in augmented truncated cube (augment/main body)
3-4 in biaugmented truncated cube (augment/main body)

171.341 (dpp + 2*dpc4 + d4)
3-5 in augmented tridiminished icosahedron (augment/main body)

171.646
3-3 in sphenomegacorona

171.755 (90 + dsp - d4/2)
3-4 in augmented sphenocorona

174.34 (dpc4 + did)
3-4 in augmented truncated dodecahedron (augment/main body)
3-4 in parabiaugmented truncated dodecahedron (augment/main body)
3-4 in metabiaugmented truncated dodecahedron (augment/main body) (3 possibilities)
3-4 in triaugmented truncated dodecahedron (augment/main body) (3 possibilities)

174.434 (da10 + dpr3)
3-3 in gyroelongated pentagonal rotunda (rotunda/antiprism) (asymmetrical)
3-3 in gyroelongated pentagonal cupolarotunda (rotunda/antiprism)
3-3 in gyroelongated pentagonal birotunda (rotunda/antiprism)

174.736 (210 - d4/2)
3-4 in augmented hexagonal prism
3-4 in parabiaugmented hexagonal prism
3-4 in metabiaugmented hexagonal prism (2 possibilities)
3-4 in triaugmented hexagonal prism

Of course, you must also take into account chiral polyhedra:
snub cube, snub dodecahedron, gyroelongated triangular bicupola, gyroelongated square bicupola, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda. Those might count twice in all cases.
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:I'm trying to compile the list of dihedral angle, but I got this idea: what if you tried to build the polychora primarily from a net of faces instead of cells? Any such net that has only regular faces could be filled in by convex cells, couldn't it?

But how would you ensure that the polygons in the net will form convex cells? And how would you guarantee they lie on a single hyperplane? You might get some polygon nets that are non-planar, then you can't make a valid CRF from it.

Marek14 wrote:So, let's look at dihedral angles. Stella is a big help here since its net mode computes dihedral angles automatically.
I include all prisms and antiprisms up to 10, for completeness.

Wow, cool! That's a long list! Hmm. That's much longer than I expected. I guess I don't know the Johnson solids well enough.

BTW, you shouldn't automatically assume that there can be only 5 faces to an edge -- dihedral angles of some Johnson solids can get quite small.

You're right! The smallest dihedral angle in your list is 31.7175, which lets us fit about 11 polyhedra around an edge. I assume that's maximal, though in this case pentagonal cupolas can't join 11 to an edge because the ridges have to alternate between square and decagon, so that means 10 of those things can fit around an edge. Wow. That means a lot more possibilities to consider if we assume more than 3 faces per edge!

Now, let's talk "possibilities". Basically, in some Johnson solids, the same dihedral angle exists at several nonequivalent places. I note this. Some of those places might be also assymetrical (i.e. joining two identical polygons, but without an axis of symmetry passing through the edge, so you have to try to fit it in both orientations).
The exact number of possiblities for CRF purposes will have to be checked, some possibilities might be chiral.
For the weird solids (sphenocorona and up) I've given up on describing the edges and counting possibilities...

Yeah, chiral Johnsons did occur to me; I was thinking that the table will be keyed by a pair of integers representing polygon degrees, then the corresponding table entry will be a list of edges. Each edge will point to a particular edge on a particular Johnson solid (the program will have to keep a face lattice for each polyhedron, so that it can easily find adjacent faces, etc.), so that takes care of the non-equivalent cases. But when both polygons are the same, you may have to try both orientations.

[...snip huge awesome list...]

Thanks for the list! Are the angle values from Stella? Is there any way of getting more digits for them? One thing I've noticed working with higher-dimensional polytopes is that they tend to be very sensitive to roundoff errors. Get the dihedral angle of one pair of cells slightly off, and the error propagates through the rest of the shape, and it might not close up anymore even though algebraically it does close up. Or it may close up when it's not supposed to due to roundoff error shifting the actual dihedral angle too close to a neighbouring value. (I note that, unfortunately, there are quite a few dihedral angles between the same polygons that are very close to each other.) This may come back to bite us with large shapes like the 600-cell diminishings.

