General Approach--can 3D methods be generalized?

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

Re: General Approach--can 3D methods be generalized?

And final base configuration, (3,5,3,5) corresponding to icosidodecahedron and related polyhedra. Same as in (3,4,5,4), both diagonal squares are (5+Sqrt[5])/2.

Solutions are:

3535-3333: hyperbolic solution.

3535-3344: Blend of 334-045 acrochoron and 334-530 acrochoron joined in pentagonal cupola.

3535-3355: Blend of 335-055 acrochoron and 335-530 acrochoron joined in pentagonal rotunda.

3535-3553: Blend of 355-530 acrochoron and 333-550 acrochoron joined in pentagonal rotunda.

3535-3504: Blend of 350-045 acrochoron and 334-550 acrochoron joined in diminished rhombicosidodecahedron, or blend of 355-030 acrochoron and 334-050 acrochoron joined in truncated dodecahedron.

3535-3003: Blend of 350-030 acrochoron and rhodomesohedral rotunda joined in truncated dodecahedron.

3535-4444: standard prismatic solution.

3535-4466: Blend of 346-065 acrochoron and expanded/truncated rhodomesohedral rotunda joined in truncated icosidodecahedron.

3535-4004: hyperbolic solution.

3535-5555: flat solution. Surrounding of icosidodecahedron with dodecahedra and tridiminished icosahedra.

3535-6666: hyperbolic solution.
Marek14
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Re: General Approach--can 3D methods be generalized?

OK, it took a while, but this should be all verfs in shape of quadragonal pyramid.
Marek14
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Re: General Approach--can 3D methods be generalized?

Recently, Marek has determined all the vertex figures CRF polychora based on
tetraehdra (type abc-def) and those based on quadragonal pyramids (44 cases,
type 3bcd-efgh). I have completed the study for the 24 cases based on
pentagonal pyramids (type abcde-fghij).

Here is the table of the possible bases:

3 3 3 3 3 0.17323532263 117.356 143.738 143.738 117.356 161.4829 2.18927 2.70949 2.70949 2.18927 2.92234 J88
3 3 3 3 3 0.18411909463 96.1983 166.441 121.743 121.743 166.441 1.66196 2.95819 2.28917 2.28917 2.95819 J84
3 3 3 3 3 0.19150678129 109.4712 144.7356 144.7356 109.4712 169.4712 2.00000 2.72474 2.72474 2.00000 2.97474 J50
3 3 3 3 3 0.19556348419 109.4712 164.2596 118.892 135.992 152.1911 2.00000 2.94375 2.22474 2.57886 2.82676 J87
3 3 3 3 3 0.19683051116 109.4712 158.572 127.552 127.552 158.572 2.00000 2.89631 2.41421 2.41421 2.89631 J10
3 3 3 3 3 0.19699313021 114.645 144.144 144.144 114.645 164.2578 2.12550 2.71573 2.71573 2.12550 2.94374 J85
3 3 3 3 3 0.19864310695 128.496 157.148 111.735 157.148 128.496 2.43369 2.88227 2.05547 2.88227 2.43369 J89
3 3 3 3 3 0.19954254423 171.755 86.7268 171.646 117.356 117.356 2.98408 1.41436 2.98408 2.18927 2.18927 J88
3 3 3 3 3 0.20088090366 118.892 159.892 118.892 143.479 143.479 2.22474 2.90857 2.22474 2.70545 2.70545 J86
3 3 3 3 3 0.20660038687 148.434 133.591 133.591 148.434 124.702 2.77806 2.53426 2.53426 2.77806 2.35396 J90
3 3 3 3 3 0.20727639431 128.496 141.31 141.31 128.496 149.565 2.43369 2.67132 2.67132 2.43369 2.79330 J89
3 3 3 3 3 0.20886393771 143.479 135.992 135.992 143.479 131.442 2.70545 2.57886 2.57886 2.70545 2.49278 J86
3 3 3 3 3 0.20965059100 138.1898 138.1898 138.1898 138.1898 138.1898 2.61803 2.61803 2.61803 2.61803 2.61803 ico
3 3 3 3 4 0.24291936021 164.25964 109.47122 171.75464 109.52403 159.89240 2.94375 2.00000 3.71415 2.57886 2.90857 J87
3 3 3 3 4 0.27332082997 171.64574 129.445 154.722 137.24 143.738 2.98408 2.45300 3.56620 3.27168 2.70949 J88
3 3 3 3 4 0.27368887139 159.187 159.187 132.624 159.095 126.964 2.90211 2.90211 3.04151 3.61803 2.51578 J24
3 3 3 3 4 0.27607141638 145.222 145.222 169.428 125.264 153.635 2.73205 2.73205 3.55189 3.00000 2.97454 J22
3 3 3 3 4 0.27721017186 166.81137 133.59119 154.41883 136.33594 148.43399 2.96044 2.53426 3.56227 3.25297 2.77806 J90
3 3 3 3 4 0.28136914499 157.14815 157.14815 141.34110 152.97558 133.97281 2.88227 2.88227 3.20259 3.54293 2.67132 J89
3 3 3 3 4 0.28253596482 164.25739 144.14362 145.44063 145.44063 144.14362 2.94374 2.71573 3.42641 3.42641 2.71573 J85
3 3 3 3 4 0.28553475395 153.962 153.962 151.33 144.736 141.595 2.84776 2.84776 3.35729 3.41421 2.81610 J23
3 3 3 3 4 0.28565364345 153.235 153.235 153.235 142.983 142.983 2.83929 2.83929 3.38298 3.38298 2.83929 snucub
3 3 3 3 5 0.35293273058 159.187 159.187 174.434 142.6227 158.682 2.90211 2.90211 3.84358 3.61803 2.99293 J25
3 3 3 3 5 0.35886935933 164.172 164.172 164.172 152.93 152.93 2.94315 2.94315 3.77584 3.77584 2.94315 Snudod

