## List of uniform honeycombs

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### List of uniform honeycombs

Hello everybody! Polychoronlover here. I joined back in 2012, but I never posted because then I kind of forgot about polytopes for a while. However, I recently made a list of all uniform tessellations of 3-dimensional space (honeycombs) that I know of, including non-convex ones. (Kind of like a 3D version of this list of uniform tilings http://en.wikipedia.org/wiki/Uniform_tiling#Expanded_lists_of_uniform_tilings) It was mostly inspired by Jonathan Bowers' work on uniform polychora, including the symbols I used. I attached my list here.
Attachments honeycombs..rtf
List of all uniform honeycombs I know about, most of which I independently discovered.
polychoronlover
Dionian

Posts: 51
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### Re: List of uniform honeycombs

Hmmm,
as I get it, I'll have to add some further symmetry groups to my page on euclidean tesselations, at least those (any other ones as well?)
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`(o(oo),o) = o3o3o3o3/2*a3*co'oo"o = o4o3o4/3oo"oo"o = o4/3o3o4/3oo8"o = o3o3o *b4/3oo'(o'o"o) = o4o4o4/3o3*bo'(o"o'o) = o4o4/3o4o3*bo"(o"o'o) = o4/3o4/3o4o3*bo(o"o~o') = o3o4/3oinfino4*bo(o,o~o) = o3o3/2oinfino3*b(oo'o"o) = o3o4o4/3o3*a(("o~(o'o)~o")') = oinfino4/3oinfino4/3*a4*c *b4*d(o"o~o') o~o = o4/3oinfino4*a oinfino(o,o~o) o~o = o3/2oinfino3*a oinfinoo'o"o o~o = o4o4/3o oinfinoo"o"o o~o = o4/3o4/3o oinfinooo^6/5o o~o = o3o6/5o oinfino(o^6o^6o,) o~o = o6o6o3/2*a oinfino(o~o^6/5o^6) o~o = oinfino6/5o6*a oinfino(o^6o^6/5o) o~o = o6o6/5o3*a oinfino(o"o~o') o = o4/3oinfino4*a o(o,o~o) o = o3/2oinfino3*a oo'o"o o = o4o4/3o oo"o"o o = o4/3o4/3o ooo^6/5o o = o3o6/5o o(o^6o^6o,) o = o6o6o3/2*a o(o"o~o') o = o4/3oinfino4*a o(o~o^6/5o^6) o = oinfino6/5o6*a o(o^6o^6/5o) o = o6o6/5o3*a oo~o o o = oinfino o oo~o o o~o = oinfino o oinfino`