Of course, you must also take into account chiral polyhedra:
snub cube, snub dodecahedron, gyroelongated triangular bicupola, gyroelongated square bicupola, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda. Those might count twice in all cases.

Yeah, chiral polyhedra all have to count as two distinct objects. Otherwise we may run into trouble (a CRF may contain both enantiomers as cells in non-equivalent places that cannot be interchanged by a mirror image operation in 4D, for example).
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

quickfur wrote:But how would you ensure that the polygons in the net will form convex cells? And how would you guarantee they lie on a single hyperplane? You might get some polygon nets that are non-planar, then you can't make a valid CRF from it.

When adding any single polygon, you could make sure it's planar. As for whether they will form convex cells... if you won't allow coplanar polygons, won't they form convex cells automatically?

Marek14 wrote:So, let's look at dihedral angles. Stella is a big help here since its net mode computes dihedral angles automatically.
I include all prisms and antiprisms up to 10, for completeness.

Wow, cool! That's a long list! Hmm. That's much longer than I expected. I guess I don't know the Johnson solids well enough.

BTW, you shouldn't automatically assume that there can be only 5 faces to an edge -- dihedral angles of some Johnson solids can get quite small.

You're right! The smallest dihedral angle in your list is 31.7175, which lets us fit about 11 polyhedra around an edge. I assume that's maximal, though in this case pentagonal cupolas can't join 11 to an edge because the ridges have to alternate between square and decagon, so that means 10 of those things can fit around an edge. Wow. That means a lot more possibilities to consider if we assume more than 3 faces per edge!

Now, let's talk "possibilities". Basically, in some Johnson solids, the same dihedral angle exists at several nonequivalent places. I note this. Some of those places might be also assymetrical (i.e. joining two identical polygons, but without an axis of symmetry passing through the edge, so you have to try to fit it in both orientations).
The exact number of possiblities for CRF purposes will have to be checked, some possibilities might be chiral.
For the weird solids (sphenocorona and up) I've given up on describing the edges and counting possibilities...

Yeah, chiral Johnsons did occur to me; I was thinking that the table will be keyed by a pair of integers representing polygon degrees, then the corresponding table entry will be a list of edges. Each edge will point to a particular edge on a particular Johnson solid (the program will have to keep a face lattice for each polyhedron, so that it can easily find adjacent faces, etc.), so that takes care of the non-equivalent cases. But when both polygons are the same, you may have to try both orientations.

[...snip huge awesome list...]

Thanks for the list! Are the angle values from Stella? Is there any way of getting more digits for them? One thing I've noticed working with higher-dimensional polytopes is that they tend to be very sensitive to roundoff errors. Get the dihedral angle of one pair of cells slightly off, and the error propagates through the rest of the shape, and it might not close up anymore even though algebraically it does close up. Or it may close up when it's not supposed to due to roundoff error shifting the actual dihedral angle too close to a neighbouring value. (I note that, unfortunately, there are quite a few dihedral angles between the same polygons that are very close to each other.) This may come back to bite us with large shapes like the 600-cell diminishings.

Of course, you must also take into account chiral polyhedra:
snub cube, snub dodecahedron, gyroelongated triangular bicupola, gyroelongated square bicupola, gyroelongated pentagonal bicupola, gyroelongated pentagonal cupolarotunda and gyroelongated pentagonal birotunda. Those might count twice in all cases.

Yeah, chiral polyhedra all have to count as two distinct objects. Otherwise we may run into trouble (a CRF may contain both enantiomers as cells in non-equivalent places that cannot be interchanged by a mirror image operation in 4D, for example).

The values are from Stella, and they are, I presume, rounded to six digits. But the exact values are actually not that important -- you can probably get more precise results by building the solids yourself. What's really important is the algebraic relations between different values (I noted this where I could) and situations where multiple solids share the same angle. For example, you notice that all snubs (including snub disphenoid and snub square antiprism) have completely unique values not related to anything else, coronas are also a world of their own, while bilunabirotunda and triangular hebesphenorotunda are actually related to the "more normal" solids.