1-5. column: edges
6. column: solid angle
7-11. column: dihedral angle between nth and n+1sr face
12-16. column: square of length of diagonal between nth and n+2nd vertex (2: square, 2.618: pentagon, 3: hexagon, 3.414: octagon, 3.618: decagon.

Trivially, the prismatic solution 3333a-44444 exists for all cases.

The other solutions are:

33333-(10)5335 for the 10.row (J90), 12. row (J86) and the 13.row (icosahedron).

33333-34(10)(10)4 exist for the icosahedron only.

Furthermore

33333-33333 (icosahedral pyramid)
33333-44444 (icosahedral prism)
33333-55555 (tridiminished icosahedra on icosahedron faces)
33333-66666 (vertex figure of truncated hexacosichoron)
johannes@itap.physik.uni-stuttgart.de
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Re: General Approach--can 3D methods be generalized?

Very interesting! I see you've managed to acquire better data for the "crown jewels" than me -- could you please look at the quadrangle pyramids with "irregularly" skewed bases to see if there are any I've overlooked?
This includes:
(3,3,3,3) vertex in J84
(3,3,3,3) vertex in J88
(3,3,3,4) vertex in J86/J87
(3,3,4,4) vertex in J86
(3,3,4,4) vertex in J88
(3,3,4,4) vertex in J89
(3,3,4,4) vertex in J90

As for your solutions, they look very interesting. 33333-05335 (two pentagonal rotundas, two pentagonal pyramids and one tetrahedron around an edge) fitting to J86 or J90 is unexpected. I presume only one specific arrangement of these cells work?

So, what next? Triangular bipyramids have the problem that there additional convexity constraints that must be checked. How about triangular prisms as verfs? Those have two triangles and three quadrangles and it might be interesting how the quadrangles fit together.
Marek14
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Re: General Approach--can 3D methods be generalized?

I recently determined some limits on the number of edges and faces of the CRF
polychora vertex figures.

There is a program named plantri (http://cs.anu.edu.au/~bdm/plantri/) with
which you can enumerate all polyhedra in three dimensions. With the qualifiers
of plantri you can limit the number of edges of the faces and the number of
edges of the polyhedra, the degree of the vertices and so on.

If you compute all the solid angles of the Johnson polyhedra you find that the
maximal number of faces is 22 for the regular tetrahedron. However, we know
that no more than 20 tetrahedra fit together in 4d to form an icosahedron. The
maximal number of vertices is much more difficult to determine. It could in
principle go up to 30=(12-2)*3, if all vertices are simple. However, in the
end I obtained 13, and my believe is that it is 12 namely the icosahedron.

What other limits do we have? The dihedral angles of the Johnson polyhedra
determine the maximal degree of the vertices, i.e. the number of faces that
can meet at a vertex. The minimal diehdral angle is 31.7175 from the
pentagonal cupola which would lead to a degree of 11. But since the vertex is
asymmetric, the maximal number of tiles has to be even, namely 10. But we find
that there is no polyhedron which fits into the gap between two pentagonal
cupolas. Thus this vertex does not exist. I have enumerated all possible edge
configurations and found further 10-degree vertices, but it was possible to
show that none of them exists. In the end the maximal degree was found to be 9
for 4 configurations which I could not eliminate.

How else could one reduce the possible vertex figures? Well by computing the
total solid angle: The smallest triangle has solid angle 00.4387, the
quadrangle 0.08774, the pentagon 0.173235 in units of the surface of a
sphere. Thus #3*0.04387+#4*0.08774+#5*0.173235 < 1.0.

So the number of possible vertex figures of a certain type is lower than given
in the following table:

#vert 4 5 6 7 8 9 10 11 12 13
1 2 7 33 249 2473 29846 394498 5528006 79919023 <- sum
#face
4 1 1
5 2 1 1
6 7 1 2 2 2
7 25 2 7 9 5 2
8 149 2 11 39 55 35 7
9 944 8 71 248 379 235 3
10 ^ 5 76 590 1976 2930 389
11 | 38 748 5290 16401 11684 3
12 sum 14 558 8309 50226 109398 409
13 219 7776 91966 449409 15892
14 50 4442 106558 1008926 213923
15 1404 78684 1400693 1296312
16 233 36528 1282828 4157926
17 9714 780953 7698651
18 1249 306470 8609942
19 70454 5875223
20 7595 2384890

Obviously, this estimate is completely useless. But it shows that it is
impossible by starting from the combinatorical vertex figure to determine the
allowed vertex figures in 4d. Because each number means only a type! There are
more than 100 vertex figures of type (4,4), 44 of type (5,5), 32 of tpye (6,6)
and maybe several hundred of the biypyramidral type (5,6) with
(#vert,#faces). I have determined the latter but not yet fully analysed them.
Although I could show that there are no infinite series for the bipyramidal
case (an the triangular pyramidal case (6,5), there seem to be vertex figures
with polygons with up to 169 edges!