Also some of the therefrom derived honeycombs are already listed in that file of polychoronlover. I would like to know whether Jonathan Bowers already has elaborated all the according OBSAs (official Bowers style abbreviations, ), or could else provide them?
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`4. (o(oo),x) = o3o3o3x3/2*a3*c | Tetrahedral hemitriangular tiling honeycomb. Cells are tets and trats (triangular tilings). Verf is an oho. Symbol is (o(oo),x). This one has hemicubic honeycomb symmetry, but is only wythoffian if it has quarter cubic honeycomb symmetry, also known as cyclotetrahedral symmetry.18. o\$'o = o3x3x *b4o, (oxxx) = o3x3x3x3*a | Truncated tetrahedral-octahedral honeycomb - also known as tatoh. Symbol is o\$'o, or (oxxx). Cells are tuts, toes, and coes. Verf is a rectangular pyramid, the rectangle is a co verf.19. (o(xx),x) = x3o3x3x3/2*a3*c | Truncated tetrahedral hemitriangular-tiling honeycomb. Symbol is (o(xx),x). Cells are tuts, hexats (as truncated trats), and ohoes. Verf is a crossed quadrilateral pyramid. This has hemicubic honeycomb symmetry.21. o'ox"x = o4o3x4/3x | Quasitruncated cubic honeycomb. Symbol is o'ox"x. Cells are quiths and octs, verf is a flat square pyramid.33. (o(xx),o) = x3o3x3o3/2*a3*c | Rectified tetrahedral hemitriangular-tiling honeycomb. Symbol is (o(xx),o). Cells are octs, ohoes, and thats. Verf is a crossed quadrilateral prism.56. o'(x'x"o) = o4x4x4/3o3*b | Retrosphenoverted cubicuboctahedral cuboctahedral tomosquare tiling honeycomb. Symbol is o'(x'x"o). Cells are soccoes, coes, and tosquats.62. o'(x"x'o) = o4x4/3x4o3*b | Wavaccocotsoth = Sphenoverted cubicuboctahedral cuboctahedral tomosquare tiling honeycomb. Symbol is o'(x"x'o). Cells are goccoes, coes, and quitsquats.63. o'xo"x = o4x3o4/3x, x6"x = x3o3x *b4/3x | Quasirhombated cubic honeycomb. Symbol is o'xo"x, or x6"x. Cells are quercoes, cubes, and coes.72. o'(x'x"x) = o4x4x4/3x3*b | Small cuboctatruncated cuboctahedral tomoctahedral truncated square tiling honeycomb. Symbol is o'(x'x"x). Cells are cotcoes, toes, and tosquats.73. o'(x"x'x) = o4x4/3x4x3*b | Great cuboctatruncated cuboctahedral tomoctahedral truncated square tiling honeycomb. Symbol is o'(x"x'x). Cells are cotcoes, toes, and quitsquats.74. o\$"x = o3x3x *b4/3x | Great quasirhombated tetrahedral-octahedral honeycomb. Symbol is o\$"x. Cells are quitcoes, quiths, and tuts.75. o'xx"x = o4x3x4/3x, x\$"x = x3x3x *b4/3x | Great quasirhombated cubic honeycomb. Symbol is o'xx"x, can also be x\$"x. Cells are quitcoes, cubes, and toes.77. x'(x'x"x) = x4x4x4/3x3*b | Small rhombicuboctahedral cubicuboctahedral prismatic tomosquare tiling honeycomb. Symbol is x'(x'x"x). Cells are gircoes, cotcoes, stops, and tosquats. Verf is an irregular tetrahedron.78. x"(x"x'x) = x4/3x4/3x4x3*b | Great rhombicuboctahedral cubicuboctahedral prismatic tomosquare tiling honeycomb. Symbol is x"(x"x'x). Cells are quitcoes, cotcoes, ops, and quitsquats. Verf is an irregular tetrahedron.79. x"xx"x = x4/3x3x4/3x | Omniquasitruncated cubic honeycomb. Symbol is x"xx"x. Cells are quitcoes and stops. Verf is a skewed disphenoid.81. x'(x'x"o) = x4x4x4/3o3*b | Small cubicuboctahedral tomocubic prismatic truncated square tiling honeycomb. Symbol is x'(x'x"o). Cells are soccoes, tics, cubes, and tosquats.83. x"(x"x'o) = x4/3x4/3x4o3*b | Goccotocpitsith = Great cubicuboctahedral tomocubic prismatic truncated square tiling honeycomb. Symbol is x"(x"x'o). Cells are goccoes, quiths, cubes, and quitsquats.84. x'ox"x = x4o3x4/3x | Prismatoquasirhombated cubic honeycomb. Symbol is x'ox"x. Cells are quercoes, quiths, cubes, and stops.86. o(o"x~x') = o3o4/3xinfinx4*b | Circumfacetocubic hemialternate square tiling honeycomb. Symbol is o(o"x~x') where ~ represents infinity. Cells are cubes and shas. Verf is a triangular crossed prism. This has doubled runcic cubic symmetry.91. o(o,x~x) = o3o3/2xinfinx3*b | Circumfacetotetrahedral hemialternate triangular tiling honeycomb. Symbol is o(o,x~x). Cells are tets and thas. Verf is a triangular crossed prism. This has doubled cyclotetrahedral symmetry.94. (oo'x"x) = o3o4x4/3x3*a | Gotactictosquath = Great tetrahedral cubic tomocubic tomosquare tiling honeycomb. Symbol is (oo'x"x). Cells are tets, cubes, quiths, and quitsquats. Verf is a flat triangular antipodium.97. o6"x = o3o3x *b4/3x | Quasirhombated tetrahedral octahedral honeycomb. Symbol is o6"x. Cells are tets, cubes, and quercoes. Verf is a triangular retropodium.102. (oo"x'x) = (ox'x"o) = o3x4x4/3o3*a | Small tetrahedral cubic tomocubic tomosquare tiling honeycomb. Symbol is (oo"x'x). Cells are tets, cubes, tics, and tosquats. Verf is a triangular retroantipodium.108. o'(o'x"x) = o4o4x4/3x3*b | Great cubicuboctahedral octahedral square tiling honeycomb. Symbol is o'(o'x"x). Cells are goccoes, octs, and squats. Verf is a flattish square podium.124. o'(o"x'x) = o4/3o4/3x4x3*b | Small cubicuboctahedral octahedral square tiling honeycomb. Symbol is o'(o"x'x). Cells are soccoes, octs, and squats. Verf is a crossed square podium.131. x'(o'x"x) = x4o4x4/3x3*b | Small rhombicuboctahedral cubicuboctahedral prismatic square tiling honeycomb. Symbol is x'(o'x"x). Cells are sircoes, goccoes, stops, and squats.140. (("x~(o'x)~x")') = oinfinx4/3xinfinx4/3*a4*c *b4*d | Discubicuboctahedral alternate square disquare apeirogonal tiling honeycomb. Symbol is (("x~(o'x)~x")') Cells are soccoes, goccoes, shas, and disquas.143. x'(o"x'x) = x4o4/3x4x3*b | Great rhombicuboctahedral cubicuboctahedral prismatic square tiling honeycomb. Symbol is x'(o"x'x). Cells are quercoes, soccoes, ops, and squats.144. x(o"x~x') = x3o4/3xinfinx4*b | Dirhombicuboctahedral apeiroprismatic alternate square tiling honeycomb. Symbol is x(o"x~x'). Cells are sircoes, quercoes, azips, and shas.168. o(o,o~x) = o3o3/2oinfinx3*b | Ditetrahedronary tetrahedral hemiditrigonal triangular tiling honeycomb. Symbol is o(o,o~x). Cells are tets and ditathas. Verf is an inflectoverted co, which looks like four tets attached at a point. Has cyclotetrahedral symmetry.178. (o"x~x') o~x = o4/3xinfinx4*a oinfinx | Square hemiapeirogonal prismatic honeycomb. Symbol is (o"x~x') o~x. Cells are cubes and azips. This is from the chon regiment, because chon can be considered as the square tiling prismatic honeycomb. Verf is a crossed quadrilateral dipyramid, which has 180 degree dihedral angles between two right triangles.182. (o,o~x) o~x = o3/2oinfinx3*a oinfinx | Ditrigonal triangular hemiapeirogonal prismatic honeycomb. Symbol is (o,o~x) o~x. Cells are trips and azips. Verf is a dipyramid of a ditrigon.185. (o,x~x) o~x = o3/2xinfinx3*a oinfinx | Triangular hemiapeirogonal prismatic honeycomb. Symbol is (o,x~x) o~x. Cells are trips and azips. Verf is a crossed quadrilateral dipyramid.190. o'x"x o~x = o4x4/3x oinfinx, x"x"x o~x = x4/3x4/3x oinfinx | Quasitruncated square prismatic honeycomb. Symbol is o'x"x o~x, also x"x"x o~x. Verf is an isosceles triangle dipyramid.192. ox^6/5x o~x = o3x6/5x oinfinx | Quasitruncated hexagonal prismatic honeycomb. Symbol is ox^6/5x o~x. Cells are trips and 12/5-ps (dodecagrammic prisms). Verf is an isosceles triangle dipyramid.194. (o^6x^6x,) o~x = o6x6x3/2*a oinfinx | Small trihexahexagonal prismatic honeycomb. Symbol is (o^6x^6x,) o~x. Cells are trips, hips, and twips. Verf is a crossed trapezoid dipyramid.196. (o~x"x') o~x = (x"x~o') o~x = x4/3xinfino4*a oinfinx | Small disquare apeirogonal prismatic honeycomb. Symbol is (o~x"x') o~x. Cells are cubes, stops, and azips. Verf is a trapezoid dipyramid, which has a 180 degree dihedral angle.198. (o~x'x") o~x = (x"o~x') o~x = x4/3oinfinx4*a oinfinx | Great disquare apeirogonal prismatic honeycomb. Symbol is (o~x'x") o~x. Cells are cubes, ops, and azips. Verf is a crossed trapezoid dipyramid, which has a 180 degree dihedral angle.201. (o~x^6/5x^6) o~x = oinfinx6/5x6*a oinfinx | Small dihexagonal apeirogonal prismatic honeycomb. Symbol is (o~x^6/5x^6) o~x. Cells are hips, 12/5-ps, and azips. Verf is a trapezoid dipyramid, which has a 180 degree dihedral angle.203. (o~x^6x^6/5) o~x = (x~o^6/5x^6) o~x = xinfino6/5x6*a oinfinx | Great dihexagonal apeirogonal prismatic honeycomb. Symbol is (o~x^6x^6/5) o~x. Cells are hips, twips, and azips. Verf is a crossed trapezoid dipyramid, which has a 180 degree dihedral angle.206. (o^6x^6/5x) o~x = o6o6/5x3*a oinfinx | Great trihexahexagonal prismatic honeycomb. Symbol is (o^6x^6/5x) o~x. Cells are trips, hips, and 12/5-ps. Verf is a trapezoid dipyramid.207. xo^6/5x o~x = x3o6/5x oinfinx | Quasirhombitrihexagonal prismatic honeycomb. Symbol is xo^6/5x o~x. Cells are trips, cubes, and hips. Verf is a crossed trapezoid dipyramid.210. (x~x"x') o~x = (x"x~x') o~x = x4/3xinfinx4*a oinfinx | Disquare apeirogonal prismatic honeycomb. Symbol is (x~x"x') o~x. Cells are ops, stops, and azips. Verf is a scalene triangle dipyramid, which has a 180 degree dihedral angle.212. (x~x^6/5x^6) o~x = xinfinx6/5x6*a oinfinx | Dihexagonal apeirogonal prismatic honeycomb. Symbol is (x~x^6/5x^6) o~x. Cells are twips, 12/5-ps, and azips. Verf is a scalene triangle dipyramid, which has a 180 degree dihedral angle.214. x'x"x o~x = x4x4/3x oinfinx | Rhombidisquare prismatic honeycomb. Symbol is x'x"x o~x. Cells are cubes, ops, and stops. Verf is a scalene triangle dipyramid.215. (x^6/5x^6x) o~x = x6x6/5x3*a oinfinx | Trihexatruncated trihexagonal prismatic honeycomb. Symbol is (x^6/5x^6x) o~x. Cells are hips, twips, and 12/5-ps. Verf is a scalene triangular dipyramid.216. xx^6/5x o~x = x3x6/5x oinfinx | Omniquasitruncated trihexagonal prismatic honeycomb. Symbol is xx^6/5x o~x. Cells are cubes, hips, and 12/5-ps. Verf is a scalene triangle dipyramid.218. o's"s o~x = o4s4/3s oinfinx, s"s"s o~x = s4/3s4/3s oinfinx | Retrosnub square prismatic honeycomb. Symbol is o's"s o~x, or s"s"s o~x. Cells are trips and cubes. Verf is an irregular pentagrammic dipyramid.219. (s~s"s') o~x = (s"s~s') o~x = s4/3sinfins4*a oinfinx | Snub square apeirogonal prismatic honeycomb. Symbol is (s~s"s') o~x. Cells are trips, cubes, and azips. Verf is an irregular nonconvex hexagonal dipyramid.225. (o"x~x') x = o4/3xinfinx4*a x | Square hemiapeirogonal tiling prism. Symbol is (o"x~x') x. Verf is a crossed quadrilateral pyramid.228. (o,o~x) x = o3/2oinfinx3*a x | Ditrigonal triangular tiling prism. Symbol is (o,o~x) x. Verf is a pyramid of a ditrigon.230. (o,x~x) x = o3/2xinfinx3*a x | Triangular hemiapeirogonal tiling prism. Symbol is (o,x~x) x. Verf is a crossed quadrilateral pyramid.233. o'x"x x = o4x4/3x x, x"x"x x = x4/3x4/3x x | Quasitruncated square tiling prism. Symbol is o'x"x x, also x"x"x x. Verf is an isosceles triangle pyramid.235. ox^6/5x x = o3x6/5x x | Quasitruncated hexagonal tiling prism. Symbol is ox^6/5x x. Verf is an isosceles triangle pyramid.237. (o^6x^6x,) x = o6x6x3/2*a x | Small trihexahexagonal tiling prism. Symbol is (o^6x^6x,) x. Verf is a crossed trapezoid pyramid.239. (o~x"x') x = (x"x~o') x = x4/3xinfino4*a x | Small disquare apeirogonal tiling prism. Symbol is (o~x"x') x. Verf is a trapezoid pyramid.240. (o~x'x") x = (x"o~x') x = x4/3oinfinx4*a x | Great disquare apeirogonal tiling prism. Symbol is (o~x'x") x. Verf is a crossed trapezoid pyramid.242. (o~x^6/5x^6) x = oinfinx6/5x6*a x | Small dihexagonal apeirogonal tiling prism. Symbol is (o~x^6/5x^6) x. Verf is a trapezoid pyramid.243. (o~x^6x^6/5) x = (x~o^6/5x^6) x = xinfino6/5x6*a x | Great dihexagonal apeirogonal tiling prism. Symbol is (o~x^6x^6/5) x. Verf is a crossed trapezoid pyramid.245. (o^6x^6/5x) x = o6x6/5x3*a x | Great trihexahexagonal tiling prism. Symbol is (o^6x^6/5x) x. Verf is a trapezoid pyramid.246. xo^6/5x x = x3o6/5x x | Quasirhombitrihexagonal tiling prism. Symbol is xo^6/5x x. Verf is a crossed trapezoid pyramid.249. (x~x"x') x = (x"x~x') x = x4/3xinfinx4*a x | Disquare apeirogonal tiling prism. Symbol is (x~x"x') x. Verf is a scalene triangle pyramid.250. (x~x^6/5x^6) x  = xinfinx6/5x6*a x | Dihexagonal apeirogonal tiling prism. Symbol is (x~x^6/5x^6) x. Verf is a scalene triangle pyramid.251. x'x"x x = x4x4/3x x | Rhombidisquare tiling prism. Symbol is x'x"x x. Verf is a scalene triangle pyramid.252. (x^6/5x^6x) x = x6x6/5x3*a x | Trihexatruncated trihexagonal tiling prism. Symbol is (x^6/5x^6x) x. Verf is a scalene triangle pyramid.253. xx^6/5x x = x3x6/5x x | Omniquasitruncated trihexagonal tiling prism. Symbol is xx^6/5x x. Verf is a scalene triangle pyramid.255. o's"s x = o4s4/3s x, s"s"s x = s4/3s4/3s x | Retrosnub square tiling prism. Symbol is o's"s x, or s"s"s x. Verf is an irregular pentagrammic pyramid.256. (s~s"s') x = (s"s~s') x = s4/3sinfins4*a x | Snub square apeirogonal tiling prism. Symbol is (s~s"s') x. Verf is an irregular nonconvex hexagonal pyramid.260. o~s s x = oinfins2s x | Apeirogonal antiduoprism. Symbol is o~s s x. Verf is a trapezoid pyramid. This is the prism of the apeirogonal antiprism.264. o~s s o~x = oinfins2s oinfinx | Apeirogonal antiprism prismatic honeycomb. Symbol is o~s s o~x. Cells are trips and azips. Verf is a trapezoid dipyramid where two of the right triangles are coplanar, this and the next honeycomb are copycats of the previous one. (263. Square tiling hemiantiprism.)`