The values, however, should definitely be exact enough to allow for decisions which edges are possible.

I guess next step would be to enumerate the edges?

I imagine that the key part of the generator will be sieve to eliminate all CRF polychora that were already discovered

Johnson solids are classified pretty well; I suspect CRF polychora will have two lists: raw list ordered, say by number of vertices -> edges -> faces -> cells, and then "taxonomic" list where we'll try to fit them.

Also, I imagine that the 7- and 9- prisms and antiprisms will actually have a huge part in the program: if our hypothesis that no new CRF polychora that use them exist, apart from infinite families and augmented duoprisms, is correct, then any new CRF that contains one of these will also have a "twin" containing the other, which could show as a hint to new infinite families. And if one is found without a twin, then our hypothesis was wrong
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Further note: I think we should start with just a "proof of concept" program with limited selection of shapes, then extend it.

Edges with more than 3 cells might present a problem because they are not "rigid" -- 3-cell edge has strictly defined dichoral angles, but 4-cell and more do not (like 4- or more- polygon vertices: compare octahedron's, triangular dipyramid's and pentagonal dipyramid's 4-triangle vertices).

I imagine the end product of the search would be a database with a program attached to search through it, for example by specifying cells which you want/don't want to be present.

This is just my hunch, but I think that the number of anomalous CRF polychora (not related to any uniform) will be bigger then in 3D because there's just significantly more shapes that can be reasonably used. I'm not sure if any are known...
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:
quickfur wrote:But how would you ensure that the polygons in the net will form convex cells? And how would you guarantee they lie on a single hyperplane? You might get some polygon nets that are non-planar, then you can't make a valid CRF from it.

When adding any single polygon, you could make sure it's planar. As for whether they will form convex cells... if you won't allow coplanar polygons, won't they form convex cells automatically?

What I meant was, how do we know the polygons that surround a cell will lie on a single hyperplane? Each polygon may be planar in itself, but I'm not sure how we can guarantee that putting a bunch of planar polygons together will guarantee cell convexity. From a programmatic point of view, the program wouldn't immediately know that it's not constructing some non-convex polyhedron that has convex faces that are mutually intersecting, or that extends outside of a hyperplane, for example.

[...]
[...] Are the angle values from Stella? Is there any way of getting more digits for them? [...]

[...]
The values are from Stella, and they are, I presume, rounded to six digits. But the exact values are actually not that important -- you can probably get more precise results by building the solids yourself. What's really important is the algebraic relations between different values (I noted this where I could) and situations where multiple solids share the same angle. For example, you notice that all snubs (including snub disphenoid and snub square antiprism) have completely unique values not related to anything else, coronas are also a world of their own, while bilunabirotunda and triangular hebesphenorotunda are actually related to the "more normal" solids.

Good point.

Hmm, I wonder if there's some way to prove that "unique" angles, say for the snub disphenoid, snub square antiprism, etc., cannot possibly be obtained from any dichoral angle between cells that only have the more "normal" angles? Some sort of algebraic independence theorem that we can use, perhaps? I'm thinking about the similarity between proving that square roots of unrelated non-square numbers cannot possibly be a linear combination of each other; in the geometric world, maybe there are analogous theorems that folding two polyhedra with "strange" angles around a face cannot produce dihedral angles around the face that can fit both "normal" and "unique" polyhedra?

The values, however, should definitely be exact enough to allow for decisions which edges are possible.

I guess next step would be to enumerate the edges?

I wonder if it will be better to have the program automatically enumerate these things, because for the algo to work, it needs to have polyhedron face adjacency data (so that after fitting in a new cell it knows which faces to look at next), and each angle for a given polygon-polygon combination in the lookup table needs to be linked to a precise edge/orientation on the polyhedron. Manually constructing these tables would be too tedious and error-prone, I think. Perhaps the best way is to just construct accurate coordinates for the CRF polyhedra, and then run it through a convex-hull algorithm to generate the face lattice for them; then the CRF search program just loads this data and constructs the requisite internal tables.