To my opinion, it would be interesting to list the combinatorial content of all vertex figures of all known CRF polyhedral. I started such work, but I have not yet finished the segmentochora. Compared to the above list one finds that there is a lot of space at the top. The maxmium vertex figure is still the icosaedron with 12 vertices and 20 faces.
johannes@itap.physik.uni-stuttgart.de
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Re: General Approach--can 3D methods be generalized?

Very interesting. I think that the basic algorithm for CRF enumeration is not that hard, just requires a lot of space:

1. Start with a polyhedron set in 4D space and mark all its faces as "unpaired". These polyhedra will be Lv. 1 open-CRF (oCRF) polychora.
2. To make a Lv. n+1 oCRF from Lv. n, add a polyhedron to an unpaired face and mark that face as "paired".
3. Check gaps between faces. If a gap has two identical polygons, attempt to pull them together (bending cells into 4D). If successful (and convex), mark that face as paired and this becomes a new Lv. n oCRF that will be checked from step 3 (this one) onward.
4. For every gap, check whether a new polyhedron can be inserted into it. If at least one gap has nothing to fit into, the oCRF is invalid and eliminated from further considerations. If all gaps can be fitted with something, it's retained (the fittings will be added later). If there are no gaps (i.e. all faces are paired), the oCRF is closed and a CRF.
5. If you have a valid oCRF or CRF at this stage, perform combinatoric comparison with others Lv. n oCRFs found so far to eliminate duplicates.

This would have to be expanded a lot (especially step 3), but theoretically, it could allow for complete enumeration. Note that this needs 4D to work: it won't enumerate Johnson solids in 3D because the vertices in 3D are not fully determined by the polygons around them; in 4D, the cells around a vertex DO determine that vertex completely since there can be no continuous changes of angles.
Marek14
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Re: General Approach--can 3D methods be generalized?

Sorry, I just see that my matrices of the combinatorial vertex figures got mixed up completely. I will post them again when I have time.

My crown jewel data are from antiprism, which again are from the NIST data base, also used in Mathematica. The data of triangular vertices, however, are computed directly. There are errors in antiprism and NIST which I corrected. Especially J65, J66, J68, J88 and J87 are wrong.

My results were derived numerically, and a comparison to Marek's results show that very high precision does not seem to be required.

Johannes
johannes@itap.physik.uni-stuttgart.de
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Re: General Approach--can 3D methods be generalized?

Looking at the possible 6-vertex verfs, there seems to be these seven:

12 edges/8 faces: Tri-tetrahedron (a tetrahedron with pyramid built on two of its faces). 8 triangles. Vertex degrees: 3,3,4,4,5,5. Could be split (in two ways) into triangular bipyramid and tetrahedron.
12 edges/8 faces: Octahedron. 8 triangles. Vertex degrees: 4,4,4,4,4,4. Could be split (in three ways) into two square pyramids.

11 edges/7 faces: Augmented square pyramid (a square pyramid with tetrahedron on one of its faces). 6 triangles + 1 square. Vertex degrees: 3,3,3,4,4,5. Could be split in square pyramid + tetrahedron, or formed in two ways as a blend of triangular bipyramid + tetrahedron.
11 edges/7 faces: Diagonal square wedge (diagonally oriented line connected to base square). 6 triangles + 1 square. Vertex degrees: 3,3,4,4,4,4. Could be formed as a blend of two square pyramids.

10 edges/6 faces: Pentagonal pyramid. 5 triangles + 1 pentagon. Vertex degrees: 3,3,3,3,3,5. Could be formed as a blend of square pyramid and tetrahedron in five ways.
10 edges/6 faces: Broken triangular prism (triangular prism with one square face broken in two triangles). 4 triangles + 2 squares. Vertex degrees: 3,3,3,3,4,4.

9 edges/5 faces: Triangular prism. 2 triangles + 3 squares. Vertex degrees: 3,3,3,3,3,3.
Marek14
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Re: General Approach--can 3D methods be generalized?

https://www.uwgb.edu/dutchs/symmetry/polynum0.htm

where there are plots of the combinatorial polyhedra.

Johannes
johannes@itap.physik.uni-stuttgart.de
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Re: General Approach--can 3D methods be generalized?

And here are the combinatorical vertex figures again.

Johannes
Attachments
enum.pdf
Vertex figure data
johannes@itap.physik.uni-stuttgart.de
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Re: General Approach--can 3D methods be generalized?

Hm, last update from 1999. I wonder if that profesor is still alive...
Marek14
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