I even found some inconsequences here: polychoronlover calls "disqua" what on my website is refered to as "satsa". - Which of those is the actual / current version of the OBSA?

--- rk
Klitzing
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### Re: List of uniform honeycombs

Hedrondude wrote:I call (x'x"x~) satsa.

I played with some of these before by sketching their verfs and getting regiment counts, although I have yet to list them in a spreadsheet.

o'ox"x = quitch
o'xx"x = gaqrich
x'xx"x = gaquapech
x'ox"x = quiprich
x'xo"x = paqrich
o'(o'x"x) = gacoca - for great cubaticubatiapeiratic honeycomb
o'(x"x'o) = wavicoca
x'(o'x"x) = skivpacoca
o(x"x~o') = wavicac - sphenoverted cubitiapeiraticubatic honeycomb

I'll coin more soon.

Jonathan B.

Above missing long names thus most probably read:
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`quitch     - quasitruncated cubic honeycombgaqrich    - great quasirhombated cubic honeycombgaquapech  - great quasiprismated cubic honeycombquiprich   - quasi prismatorhombated cubic honeycombpaqrich    - prismatoquasirhombated cubic honeycombwavicoca   - sphenoverted cubaticubatiapeiratic honeycomb skivpacoca - small skewverted prismatocubaticubatiapeiratic honeycomb`

Btw. "satsa" is the abbreviation of: squariapeirotruncated squariapeirogonal tiling

--- rk
Klitzing
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### Re: List of uniform honeycombs

Oh yeah, I should disclaim that some of the acronyms I listed, including the ones for nonconvex honeycombs, were made by me because I didn't know what the real acronyms were.
polychoronlover
Dionian

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Joined: Wed Feb 29, 2012 1:15 am

### Re: List of uniform honeycombs

The convex uniform apeieotera are as follows.