I imagine that the key part of the generator will be sieve to eliminate all CRF polychora that were already discovered

Yeah, that is a problem that I've been thinking about. It's a tough one. Basically it's the face lattice isomorphism problem, which according to this paper is generally as hard as the graph isomorphism problem, which is in NP (requires exponential time algo). Fortunately, it seems that if we bound the number of dimensions (in this case d=4), a polynomial time algorithm is possible.

Johnson solids are classified pretty well; I suspect CRF polychora will have two lists: raw list ordered, say by number of vertices -> edges -> faces -> cells, and then "taxonomic" list where we'll try to fit them.

I've actually thought a lot about classification of CRF polychora following the classification of the Johnson solids. I haven't finalized the full classification scheme yet, but basically we can break it down as follows:

I. Monostratic polychora (includes the Klitzing segmentotopes, but generalized to admit stuff that doesn't inscribe a 3-sphere -- note that Klitzing's list doesn't include things like the snub disphenoid prism because it doesn't inscribe a 3-sphere, so there does remain more monostratic CRFs to be discovered).

II. Laminochora (made by stacking monostratic polychora on top of each other)

III. Rotundae (basically cup-shaped objects with a large cell on the bottom and a small cell/face/edge/vertex on the top, that aren't listed in I and II), and their augmentations with monostratic polychora or laminochora. Maybe diminishings as well, if those aren't already covered.

IV. Modified uniform polychora: diminishings and augmentations with I, II, and/or III. This category includes the duoprisms and their modifications. I've recently started to search for maximally-diminished uniforms, and found a few interesting shapes that aren't in categories I-III, derived from the 5-cell and tesseract families. Currently just starting with the 24-cell family, which promises to have a few interesting cases as well.

V. Crown jewels: stuff that can't be directly derived from the above categories.

There may be more simple categories that I missed, that shouldn't be put in the crown jewels category. As far as I know, we don't know any crown jewels yet. I also don't know how to search academic research papers for them, because of the lack of a common keyword for them (we're the only ones who call them crown jewels AFAIK).

Also, recently I realized that the duoprisms can be augmented not just by prism pyramids (which restricts consideration of duoprisms to those with trigonal, square, pentagonal prism cells), but that for each m,n-duoprism augmented with an m-gonal prism pyramid, there is a corresponding augmentation of a 2m,n-duoprism with an m-gonal cupola (i.e., m-gon||2m-prism). This means that in addition to the 1600+ augmentations of 3,n-duoprisms, 4,n-duoprisms, and 5,n-duoprisms, we also have augmentations of 6,n-duoprims, 8,n-duoprisms, and 10,n-duoprisms, which is at least another 1600+ augmentations, probably much more because the m-gonal cupola induce a symmetry-breaking orientation on the 2n,m-duoprism, which means many more unique combinations. Also, there are more unique positions to augment in a 8,n-duoprism than the 4,n-duoprism (choice from 8 positions instead of just 4, to place the augment).

Also, I imagine that the 7- and 9- prisms and antiprisms will actually have a huge part in the program: if our hypothesis that no new CRF polychora that use them exist, apart from infinite families and augmented duoprisms, is correct, then any new CRF that contains one of these will also have a "twin" containing the other, which could show as a hint to new infinite families. And if one is found without a twin, then our hypothesis was wrong

True!

Marek14 wrote:Further note: I think we should start with just a "proof of concept" program with limited selection of shapes, then extend it.

Good idea, I was thinking the same thing. This goes along with what I said above, that we should precompute the face lattice of all the 3D CRFs, and then the program can just load them (for the initial trial run, we can select a limited subset of them), automatically precompute the tables, etc., and then run the search algo. For testing purposes, we can place an upper limit on cell count, so that the program won't spend huge amounts of time cranking out things like 600-cell diminishings -- those can be temporarily rejected by the upper limit, so that the program will discover smaller interesting CRFs first. Later, once we're confident with the algo, we can let it loose on the full set of 3D CRFs and see what turns up.

