## Uniform Euclidean honeycombs

Higher-dimensional geometry (previously "Polyshapes").

### Uniform Euclidean honeycombs

I browsed through the article on uniform tesselations and noticed one interesting things -- the list of 28 Euclidean honeycombs was never proved complete.

I looked at my list of dihedral angles of polyhedra and tried to determine whether it can be used to finish the proof.

Basically, all dihedral angles around an edge in 3D uniform honeycomb must add to 360 degrees. This means we can make a complete list of edges, just like we can make a list of allowed vertices for 2D tilings.

So I wrote a simple Python script that would find valid combinations. I listed prisms and antiprisms up to 12. Since my list of dihedral angles is decimal and not exact, I instructed it to output all combinations with sum between 359.99 and 360.01. Here is what I found:

Type 3_1: 60, 150, 150 - this corresponds to triangular prism, dodecagonal prism and dodecagonal prism (4-4-4). It occurs in O7-thaph.

Type 3_2: 70.5288, 144.736, 144.736 - this corresponds to tetrahedron, rhombicuboctahedron and rhombicuboctahedron (3-3-4). It occurs in O26-ratoh.
Or it can correspond to truncated tetrahedron, truncated cuboctahedron and truncated cuboctahedron (6-6-4). It occurs in O28-gratoh.

Type 3_3: 90, 120, 150 - this corresponds to cube, hexagonal prism and dodecagonal prism (4-4-4). It occurs in O9-otathaph.

Type 3_4: 90, 125.264, 144.736 - this corresponds to cube, cuboctahedron and rhombicuboctahedron (4-4-3). It occurs in O17-srich.
Or it can correspond to cube, truncated octahedron and truncated cuboctahedron (4-4-6). It occurs in O18-grich.
Or it can correspond to octagonal prism, truncated cube and rhombicuboctahedron (4-8-3). It occurs in O19-prich.
Or it can correspond to octagonal prism, truncated cuboctahedron and truncated cuboctahedron (4-8-6). It occurs in O20-otch.

Type 3_5: 90, 135, 135 - this corresponds to cube, rhombicuboctahedron and rhombicuboctahedron (4-4-4). It occurs in O17-srich and O26-ratoh.
Or it can correspond to cube, rhombicuboctahedron and octagonal prism (4-4-4). It occurs in O19-prich.
Or it can correspond to cube, truncated cuboctahedron and truncated cuboctahedron (4-4-8). It occurs in O18-grich.
Or it can correspond to cube, octagonal prism and octagonal prism (4-4-4). It occurs in O6-tassiph.
Or it can correspond to truncated cube, truncated cuboctahedron and truncated cuboctahedron (8-8-4). It occurs in O28-gratoh.
Or it can correspond to octagonal prism, rhombicuboctahedron and truncated cuboctahedron (8-4-4).
Or it can correspond to octagonal prism, truncated cuboctahedron and octagonal prism (4-8-4). It occurs in O20-otch.

Type 3_6: 100.812, 116.565, 142.623 - this corresponds to pentagonal antiprism, dodecahedron and icosidodecahedron (3-5-5).

Type 3_7: 108, 108, 144 - this corresponds to pentagonal prism, pentagonal prism and decagonal prism (4-4-4).

Type 3_8: 109.471, 125.264, 125.264 - this corresponds to octahedron, cuboctahedron and cuboctahedron (3-3-4). It occurs in O15-rich.
Or it can correspond to octahedron, truncated cube and truncated cube (3-3-8). It occurs in O14-tich.
Or it can correspond to truncated tetrahedron, cuboctahedron and truncated octahedron (6-3-4). It occurs in O25-tatoh.
Or it can correspond to truncated tetrahedron, truncated cube and truncated cuboctahedron (6-3-8). It occurs in O28-gratoh.
Or it can correspond to truncated octahedron, truncated octahedron and truncated octahedron (6-6-4). It occurs in O16-batch.
Or it can correspond to truncated octahedron, truncated cuboctahedron and truncated cuboctahedron (6-6-8). It occurs in O18-grich.

Type 3_9: 120, 120, 120 - this corresponds to hexagonal prism, hexagonal prism and hexagonal prism (4-4-4). It occurs in O3-hiph.