1. Wythoffian.
2. Wythoff snub (1) s3s4o3o3o
3. Laminates LB3, LC3, LPA3, LPB3, LPC3, LC1A2, LC1B2, LC1C3.

Among the non-convex ones, one has the isomorphs in 4/3. the Wythoff altisnub s3s4/3o3o3o, and the laminaes LPx3, LCA1 and LCA3 have isomorphs.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
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### Re: List of uniform honeycombs

Hehe Wendy, polychoronlover and I just were after the non-convex apeirochora, not the apeirotera ...
But good to know - even so your lamiate namings surely are hard to decode ...
--- rk
Klitzing
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### Re: List of uniform honeycombs

Klitzing wrote:Hehe Wendy, polychoronlover and I just were after the non-convex apeirochora, not the apeirotera ...
But good to know - even so your lamiate namings surely are hard to decode ...
--- rk

I presume "lamiate" is a tiling that utilizes a mythological human/snake hybrid? Marek14
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### Re: List of uniform honeycombs

I make there to be ten starry apeirochora, that is, tilings in 3D, not counting the prismic groups.

.4.3.4. gives xxox xxxx xxoo xoxo xxxo all together 5
.4.3.A. gives xoxo and xxoo all thgether two
The laminates give LLPA2, LLPB2, LLPC3, all together three.
The dream you dream alone is only a dream
the dream we dream together is reality.

\(\LaTeX\ \) at https://greasyfork.org/en/users/188714-wendy-krieger wendy
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### Re: List of uniform honeycombs

Ha,
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`4. (o(oo),x) = o3o3o3x3/2*a3*c | Tetrahedral hemitriangular tiling honeycomb. Cells are tets and trats (triangular tilings). Verf is an oho. Symbol is (o(oo),x). This one has hemicubic honeycomb symmetry, but is only wythoffian if it has quarter cubic honeycomb symmetry, also known as cyclotetrahedral symmetry.19. (o(xx),x) = x3o3x3x3/2*a3*c | Truncated tetrahedral hemitriangular-tiling honeycomb. Symbol is (o(xx),x). Cells are tuts, hexats (as truncated trats), and ohoes. Verf is a crossed quadrilateral pyramid. This has hemicubic honeycomb symmetry.33. (o(xx),o) = x3o3x3o3/2*a3*c | Rectified tetrahedral hemitriangular-tiling honeycomb. Symbol is (o(xx),o). Cells are octs, ohoes, and thats. Verf is a crossed quadrilateral prism.`

the first 2 of these 3 Dynkin symbols are wrong. Consider
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`(B(CA),D) = A3B3C3D3/2*a3*c =      _A_        _-3- | 3/2_   B_    3    _D    -3-_|_-3-          C       `

They rather should be
• (x(oo),o) = o3x3o3o3/2*a3*c with cells o3x3o3.3/2*a3*c = trat, o3x3.3o3/2*a3*c = tet, o3.3o3o3/2*a3*c = vertex (verf=oho), .3x3o3o3/2*a3*c = tet - thus indeed what no. 4 described (but not the there given symbol)
• (x(xx),o) = x3x3x3o3/2*a3*c with cells x3x3x3.3/2*a3*c = hexat, x3x3.3o3/2*a3*c = tut, x3.3x3o3/2*a3*c = oho, .3x3x3o3/2*a3*c = tut - thus indeed what no. 19 described (but not the there given symbol)
• (o(xx),o) = x3o3x3o3/2*a3*c with cells x3o3x3.3/2*a3*c = that, x3o3.3o3/2*a3*c = oct, x3.3x3o3/2*a3*c = oho, .3o3x3o3/2*a3*c = oct - thus indeed what no. 33 described
If I not misread or even overread something, so the further here possible case is missing:
• (o(xo),x) = o3o3x3x3/2*a3*c with cells o3o3x3.3/2*a3*c = trat, o3o3.3x3/2*a3*c = tet, o3.3x3x3/2*a3*c = oho, .3o3x3x3/2*a3*c = tut
(All others of this specific symmetry (o(oo),o) would ask for some Grünbaumian cells like a double cover of tet, of oct, of tut, or of thah, if not even severals of these.)

--- rk
Klitzing
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### Re: List of uniform honeycombs

Thanks for pointing out my errors! Were there any other errors, or known honeycombs that I missed?
Also, are Paqrich and Quiprich distinct? The only nonconvex prismatorhombate regiment I could find contained something I called the prismatoquasirhombated cubic honeycomb as the second member. But that was x"ox"x, not x"ox'x or x'ox"x.

EDIT: When I wrote x'ox"x as the symbol for the honeycomb in my document, I meant x"ox"x.
polychoronlover
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### Re: List of uniform honeycombs

I have a listing of uniform apeirotopes, based on the claim that the N dimensional ones move onto the N+1, by way of the comb product with the horogon {W4}.

The ones marked with a # have an isomorph made by reversing the square-root sign. Different presentations of the same tiling fall on the same row.

Code: Select all
`1D   1  horogon, thus the square, cubic, and tesseractic tilings.   2D   Wythoffian         1  x4o4o   o4x4o  x4o4x   2  x3o6o   s3s3s3z  LA1  LB1   3  x6o3o   x3x6o   x3x3x3z   4  o3x6o   x3x3o3z   5  x4x4o   x4x4x   #   6  x6x3o   #   7  x3o6x   #   8  x3x6x   #   9  s3s6s     10  s4s4s   #    Laminates    11  LPC1  #3D  Wythoffian     1  x4o3o4o  x4o3o4x  12  o4x3o4o  13  x3o4oAo     x3o3x3o3z LA2    14  x3x3o3o3z  15  x3x4oAo     x3x3x3o3z  16  x3o4xAo     #  17  x4x3o4o     #  18  x4o3x4o     #  19  o4x3x4o     x3x4o3x   x3x3x3x3z  20  x3x4xAo     #  21  x4x3x4o     #  22  x4x3o4x     #  23  x4x3x4x     #  Laminates   24  LB2      25  LC2  26  LPA2    #  27  LPB2    #  28  LPC2    #  4D  29  55 products, formed from elements 2-11    Wythoffian    ...    Wythoff - snub    s3s4o3o3o = s3s3s4o3o  #    Laminates         LPA3  LC1A2  -##  LB3  LPB3  LC1B2  .##  LC3  LPB3  LC1C2  .#.        `
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the dream we dream together is reality.

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### Re: List of uniform honeycombs

polychoronlover wrote:Thanks for pointing out my errors! Were there any other errors, or known honeycombs that I missed?
Also, are Paqrich and Quiprich distinct? The only nonconvex prismatorhombate regiment I could find contained something I called the prismatoquasirhombated cubic honeycomb as the second member. But that was x"ox"x, not x"ox'x or x'ox"x.

EDIT: When I wrote x'ox"x as the symbol for the honeycomb in my document, I meant x"ox"x.

Actually all of these are distinct: x'ox'x (convex), x'ox"x (starry with convex verf), x"ox'x (starry, crossed verf), x"ox"x (very starry, crossed verf) - we could call the last one quipqrich for quasiprismatoquasirhombated cubic honeycomb - its a bit more "quasier" . We can also call x"xx"x - quequapech for quasiquasiprismated cubic honeycomb.