Edges with more than 3 cells might present a problem because they are not "rigid" -- 3-cell edge has strictly defined dichoral angles, but 4-cell and more do not (like 4- or more- polygon vertices: compare octahedron's, triangular dipyramid's and pentagonal dipyramid's 4-triangle vertices).

Hmm, you're right! In fact, I've used this very fact to understand how the snub disphenoid can be made from conjoining two octahedra: first you cut off a quadrant from each octahedron, then squeeze them so that the skew polygon around the cut becomes identical to itself rotated 90° then flipped; then you can glue the two 3/4 octahedra together and the result is a snub disphenoid. I tried to discover a similar construction in 4D, but cuttings of the 16-cell produce a skew octahedron at the cut, which can't be made equal to itself with the 90° rotation trick, so you can't glue two 3/4 16-cells this way. So no luck in that direction at producing a crown jewel so far.

In any case, this is bad news for the algorithm. This means it's only feasible when restricted to 3 cells per edge. Non-rigid edges will greatly increase the complexity of the problem, and may even make it incrementally unsolvable because we may not be able to fix any angles until most of the rest of the shape is constructed. Unfortunately, this also appears to be where a lot of crown jewels are likely to be -- look at the snub disphenoid, for example. It has vertices with 4 triangles that are distorted from the octahedral angles. And the various coronas, which have dihedral angles slightly shifted from the "normal" angles in the platonic solids.

I imagine the end product of the search would be a database with a program attached to search through it, for example by specifying cells which you want/don't want to be present.

Well, assuming we fix a way of canonicalizing CRFs (see face lattice isomorphism above), we can simply have a central database to which we can add newly discovered CRFs. The database will then act as a "canonical listing" of CRFs, perhaps with a front-end that we can use to name/classify the various discovered CRFs. Separating the database from the search program also lets us do incremental searches, e.g., search for CRFs constructed from different subsets of 3D CRFs. It also lets us do SETI-like distributed brute force searches.

This is just my hunch, but I think that the number of anomalous CRF polychora (not related to any uniform) will be bigger then in 3D because there's just significantly more shapes that can be reasonably used. I'm not sure if any are known...

Yeah I don't know of any crown jewels so far.

BTW I found this interesting page that explains how some of the Johnson "crown jewels" are actually members of a series of mainly-nonconvex polyhedra that just happen to be convex. I wonder if some of the 4D crown jewels will turn out to be convex members of similar series, too.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Automatizing the process of generating edges is good idea in any case. As for the nonrigid edges, I wonder if there's a way around that...
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:Automatizing the process of generating edges is good idea in any case. As for the nonrigid edges, I wonder if there's a way around that...

I was just thinking, that non-rigid edges only happen in certain cases. For example, in 3D, there is only one way to put 5 equilateral triangles together; you can't distort the dihedral angles at all. When 4 equilateral triangles are put around a vertex then it can be distorted, because the ridges between the faces just happen to be lined up correctly in two planes, such that when you squeeze the angle in one plane, the change gets "shunted" into a corresponding change in the other plane.

I'm not 100%, but is pentagon-triangle-triangle-triangle rigid or non-rigid? What's the criteria for non-rigidness? Is it because in 3D, vertices with 4 faces around it is non-rigid, or is there a more specific condition, like there must be two pairs of edges that lie in two planes? Let's see... If we make a hexagonal pyramid with isosceles triangles and take off the hexagonal base, then the resulting fan of 6 triangles can be squeezed, so the vertex is non-rigid. Hmm. Does this mean that vertices of even order are non-rigid, and vertices of odd order are rigid?

I suspect in 4D there should be analogous criteria for this. AFAIK, an edge surrounded by 5 tetrahedra can only have one configuration - it's rigid. An edge surrounded by 4 tetrahedra is non-rigid. Maybe the criterion is that edges surrounded by an even number of cells are non-rigid? If so, the algo is still usable without modification in CRFs with only odd-degree edges.

(I'm not sure about this though. We need to verify that even/odd degree really is the criterion at work here.)
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

P.S. Found this paper that discusses rigidity of graphs. I only skimmed it briefly, but it seems to describe the necessary conditions for non-rigidness. It seems to be a combinatorial, not a geometric, property. Maybe this will help us identify which cell configurations are non-rigid and which are rigid. Hopefully this will let us use the simple version of the algo to search for CRFs in the rigid cases.