Type 4_1a: 60, 60, 90, 150 - this corresponds to triangular prism, triangular prism, cube and dodecagonal prism (4-4-4-4).

Type 4_1b: 60, 90, 60, 150 - this corresponds to triangular prism, cube, triangular prism and dodecagonal prism (4-4-4-4).

Type 4_2a: 60, 60, 120, 120 - this corresponds to triangular prism, triangular prism, hexagonal prism and hexagonal prism (4-4-4-4).

Type 4_2b: 60, 120, 60, 120 - this corresponds to triangular prism, hexagonal prism, triangular prism and hexagonal prism (4-4-4-4). It occurs in O5-thiph.

Type 4_3a: 60, 70.5288, 109.471, 120 - no legal combination of faces exists.
Type 4_3b: 60, 109.471, 70.5288, 120 - no legal combination of faces exists.
Type 4_3c: 60, 70.5288, 120, 109.471 - no legal combination of faces exists.
These three would always require dihedral angle 60 (two square faces of triangular prism) next to either 70.5288 or 109.471 which can both only have triangular or hexagonal faces.

Type 4_4a: 60, 90, 90, 120 - this corresponds to triangular prism, n-gonal prism, n-gonal prism and hexagonal prism (4,n,n,4).

Type 4_4b: 60, 90, 120, 90 - this corresponds to triangular prism, cube, hexagonal prism and cube (4,4,4,4). It occurs in O8-rothaph.

Type 4_5a: 70.5288, 70.5288, 109.471, 109.471 - this corresponds to tetrahedron, tetrahedron, octahedron, octahedron (3-3-3-3). It occurs in O22-gytoh.
Or it can correspond to tetrahedron, tetrahedron, truncated tetrahedron, truncated tetrahedron (3-3-3-6).
Or it can correspond to truncated tetrahedron, truncated tetrahedron, truncated tetrahedron, truncated tetrahedron (6-6-6-3).
Or it can correspond to truncated tetrahedron, truncated tetrahedron, truncated octahedron, truncated octahedron (6-6-6-6).

Type 4_5b: 70.5288, 109.471, 70.5288, 109.471 - this corresponds to tetrahedron, octahedron, tetrahedron, octahedron (3-3-3-3). It occurs in O21-octet, in O22-gytoh, in O23-etoh, in O24-gyetoh.
Or it can correspond to tetrahedron, truncated tetrahedron, truncated tetrahedron, truncated tetrahedron (3-3-6-6). It occurs in O27-batatoh.
Or it can correspond to truncated tetrahedron, truncated octahedron, truncated tetrahedron, truncated octahedron (6-6-6-6). It occurs in O25-tatoh.

Type 4_6a: 70.5288, 90, 90, 109.471 - this corresponds to tetrahedron, triangular prism, triangular prism, octahedron (3-3-4-3). It occurs in O23-etoh, in O24-gyetoh.
Or it can correspond to tetrahedron, triangular prism, hexagonal prism, truncated tetrahedron (3-3-4-6).
Or it can correspond to truncated tetrahedron, hexagonal prism, triangular prism, truncated tetrahedron (6-6-4-3).
Or it can correspond to truncated tetrahedron, hexagonal prism, hexagonal prism, truncated otahedron (6-6-4-6).

Type 4_6b: 70.5288, 90, 109.471, 90 - no legal combination of faces exists.
90 edge must always either have one square or (in case of truncated cube) two octagons, but none of these would fit between 70.5288 edge and 109.471 edge which both can only have triangles and hexagons.