I noticed that number 25 and 28 were the same, one of them should have coes and thas in it. Number 26 should have octs, hohas, and ohoes. It also appears that the rich regiment has 45 members, this is due to the alternate symbol (oxox) where the two squares in the verf can now be considered as two different groups of octs in the honeycomb, making the rich regiment behave similar to the afdec regiment. Unlike the afdec regiment, the rich regiment has hemicells, therefore a different regiment count. The regiment with a square frustrum verf will also behave like afdec, I suspect it has 58 members due to the top and bottom square being different sizes.

Also the ionic ones in the octet regiment will crash , this is because it will cause the octahedra to have one square face active and all edges active, which will have a verf shaped like a percentage symbol % (bar for the square, two points for two edges) at four vertices, but would look like four points :: at two vertices - so the verfs of the octet members would need to have four of the six squares of co to look like % and two like :: in order to have any chance of working, instead of all of them like %.
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### Re: List of uniform honeycombs

Code: Select all
`140. (("x~(o'x)~x")') = oinfinx4/3xinfinx4/3*a4*c *b4*d | Discubicuboctahedral alternate square disquare apeirogonal tiling honeycomb. Symbol is (("x~(o'x)~x")') Cells are soccoes, goccoes, shas, and disquas.`

does not work either.

Sure, you well can consider the symmetry groups
Code: Select all
`         o         where        /|\               / P*\       P =  3,   4,   6      /  |  \      P* = 3/2, 4/3, 6/5 (resp.)     U  _o_  P     U = infin (always)    / _P   U_ \       /_-       -_\     o------P*-----o  `

provided one ringed node at any U-link at least. But this then becomes a degenerate group: none of the cells would be truely 3D, all being flat tilings only!

Btw. one likewise can consider the following groups generally:
Code: Select all
`         o         where         |                  3         P =  3,   4,   6         |         P* = 3/2, 4/3, 6/5 (resp.)        _o_        U = infin (always)      _P   P*          _-       -_      o------U------o  `

again provided one ringed node at the U-link at least. This works well for P=3 and P=4, as then at least some true 3D polyhedra occur for cells. But the case P=6 similarily does produce only tilings for "cells". Therefore this latter case again is degenerate!

Btw., I did not say anything about the existance of an honeycomb with soccoes, goccoes, shas, and satsas (= disquas). I just refer to that provided symbol, which definitely does neither produce these cells, nor even will produce any non-degenerate honeycomb.

--- rk
Klitzing
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### Re: List of uniform honeycombs

I am working on an updated list of uniform honeycombs now. It will probably be ready in a few days. Meanwhile, is there a reason the Wikipedia page "Uniform tiling" doesn't seem to list the rhombisnub uniform tilings on Klitzing's site?
polychoronlover
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### Re: List of uniform honeycombs

Sadly there are 2 contracurrent streamings on Wikipedia:

- On the one hand it is open for anyone to add content, even new pages. At best when additionally be logged in, i.e. having an personal account, so that the changes can be tracked to a person instead of an IP-address only (which furthermore might change, depending on one's internet provider).
- On the other hand there have been lately lots of hardliner remakes, allowing only for assured content, i.e. only such which have any printed reference. Websites currently are accepted only in addition as a crossreference, but not as its sole reference.

To my personal knowledge e.g. Tom Ruen generally struggles with diligently adding lots of polytopal contents to wiki, whereas e.g. Guy Inchbald rather takes on the opposite position ...

Thus it is more a question of speed: contributing much more content as can be deleted again. And hoping on change of politics ...

I.e. feel free to provide content to Wikipedia whenever you feel like there is something missing! But likewise it might also occur that your contributions gets edited by others as well or even get deleted again ...

At least you should know that nothing on any website so ever is static. Content gets changed every now and then, URLs might no longer work, etc.

Wrt. capirsit = celliprismatorhombisnub icositetrachoral tetracomb = s3s3s4o3x the corresponding problem would be that my formal extension of snubbing devices, i.e. the alternated facetings (which since then allows for arbitrary mixtures of s, o, and x nodes in Dynkin symbol decorations) well were described in a printed reference (cf. "Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams", published at: "Symmetry: Culture and Science", vol. 21, no. 4, 329-344, (2010) - also available in electronic form here), but this very paper explicitely deals with its application onto the group o3o4o only (for obvious publishing reasons). So s3s3s4o3x therefore would not be captured thereby explicitely ...

Btw., you speak of "rhombisnub uniform tilings" (plural). Which else do you refer to?

--- rk
Klitzing
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### Re: List of uniform honeycombs

Klitzing wrote:Btw., you speak of "rhombisnub uniform tilings" (plural). Which else do you refer to?  Here are two images from the Wikipedia page "Uniform tiling". The nonconvex tilings they list, in order, seem to be:
Ditatha, sha, tha, hoha, shothat, sraht, sossa, sost, gossa, shaha, ghaha, huht, ghothat, graht, qrothat, satsa, hatha, qrasquit, quitothit, thotithit, quitsquat, quothat, retrat, rasisquat
which skips rassersa, rarsisresa, rosassa, and rorisassa from bendwavy.org, as well as chatit, which I think is degenerate.
By the way, here is my list of antifrustumverts from the rich regiment, but I'm not sure that I got them all. I used Stella to step through facetings of the verf. Did I miss any?
Code: Select all
`283. Octahemioctahedral tomoctahedral hemitrihexagonal tiling honeycomb. Cells are ohoes, toes, and thats.284. Tomoctahedral intercepted disoctahedral cuboctahedral cubohemioctahedral hemitrihexagonal tiling honeycomb. Cells are octs (in two orientations), coes, choes, toes, and thats.285. Tomoctahedral intercepted disoctahedral octahemioctahedral hemialternate hexagonal tiling hemialternate triangular tiling honeycomb. Cells are octs in two different orientations, ohoes, toes, hohas, and thas.286. Cuboctahedral cubohemioctahedral tomoctahedral hemialternate hexagonal tiling hemialternate triangular tiling honeycomb. Cells are octs in two different orientations, coes, choes, toes, hohas, and thas.287. Tomoctahedral intercepted disoctahedral cuboctahedral octahemioctahedral hemialternate square hemialternate hexagonal tiling honeycomb. Cells are octs in two different orientations, coes, ohoes, toes, shas, and hohas.288. Tomoctahedral intercepted disoctahedral cuboctahedral hemitrihexagonal tiling hemialternate square tiling hemialternate hexagonal tiling honeycomb. Cells are octs in two different orientations, coes, toes, thats, shas, and hohas.289. Tomoctahedral intercepted disoctahedral octahemioctahedral cubohemioctahedral hemialternate square hemialternate triangular tiling honeycomb. Cells are octs in two different orientations, ohoes, choes, toes, shas, and thas.290. Tomoctahedral intercepted disoctahedral cubohemioctahedral hemialternate square tiling hemialternate triangular tiling hemitrihexagonal tiling honeycomb. Cells are octs in two different orientations, choes, toes, thats, thas, and shas.291. Tomoctahedral intercepted cuboctahedral hemitrihexagonal tiling hemialternate square tiling honeycomb. Cells are coes, toes, thats, and shas.292. Octahemioctahedral cubohemioctahedral tomoctahedral hemialternate square tiling hemialternate triangular tiling honeycomb. Cells are ohoes, choes, toes, shas, and thas.293. Cubohemioctahedral tomoctahedral hemialternate square tiling hemialternate hexagonal tiling honeycomb. Cells are choes, toes, shas, and hohas.294. Cubohemioctahedral octahemioctahedral hemialternate square tiling hemitrihexagonal tiling hemialternate hexagonal tiling honeycomb. Cells are choes, ohoes, shas, thats, and hohas.295. Cuboctahedral apeiroprismatic hemialternate triangular tiling honeycomb. Cells are coes, azips, and thas.`
polychoronlover
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### Re: List of uniform honeycombs