Non-rigid edges are a pain because they can be deformed continuously; it makes incremental algos very hard or maybe impossible.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Maybe it would be worth it to make the program run in 3D at first and generate Johnson solids? We need a way to generate snub square antiprism from scratch...

One way could be to have a system that could, apart from normal polychora also tolerate "incomplete" polychora, and you could try to fit two incomplete polychora together if the "missing piece" had the same combinatoric shape on both. That's how you tried the snub disphenoid.

I'm trying to imagine the nonrigid edges. If I imagine a 3D case of 4 triangles at a vertex, then if I keep one triangle fixed and let the two neighbouring triangles "flap around", each of their third vertices will lie somewhere on a circle. Fourth triangle can then fit exactly when the distance between these two points is 1. That's nothing more than a set of quadratic equations, which can be solved. We have one degree of freedom here since for every position of one triangle, there is only finite number of positions where the second one can be.

If we use different polygons (not 4 triangles), then we find that since every polygon is rigid, then any 4-polygon vertex also has only 1 degree of freedom.

In case of 5 polygons per vertex, how many degrees of freedom there is? Two. If polygons are, clockwise from base, b, c, d, and e, then we can flap b arbitrarily, flap c from b, and then fix e so that d exists.

Principially, the nonrigid edges in 4D work the same way. The deformation of the edge can be described with help of quadratic equations. This means that we might write the coordinates not as absolute numbers, but also permit functions there. A (3+n)-face edge would have n-1 degrees of freedom so cells around it would have coordinates that could be described with help of n-1 variables.
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

So, imagine a general tetrahedron lying in xyz hyperplane:

A = [0,0,0,0]
B = [1,0,0,0]
C = [1/2,sqrt(3)/2,0,0)
D = [1/2,sqrt(3)/6,sqrt(6)/3,0]

Building a second tetrahedron over ABC face leads to point E lying on a circle in hyperplane x=1/2, y=sqrt(3)/2 satisfying condition 3z^2 + 3w^2 = 2
Building a third tetrahedron over face ABD leads to point F lying on another circle (harder to express). Now, if we want to close these two into three-at-edge, we can solve for case where E and F are identical. If we want to go into four-at edge, we can solve to put distance between E and F equal to a specific number corresponding to a polyhedron we want to fit between them, and we'll end up with one-variable system. And so on.
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Hmm you're right, any vertex of degree >3 will be non-rigid in 3D, and any edge of degree >4 will be non-rigid in 4D. So the algo really only works for degree-3 edges.

Representing vertex coordinates as (quadratic) functions instead of numbers is possible, but very hard to deal with algorithmically, mainly because a system of quadratic equations is equivalent to a system of polynomial equations of arbitrary degree: you can repeatedly introduce new variables and add equations of the form u=x2 and you can reduce any arbitrary-degree polynomial system to a system of quadratic equations. This is why the snub cube's coordinates involves solving cubics, even though the initial constraints are only quadratic. The problem is, I don't think any general method of solution is known for such systems.

There might be solution methods for special subclasses of these systems, maybe there is a method for systems that arise from geometry. But I don't have much hope for it, because even though we start off with simple equations based on circles, once you start combining them algebraically to solve for variables they quickly lose their simple circle-based appearance and start looking like an arbitrary system of quadratic equations.

Looks like we don't really have much of a choice except to build a 4D lego simulator that lets you fit things together arbitrarily.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

Too bad... of course, even the 3-edge enumeration would be a step forward...
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

### Re: Johnsonian Polytopes

Marek14 wrote:Too bad... of course, even the 3-edge enumeration would be a step forward...