Type 4_7: 90, 90, 90, 90 - this corresponds to cube, cube, cube and cube (4,4,4,4). It occurs in O1-chon, O4-etoph, O13-gyetaph.
Or it can (more generally) correspond to m-gonal prism, m-gonal prism, n-gonal prism and n-gonal prism (4, m, 4, n). It occurs in O2-tiph for m=n=3, in O3-hiph for m=n=6, in O4-etoph for m=n=3 and m=3, n=4, in O5-thiph for m=3, n=6, in O6-tassiph for m=4, n=8 and m=n=8, in O7-thaph for m=3, n=12 and m=n=12, in O8-rothaph for m=3, n=4 and m=4, n=6, in O9-otathaph for m=4, n=6, m=4, n=12 and m=6, n=12, in O10-sassiph for m=n=3 or m=3, n=4, in O11-snathaph for m=n=3 or m=3, n=6, in O12-gytoph for m=n=3, in O13-gyetaph for m=n=3 and m=3, n=4.
Or it can correspond to cube, octagonal prism, truncated cube and octagonal prism (4,4,8,8). It occurs in O19-prich.
Or it can correspond to truncated cube, truncated cube, truncated cube and truncated cube (8,8,8,8). It occurs in O14-tich.
Or it can correspond to truncated cube, truncated cube, octagonal prism and octagonal prism (8,8,8,4).

Type 5_1: 60, 60, 60, 60, 120 - this corresponds to triangular prism, triangular prism, triangular prism, triangular prism and hexagonal prism (4-4-4-4-4). It occurs in O11-snathaph.

Type 5_2a: 60, 60, 60, 70.5288, 109.471 - no legal combination of faces exists.
Type 5_2b: 60, 60, 70.5288, 60, 109.471 - no legal combination of faces exists.
For the same reason as in Type 4_3.

Type 5_3a: 60, 60, 60, 90, 90 - this corresponds to triangular prism, triangular prism, triangular prism, n-gonal prism and n-gonal prism (4-4-4-4-n). It occurs in O4-etoph for n=4, in O12-gytoph for n=3.

Type 5_3b: 60, 60, 90, 60, 90 - this corresponds to triangular prism, triangular prism, cube, triangular prism and triangular prism (4-4-4-4-4). It occurs in O10-sassiph.

Type 6: 60, 60, 60, 60, 60, 60 - this corresponds to triangular prism, triangular prism, triangular prism, triangular prism, triangular prism and triangular prism (4-4-4-4-4-4). It occurs in O2-tiph, in O23-etoh, in O24-gyetoh.

So, what can we glean from this list? It suggests some lines of research.

First let's have a look at edge combinations that don't occur in any known honeycomb:

Type 3_5 corresponding to octagonal prism, rhombicuboctahedron and truncated cuboctahedron (8-4-4). This doesn't seem to work. The 4-6 angle thus created is smaller than 90 degrees, and that's not allower.

Type 3_6 corresponding to to pentagonal antiprism, dodecahedron and icosidodecahedron (3-5-5). This is a weird edge. Stella confirms that the sum IS exact, but unfortunately, any uniform tiling would also need an edge corresponding to 3-3 in pentagonal antiprism (or icosahedron), which is 138.19 and doesn't occur in any legal combination.

Type 3_7 corresponding to pentagonal prism, pentagonal prism and decagonal prism (4-4-4). No uniform tesselation with this edge is possible as you'd have to separate lateral faces of pentagonal prism into alternating sets, which is impossible.

Type 4_1a and 4_1b -- can't be made uniform. So can't Type 4_2a.

Type 4_4a corresponding to triangular prism, n-gonal prism, n-gonal prism and hexagonal prism (4,n,n,4). Looks very weird. Most likely can't be made uniform.

Type 4_5a:
Three possible correspondences don't normally occur:
Tetrahedron, tetrahedron, truncated tetrahedron, truncated tetrahedron (3-3-3-6) -- probably can't be made uniform.
Truncated tetrahedron, truncated tetrahedron, truncated tetrahedron, truncated tetrahedron (6-6-6-3) -- would belong to the same tiling as the previous possibility, but it looks like it won't work.
Truncated tetrahedron, truncated tetrahedron, truncated octahedron, truncated octahedron (6-6-6-6) -- seems this doesn't work either.

Type 4_6a:
Also three possible correspondences that don't normally occur:
Tetrahedron, triangular prism, hexagonal prism, truncated tetrahedron (3-3-4-6).
Truncated tetrahedron, hexagonal prism, triangular prism, truncated tetrahedron (6-6-4-3).
Truncated tetrahedron, hexagonal prism, hexagonal prism, truncated otahedron (6-6-4-6).
The first two suggest an elongated honeycomb together which, unfortunately, is not uniform.
The third one also suggests a nonuniform elongated honeycomb.