Attached is my finished updated list of honeycombs.
Attachments honeycombs2.rtf
Added antifrustum-verted extensions to regiments; extra prismatic honeycombs.
polychoronlover
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### Re: List of uniform honeycombs

I think I discovered some more of these. Specifically:
o3x4o~x4/3*b (sphenovert)
x3x4o~x4/3*b (prismatorhombate)
o3x4x~o4/3*b (o3x4o~x4/3*b regiment)
x3x4x~o4/3*b (x3x4o~x4/3*b regiment)
o3x4x~x4/3*b (great rhombate)
x3x4x~x4/3*b (omnitruncate)
x4o3o3x4/3*a *b3d (trigonal cupolivert)
x4x3o3o4/3*a *b3d (x4o3o3x4/3*a *b3d regiment)
x4x3o3x4/3*a *b3d (prismatorhombate)
x4o3x3x4/3*a *b3d (skewvert)
x4x3x3o4/3*a *b3d (x4o3x3x4/3*a *b3d regiment)
x4x3x3x4/3*a *b3d (omnitruncate)
x4x3o3/2o4/3*a *b~d (x4o3o3x4/3*a *b3d regiment)
x4o3o3/2x4/3*a *b~d (x4o3o3x4/3*a *b3d regiment)
x4x3o3/2x4/3*a *b~d (x4x3o3x4/3*a *b3d regiment)
x4o3/2o~x3*b (x4o3o3x4/3*a *b3d regiment)
x4/3o3/2o~x3*b (x4o3o3x4/3*a *b3d regiment)
x4/3x4x3/2o3*a *a~c *b4d (x4o3x3x4/3*a *b3d regiment)
x4/3x4x3/2x3*a *a~c *b4d (omnitruncate)
polychoronlover
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### Re: List of uniform honeycombs

I know it's bumping an old thread, but I have a question related to this: Has anyone investigated the starry 4D honeycombs yet? There are some interesting regiments there that I'm somewhat curious how to count. For reference, I will post the convex 4D honeycombs and what I know of their regiments...

o'ooo'x = x'ooo'x - test. Verf is hex. Should act like the scad regiment with 11 main members, but also some prismatic and duoprismatic ones.

o'xoo'o - rittit. Verf is taller octahedral prism. Should act like the rat regiment, at least 44 members. However, there should also be a subsymmetry (o'oo9x, I think) whose verf is an oct podium, which probably adds more.

o'oxo'o - icot (birectified test). Verf is a tes, not sure how many here.

o'oox'x - tattit. Verf is oct pyramid. 2 members.

o'oxx'o - batitit. Verf is square disphenoid. 2 members.

o'oxo'x - srittit. Verf is a square prismatic wedge, acts like the sart regiment. 93 members.

o'xox'o - ricot (birhombated test). Verf is duowedge, but should have fore than sibrant, as not only does it have a half symmetry, but also quarter symmetry (\$o\$).

o'xoo'x - sidpitit. Verf is a trigonal antifastegium, due to half-symmetric representation it acts like spat. 85 members.

o'oxx'x - grittit. Verf is square scalene. 2 members.

o'xxx'o - ticot (great birhombated test). Verf is disphenoid pyramid. 1 member.

o'xxx'x - gippittit. Verf is line tettene. 1 member.

x'xxx'x - otatit. Verf is irregular pen. 1 member.

o'xox'x - potatit. Verf is wedge pyramid, but can be given half symmetry. 11 members.

o'xxo'x - prittit. Verf is trapezoidal disphenoid. 3 members.

x'xoo'x - capotat. Verf is trigonal antipodic pyramid. 7 members.

x'oxo'x - scartit (@hedrondude: I suggest changing this one to "cartit" to match with the polytera). Verf is antiduowedge. 18 members.

x'oxx'x - gicartit (@hedrondude: I suggest chaning it to cogratit to match with polytera). Verf is trapezoidal scalene. 3 members.

x'xox'x - captatit. Verf is trapezoidal scalene. 3 members.

Any information on missing ones? If I get feedback, I will analyze the rest of the convex ones.
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### Re: List of uniform honeycombs

Welcome to this forum, "username5243"! You'd maybe interested in this page. Or in this enlisting of Dinogeorge, which was set up in the years 2003-2006.
--- rk
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### Re: List of uniform honeycombs

username5243 wrote:I know it's bumping an old thread, but I have a question related to this: Has anyone investigated the starry 4D honeycombs yet? There are some interesting regiments there that I'm somewhat curious how to count. For reference, I will post the convex 4D honeycombs and what I know of their regiments...

I was looking into these a bit. One of the more interesting things I found was a family with diagram like so: o3o3o4o3o4/3 *a. It includes the lieutenant of the x3o3o4o3x regiment, as well as the head of its conjugate regiment, a "double-covered" version of a o3o3o4x3o, and even some afdec-like members with vertex figures of the form a b3o || c o3d. These regiments seem to even include afdec members as potential facets, and may have over 1000 members!

Also, I found a "novel" tetracomb (one that does not have anything with a linear diagram in its regiment), o4o3o4x4/3x3 *c. It is interesting because the vertex figure is an octahedral podium whose "crossed" faceted form creates the conjugate, while a similar spherical polyteron, o3o3o4x4/3x3 *c, has a tetrahedral podium as its verf, whose crossed form is an antipodium instead.

oNo3o4x4/3x3 *c is also the only member of the oNo3o4o4/3o3 *c family which is not "skewed", does not contain cotcoes, and does not share its regiment with a polytope of the form ...x3o4/3x. It's interesting how it's so different in each case.
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### Re: List of uniform honeycombs

polychoronlover wrote:
username5243 wrote:I know it's bumping an old thread, but I have a question related to this: Has anyone investigated the starry 4D honeycombs yet? There are some interesting regiments there that I'm somewhat curious how to count. For reference, I will post the convex 4D honeycombs and what I know of their regiments...

I was looking into these a bit. One of the more interesting things I found was a family with diagram like so: o3o3o4o3o4/3 *a. It includes the lieutenant of the x3o3o4o3x regiment, as well as the head of its conjugate regiment, a "double-covered" version of a o3o3o4x3o, and even some afdec-like members with vertex figures of the form a b3o || c o3d. These regiments seem to even include afdec members as potential facets, and may have over 1000 members!

Also, I found a "novel" tetracomb (one that does not have anything with a linear diagram in its regiment), o4o3o4x4/3x3 *c. It is interesting because the vertex figure is an octahedral podium whose "crossed" faceted form creates the conjugate, while a similar spherical polyteron, o3o3o4x4/3x3 *c, has a tetrahedral podium as its verf, whose crossed form is an antipodium instead.

oNo3o4x4/3x3 *c is also the only member of the oNo3o4o4/3o3 *c family which is not "skewed", does not contain cotcoes, and does not share its regiment with a polytope of the form ...x3o4/3x. It's interesting how it's so different in each case.