Perhaps an interactive application is the way to go. It will have built-in 3-edge enumeration integrated, so it can display a list of possible 3-edge configurations between two given cells. So you can select one of the configurations found, or you can elect to have a 4-edge or 5-edge (or higher), in which case you'll have to manually set the angles. Or maybe leave those edges unassigned temporarily, and then once we construct enough of the shape, it will hopefully constrain them to a particular configuration. Of course, this wouldn't work for shapes that only have 4-edges or higher, or shapes that don't have enough 3-edges to constrain all the higher-order edges. So this approach won't be able to discover things like the 16-cell without manually setting angles.

But I suspect that the interesting CRF cases, the crown jewels, will depend on some unusual higher-order edge configurations, like the 3D crown jewels do. In which case the only method of discovery seems to be the Lego simulator. Or maybe a ball-and-springs simulator that generates candidate cell combinations and checks if the system converges to a CRF solution. Such an approach may not be feasible, though, since the number of possibilities just grows too fast, like the 600-cell diminishings or duoprism augmentations.
quickfur
Pentonian

Posts: 2173
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

### Re: Johnsonian Polytopes

wendy wrote:A Johnson polyhedron is made of regular polygons, and is convex, but is not elsewhere included in platonic/uniform/etc.

The current thinking of uniform is equal edge + vertex transitive + uniform surtopes.

The hexagonal pyramid, or the hexagonal

Uniform johnson polychora, for example, would be all convex polychora made of regular polygons, except for the sixty-seven regulars, and their classes.

There are of course, the 92 prisms of the three-dimensional ones.

One might here include joys like xo3of3ox, a polytope made of 4 tri-diminished icosahedra, five tetrahedra, and an octahedron, the various diminished 500chora, the various augmented tesseracts (which are indeed convex, giving diminished, or all of the segmentotopes enumerated by Richard Klitzing (eg cube || icosahedron), and various sectionings of the 500ch.

Beside this, there are the figures like point | x-diminished icosa | x-diminished icosa | point, and all various sections thereof, for which gives at least four separate figures.

Yep, 4D Johnsons is quite well a task. I also once went into that direction. But then settled to a more manageable sub-class: the monostratic orbiform ones. That research, Wendy was citing, already has been published:
"Convex Segmentochora", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 11, 139-181, 2000
Cf. also:
http://bendwavy.org/klitzing/explain/segmentochora.htm
http://bendwavy.org/klitzing/pdf/artConvSeg_7.pdf
In that sub-class the (manual) research resulted in 177 such polychora.
There I found a single crown jewel only: cube || icosahedron.

--- rk
Klitzing
Tetronian

Posts: 877
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: Johnsonian Polytopes

Combinatoric approach would work like this:

With list of abstract polychora on n vertices, you make abstract polychora on n+1 vertices like this:

1. Add a pyramid to of n-vertex polychora.
2. Then you can join two cells together, removing a face between them, which may also result in removing an edge that is no longer needed. All these joins would use just faces/edges of the new pyramid and its base, since any changes on the rest are already included in the original list of n-vertex polychora.
3. Once you have a polychoron, you can try johnsonizing it.

So, there is one 5-vertex polychoron (a pentachoron). The basic 6-vertex polychoron would be tetrahedral dipyramid made by building a pentachoron on cell of another pentachoron.

Tetrahedral dipyramid has 8 tetrahedral cells, 16 triangular faces, 14 edges and 6 vertices. If we remove an equatorial triangle, we'll get triangular-dipyramidal pyramid (6 tetrahedra + 1 triangular dipyramid), which is not a CRF.
But now, if we remove another equatorial triangle and create another triangular dipyramid, we will be left with a "free" equatorial edge that exists between just two faces. That edge can be removed and the result is square-pyramidal pyramid (2 square pyramids + 4 tetrahedra) , which can be CRF.

If we removed an axial triangle from the original tetrahedral bipyramid, it would lead to a "free" edge between triangular dipyramid and tetrahedron -- this couldn't be removed because tetrahedron would cease to be a valid polyhedron without that edge.

And no more faces can be removed from square-pyramidal pyramid, so the first step concludes that there are 3 abstract polychora on 6 vertices, 2 of which belong to CRF group.

Theoretically, this could continue...
Marek14
Tetronian

Posts: 881
Joined: Sat Jul 16, 2005 6:40 pm

PreviousNext