Type 4_7 corresponding to truncated cube, truncated cube, octagonal prism and octagonal prism (8,8,8,4). Can't be made uniform, would lead to elongated honeycomb that uses Johnson solids.

So these are all exhausted. So if there is another honeycomb, it would have to either use one of the existing edges in a new way (I suppose that was checked) or it would have to use prisms/antiprisms based on 13-gon or more. That doesn't look too likely. Perhaps a general math argument can be developed to exclude it?

There's one more anomaly, and that's Type 3_5 which, in its "cube, rhombicuboctahedron, rhombicuboctahedron" form occurs in two different honeycombs that are not obviously related: O17-srich and O26-ratoh. I wonder if that could be used to create some sort of mix between them...
Marek14
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### Re: Uniform Euclidean honeycombs

One thing here is that vertex figures themselves do not suffice here. LPA2 and LPB2 have the same vertex figure. viz x3o || x3x |q| o3o , ie a triangular cupola attatched to a hexagonal pyramid, the sloping edge as q.

But the list as i see is is AAA 1 (cubic), AB = 10 2d*1d, C gives q 1, qr 4, qrr 7, and the laminates 5 all together 17, the total is 28.

There are 12 wythoffian ones and 5 non-wythoffian laminates. LA2, and LPP are both wythoffian laminates.
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wendy
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### Re: Uniform Euclidean honeycombs

wendy wrote:One thing here is that vertex figures themselves do not suffice here. LPA2 and LPB2 have the same vertex figure. viz x3o || x3x |q| o3o , ie a triangular cupola attatched to a hexagonal pyramid, the sloping edge as q.

But the list as i see is is AAA 1 (cubic), AB = 10 2d*1d, C gives q 1, qr 4, qrr 7, and the laminates 5 all together 17, the total is 28.

There are 12 wythoffian ones and 5 non-wythoffian laminates. LA2, and LPP are both wythoffian laminates.

I know vertex figures don't suffice by themselves. What I was trying to determine was if there are any possible vertex figures that were not taken into account before.
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### Re: Uniform Euclidean honeycombs

I wonder if these could reveal some interesting "CRF" honeycombs that wouldn't need to be uniform, or even periodical... Dodecahedra, icosidodecahedra and suitable parts of icosahedra might lead to something.
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### Re: Uniform Euclidean honeycombs

I don't think that interesting CRF polytopes with dodecahedra etc would come naturally, but it's always a good foray into that area.

The main class-2 symmetry in 3D is one i designate '4V', or o5o5/2oAo. I don't think it has any other wythoff symbols, but the four-dimensional 5V abounds in them (eg {5,3,3,5/2} or {3,3,5/2:3,3}.

The octahedral groups might lead to something more interesting, especially when one notes a tiling including xxxx4oxxxo&#xt could be co-erced into xxxo4oxxx&#xt (a johnson-figure).

A chap down in Magyar named Sándor Kabai has been doing some interesting things with Mathematica and all things icosahedral, such as nested fractals, etc. One area that might prove fruitful is to construct a kind of tiling out of the prolate and oblate 'golden rhombohedra', these make a rhombo-tricontahedron o3m5o.
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### Re: Uniform Euclidean honeycombs

Take it that "CRF" wrt. euclidean honeycombs should ask for convex and regular-faced cells, which then tile the 3-space without gaps nor overlaps.

Of special interest in this context then would be those, which not only incorporate platonics, archimedeans, and/or (anti)prisms, but whenever some johnsonians play in.