One other interesting one should be x3o4o3x3o, which has a square antifastegium verf. I'm pretty sure this one has a conjugate, but not sure about the symbol.

Indeed, that's curious, but a similar thing happens in lower dimensions. Take o3o3x4/3x4*a, which has a triangle podium, its crossed faceting is a trigonal crossed antipodium. o4o3x4/3x4*a has a square podium verf, and its crossed version is a square crossed podium. (In spherical 4D there is a regiment with a pentagon podium verf, and one of that one's crossed members has a pentagonal crossed antipodium verf.)
Trionian

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### Re: List of uniform honeycombs

Fun fact, the edges of o3x3o4o3x are a subset of the edges of x3o3o4o3x! This is similar to how the edges of o3x3o4x are a subset of those of x3o4o3x. Adding more edges increases the symmetry; however, when applied to tesselations in affine geometry, the symmetry can be "increased" or "decreased" just by zooming in or out. And that's what happens in the case of o3x3o4o3x and x3o3o4o3x -- they have the same symmetry, but when the edge lengths are the same, x3o3o4o3x is more densely packed with symmetrical modules. If we were to superimpose those tetracombs over each other, the size of x3o3o4o3x would have to increase for their symmetries to line up.

This is because the square antifastegium vertex figure of o3x3o4o3x is just the cube - octahedron antipodium vertex figure of x3o3o4o3x, minus two vertices.

username5243 wrote:One other interesting one should be x3o4o3x3o, which has a square antifastegium verf. I'm pretty sure this one has a conjugate, but not sure about the symbol.

Indeed, that's curious, but a similar thing happens in lower dimensions. Take o3o3x4/3x4*a, which has a triangle podium, its crossed faceting is a trigonal crossed antipodium. o4o3x4/3x4*a has a square podium verf, and its crossed version is a square crossed podium. (In spherical 4D there is a regiment with a pentagon podium verf, and one of that one's crossed members has a pentagonal crossed antipodium verf.)

The conjugate of o3x3o4o3x is o3x3o4/3o3x. The head of its regiment also has a square antifastegium vertex figure, and the symbol o3x3o4o3x4/3 *b.
polychoronlover
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### Re: List of uniform honeycombs

Neat one. Wonder if there are any other relationships like that...

I think icot (tes verf) and hext (ico verf) may be related like that, since tes's edges are a subset of ico's. There are probably a few more like that.

I think I figured out what one of the afdec-types is. Its CD diagram is (I'm not sure of the linearization, so here's a rough idea):

Code: Select all
`   o   3   o 3   4x 4/3 x 4   3   o`

This thing has as facets rits, ricoes, gittiths, and afdecs. The verf is a q x3o || x q3o (the bases being rit and rico verfs, and connected by the gittith and afdec verfs).

One thing is, I'm fairly sure here that afdec acts like it has more members than it actually does. The two sets of 24 coes in the afdecs behave differently (one set connects to rits while the other connects to ricoes), and the goccoes are simila"r (one pair connects to gittiths, the other connects to more afdecs). This is why the "a" and "b" sets of goccoes (as listed on Hedrondude's afdec page) are different, so the afdecs here should act closer to affixthi and have 99 members.

BTW, I noticed after your latest revision of the list of honeycombs, you posted a list of new ones. Did you ever get around to adding those to your list of honeycombs? In particular I'm curious about the one with a trigonal cupola verf - does it act like the stut phiddix regiment of 4D?
Trionian

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### Re: List of uniform honeycombs

username5243 wrote:I think icot (tes verf) and hext (ico verf) may be related like that, since tes's edges are a subset of ico's.

Yes indeed, the hext regiment is home to a compound of (I think 4) icots! Its vertex figure is a gico, a compound of 3 tesseracts. Furthermore, there is a compound of tesseractic tetracombs (tests) in the icot regiment, two meeting at each vertex. They act like they have demitesseractic symmetry, though. There is also a compound of tests in the hext regiment, this time with full symmetry and meeting 3 to a vertex.

About the afdec-containing regiments, you're right about afdec acting like affixthi, because the contic symmetry doesn't show in this regiment. There should be 98 possible types of cells in the vertex figure that can appear in place of afdec's verf, and sometimes nothing will be there at all.

The vertex figure looks like nit's vertex figure, a tisdip, but with less symmetry. That means it will have more facetings, which leads to more regiment members. There should be one figure corresponding to each of the 24 nit regiment members with pentic symmetry and two corresponding to each of the 125 members with demipentic symmetry. So there are at least 274 tetracombs in this regiment, and probably a lot more.

username5243 wrote:BTW, I noticed after your latest revision of the list of honeycombs, you posted a list of new ones. Did you ever get around to adding those to your list of honeycombs?

I've given up on updating that list. It was becoming too complicated trying to fill the gaps in my numbering with newly discovered honeycombs, and then I discovered so many more that I was too lazy to add. Plus, I kept noticing errors that had crept into my original list. I'll make a new one eventually, though, with brand new numbering.

username5243 wrote:In particular I'm curious about the one with a trigonal cupola verf - does it act like the stut phiddix regiment of 4D?

If you mean the triangular tiling antiprism, then no. This is because the edges in the verf that make up the lacing between the triangle and the hexagon don't correspond to a face with an even number of sides. Also, where the stut phiddix regiment has srids as possible cells, this figure has apeirogonal antiprisms (azip). Since azip does not have any other regiment members, this forces the honeycomb's regiment to be smaller. (For a 3-dimensional analogy, think of how the antiprisms and cantellates both have trapezoidal verfs, but only the cantellates have regiments.)

I'm pretty sure the triangular tiling antiprism does have some regiment members besides the 3 I listed, though. I just need to figure out what they are.
polychoronlover
Dionian

Posts: 51
Joined: Wed Feb 29, 2012 1:15 am

### Re: List of uniform honeycombs

polychoronlover wrote:x4o3o3x4/3*a *b3d (trigonal cupolivert)

And agreed, just make a new one, with a new numbering (I'd probably be willing to help, at least with some of the simplaer regiments).

Indeed, that regiment (the afdec-type) should have a lot more than 274 (and at least that many fissaries). One thing I find interesting is, looking at hedrondude's polyteron spreadsheet that it seems all the nit members either have icoes, rathoes, gicoes, or no ico member at all (and the same goes for the sart regiment). There is not a single one with, say, shahohs as cells. I wonder why this is...

I've also been thinking of that spreadsheet, and noticed that one regiment had most of its members not listed (it was skatbacadint IIRC). I spent some time looking at those, even came up with some possible names for some of them. Also, I started looking more at 6D ones, especially the simpler ones, and coming up with names for them.
Trionian

Posts: 111
Joined: Sat Mar 18, 2017 1:42 pm

### Re: List of uniform honeycombs

Oh, I had forgotten about that one! I was never 100% sure that that was even a valid uniform honeycomb, but the verf doesn't seem to be cut from a larger verf, so the vertices probably don't coincide, so probably nothing coincides... and if that's true then this is valid, and its regiment probably has 79 members just like the Stut Phiddix regiment.