Therein you might be interested then e.g. in the following small collection (all being periodic, with N → ∞)
1. pextoh - with 2N vertices, resp.
N elongated square dipyramids (J15, esquidpy), and
2N tetrahedra (tet)
2. ditoh - with N vertices, resp.
N octahedra (oct), and
N trigonal dipyramids (J12, tridpy)
3. gyeditoh - with 2N vertices, resp.
2N elongated trigonal pyramids (J7, etripy), and
N octahedra (oct)
4. editoh - with 2N vertices, resp.
N elongated trigonal dipyramids (J14, etidpy),
N octahedra (oct), and
N trigonal prisms (trip)
5. erich - with 6N vertices, resp.
N elongated trigonal gyrobicupolae (J36, etigybcu),
N octahedra (oct), and
2N trigonal prisms (trip)
6. gyerich - with 6N vertices, resp.
N elongated trigonal bicupolae (J35, etobcu),
N octahedra (oct), and
2N trigonal prisms (trip)
7. gyrich - with 3N vertices, resp.
N octahedra (oct), and
N trigonal orthobicupolae (J27, tobcu)
8. pexrich - with 5N vertices, resp.
N cuboctahedra (co),
N cubes (cube), and
N elongated square dipyramids (J15, esquidpy)
9. pacsrich - with 8N vertices, resp.
N cuboctahedra (co),
2N cubes (cube), and
N square orthobicupolae (J28, squobcu)
10. 3Q4-T-2P8-P4 - with 8N vertices, resp.
N cubes (cube),
N octagonal prisms (op),
2N square cupolae (J4, squacu), and
2N tetrahedra (tet)
11. 6Q4-2T - with 4N vertices, resp.
2N square cupolae (J4, squacu), and
2N tetrahedra (tet)
12. pacratoh - with 4N vertices, resp.
N square orthobicupolae (J28, squobcu), and
2N tetrahedra (tet)
13. 6Q3-2S3-gyro - with 3N vertices, resp.
N octahedra (oct), and
2N trigonal cupolae (J3, tricu)
14. 6Q3-2S3-ortho - (same statistics, but different arrangement)
15. 3Q3-S3-2P6-2P3-gyro - with 6N vertices, resp.
N hexagonal prisms (hip),
N octahedra (oct),
2N trigonal cupolae (J3, tricu), and
2N trigonal prisms (trip)
16. 3Q3-S3-2P6-2P3-ortho - (same statistics, but different arrangement)
17. cube-doe-bilbiro - with 14N vertices, resp.
N cubes (cube),
N dodecahedra (doe), and
3N bilunabirotundae (J91, bilbiro)
18. 5Y4-4T-4P4 - with 2N vertices, resp.
N cubes (cube),
2N square pyramids (J1, squippy), and
2N tetrahedra (tet)
19. 5Y4-4T-6P3-sq-para - with N vertices, resp.
N square pyramids (J1, squippy),
N tetrahedra (tet), and
N trigonal prisms (trip)
20. 5Y4-4T-6P3-sq-skew - (same statistics, but different arrangement)
21. 10Y4-8T-0 - with N vertices, resp.
2N square pyramids (J1, squippy), and
2N tetrahedra (tet)
22. 10Y4-8T-1-alt - (same statistics, but different arrangement)
23. 10Y4-8T-1-hel - (same statistics, but different arrangement, itself occurs in 2 enantiomeric forms!)
24. 10Y4-8T-2-alt - (same statistics, but different arrangement)
25. 10Y4-8T-2-hel - (same statistics, but different arrangement, itself occurs in 2 enantiomeric forms!)
26. 10Y4-8T-3 - (same statistics, but different arrangement)
27. 5Y4-4T-6P3-tri-0 - with 2N vertices, resp.
2N square pyramids (J1, squippy),
2N tetrahedra (tet), and
2N trigonal prisms (trip)
28. 5Y4-4T-6P3-tri-1-alt - (same statistics, but different arrangement)
29. 5Y4-4T-6P3-tri-1-hel - (same statistics, but different arrangement, itself occurs in 2 enantiomeric forms!)
30. 5Y4-4T-6P3-tri-2-alt - (same statistics, but different arrangement)
31. 5Y4-4T-6P3-tri-2-hel - (same statistics, but different arrangement, itself occurs in 2 enantiomeric forms!)
32. 5Y4-4T-6P3-tri-3 - (same statistics, but different arrangement)

(anyone aware of further ones?)

Wendy wrote:I don't think that interesting CRF polytopes with dodecahedra etc would come naturally, but it's always a good foray into that area.
Hehe, what about Weimholt's one, mentioned here as nr. 17?

--- rk
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