I've been thinking of a plan for my new list of uniform honeycombs. Here is what I have so far:
Code: Select all
`chon regiment members w/ A(3) or C(3) verfsoctet regiment members w/ C(3) verfstich regimenttatoh regimentquitch regimentrich regiment members w/ C(2)*A(1) or A(1)^3 verfssrich regiment members w/ A(1)^2 verfswavicoca regiment members w/ A(1)^2 verfso3x4o~x4/3*b (wavicac) regimentbatchgrich, gratoh, cuteca, o3x4x~x4/3*b, gaqrich, gaqrahch, caquitecaotch, cutpica, gactipeca, x3x4x~x4/3*b, x4x3x3x4/3*a *b3d, x4/3x4x3/2x3*a *a~c *b4d, gaquapech, quequapech, caquitpica, gacquitpicaprich regimentquiprich regimentgepdica regimentgapdica regimentx3x4o~x4/3*b regimentx4x3o3x4/3*a *b3d regimentchon regiment members w/ A(2) verfsbatatoh regiment'gotactictosquath' regimentratoh regimentrich regiment members w/ C(2) verfsgacoco regiment members w/ C(2) verfssrich regiment members w/ A(1) verfsskivpacoca regimentwavicoca regiment members w/ A(1) verfsx4o3x3x4/3*a *b3d regimentrich regiment members w/ A(1)^2 verfsgacoco regiment members w/ A(1)^2 verfsoctet regiment members w/ A(3) verfsx4o3o3x4/3*a *b3d regiment membersprismatics and their regimentsprismsregiment members of prismsother slabs and their regimentsgyrates and elongates`
polychoronlover
Dionian

Posts: 51
Joined: Wed Feb 29, 2012 1:15 am

### Re: List of uniform honeycombs

I have some ideas for names:

polychoronlover wrote:o3x4x~x4/3*b

Possible name: Dacta (dicubatitruncated apeiratic honeycomb)

polychoronlover wrote:x3x4x~x4/3*b

Possible name: Dactapa (dicubiatitruncated prismato-apeiratic honeybomc)

polychoronlover wrote:x4x3x3x4/3*a *b3d

Possible name: dicroch (dicubatirhombated cubihexagonal honeycomb)

polychoronlover wrote:x4/3x4x3/2x3*a *a~c *b4d

wait, this one contains x3/2x and is degenerate. Not sure what you meant to write here...

polychoronlover wrote:x3x4o~x4/3*b

Possible name: Sacpaca (Small cubatiprismatocubatiapeiratic honeycomb)

polychoronlover wrote:x4x3o3x4/3*a *b3d

Possible name: Dichac (dicubatihexacubatic honeycomb)

polychoronlover wrote:'gotactictosquath'

Possible shorter name: Getdoca (great tetradicubatiapeiratic honeycomb)

polychoronlover wrote:x4o3x3x4/3*a *b3d

Possible name: Skivcadach (small skewverted cubatiapeiroducubatic honeycomb)

polychoronlover wrote:x4o3o3x4/3*a *b3d

Possible name: Stut Cadoca (small tritrigonary cubatidicuubatiapeiratic honeycomb)

Feel free to adopt some of these names. Or none at all. I'm just giving ideas.
Trionian

Posts: 111
Joined: Sat Mar 18, 2017 1:42 pm

### Re: List of uniform honeycombs

Code: Select all
`          _o_      _3-  |  -4_   o<      3      >o      -3_  |  4/3          -o-`

First of all: the linearisation symbol of the above symmetry group you gave as "o4o3o3o4/3*a *b3d" is wrong, as the letter "d" likewise isn't a real node, rather it is a virtaul one too, which re-reffers to the d-th so far already provided real node from the left. Accordingly it ought be prefixed by an asterisk as well. So you'd write rather "o4o3o3o4/3*a *b3*d".

But you could try a different linearisation here as well, cutting open at a threefold node (instead of a normal twofold one) and you'd get e.g. the simpler symbol "o3o3o4/3o4*a3*c" instead.

Next I looked through your according decoration cases.
• x3x3x4/3x4*a3*c - dicroch = dicubatirhombated cubihexagonal HC, cells are: ^ girco, <| hexat, v quitco, |> cotco
• o3x3x4/3x4*a3*c - ???, cells are: ^ sirco, <| that, v quitco, |> gocco
• x3o3x4/3x4*a3*c - dichac = dicubatihexacubatic HC, cells are: ^ tic, <| that, v quith, |> cotco
• x3x3o4/3x4*a3*c - skivcadach = small skewverted cubatiapeiroducubatic HC, cells are: ^ girco, <| that, v querco, |> socco
• o3o3x4/3x4*a3*c - stut cadoca = small tritrigonary cubatidicuubatiapeiratic HC, cells are: ^ cube, <| trat, v quith, |> gocco
• x3o3o4/3x4*a3*c - ???, cells are: ^ tic, <| trat, v cube, |> socco
(Here the symbols ^, <|, v, and |> refer to the accordingly shaped subdiagram. - All other decorations would use some Grünbaumian figure either as cell or as verf.)

You overlooked the "???" cases? At least the according namings are missing.

--- rk
Klitzing
Pentonian

Posts: 1555
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany

### Re: List of uniform honeycombs

Klitzing wrote:
Code: Select all
`          _o_      _3-  |  -4_   o<      3      >o      -3_  |  4/3          -o-`

First of all: the linearisation symbol of the above symmetry group you gave as "o4o3o3o4/3*a *b3d" is wrong, as the letter "d" likewise isn't a real node, rather it is a virtaul one too, which re-reffers to the d-th so far already provided real node from the left. Accordingly it ought be prefixed by an asterisk as well. So you'd write rather "o4o3o3o4/3*a *b3*d".

But you could try a different linearisation here as well, cutting open at a threefold node (instead of a normal twofold one) and you'd get e.g. the simpler symbol "o3o3o4/3o4*a3*c" instead.

Next I looked through your according decoration cases.
• x3x3x4/3x4*a3*c - dicroch = dicubatirhombated cubihexagonal HC, cells are: ^ girco, <| hexat, v quitco, |> cotco
• o3x3x4/3x4*a3*c - ???, cells are: ^ sirco, <| that, v quitco, |> gocco
• x3o3x4/3x4*a3*c - dichac = dicubatihexacubatic HC, cells are: ^ tic, <| that, v quith, |> cotco
• x3x3o4/3x4*a3*c - skivcadach = small skewverted cubatiapeiroducubatic HC, cells are: ^ girco, <| that, v querco, |> socco
• o3o3x4/3x4*a3*c - stut cadoca = small tritrigonary cubatidicuubatiapeiratic HC, cells are: ^ cube, <| trat, v quith, |> gocco
• x3o3o4/3x4*a3*c - ???, cells are: ^ tic, <| trat, v cube, |> socco
(Here the symbols ^, <|, v, and |> refer to the accordingly shaped subdiagram. - All other decorations would use some Grünbaumian figure either as cell or as verf.)

You overlooked the "???" cases? At least the according namings are missing.

--- rk

He was only listing regiment colonels. So, I didn't bother with names for the ones with non-convex verf either.

I'll give some ideas for the other ones here:

i think you have x3x3o4/3x4*a3*c and o3x3x4/3x4*a3*c wrong - the one polychoronlover listed was x4o3x3x4/3*a *b3d, which would be o3x3x4/3x4*a3*c, the one I called Skivcadach. The other one, being its conjugate, shouuld be called Gikkivcadach, switching that "small" for "great".

As for x3o3o4/3x4*a3*c, that one is just Stut Cadoca's conjugate and would be Getit cadoca (again with a "great" instead of "small").