Talk:Convex uniform honeycomb.html



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Mathematics rating:	B Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. More introduction needed. Geometry guy 20:50, 20 May 2007 (UTC)

I did a Google search on: http://www.google.com/search?q=Andreini+tessellation and got no results? Is Andreini the proper name? --css

Some searches for some of the tilings suggest that none of the google pages mention the concept. Andreini is the name used by books I have, but someone else is welcome to change it if they have a better name.

Since web search delivers no clue, could you add the "hardcopy" refrences of books you mentioned please? Mikkalai 08:03, 14 Dec 2003 (UTC)

The article claims that "All of these are found in crystal arrangements." Really ? http://www.iucr.org/iucr-top/comm/cteach/pamphlets/21/node4.html says only 17 crystal space groups are known to exist in real crystals (out of the theoretically 230 mathematical space groups) ... but perhaps some "space groups" include more than one "Andreini tesselation" ? --DavidCary 04:33, 8 Jan 2005 (UTC)

Likely it's impossible in some (or most) of the 230 space groups to find a point which has equal distance to enough of its images to form closed cells. —Tamfang 08:04, 24 February 2006 (UTC)

most of these tesselations actually lie in just three space groups: body-centered cubic, the cubic, and face-centered cubic symmetries. There is one (maybe two) with the symmetry of the diamond lattice, and a few with some of the uninteresting reflection groups. In short, none of the bizarro symmetries are represented, so the claim may be more plausible than it sounds. On the other hand, what does it mean to say that a tesselation appears in a crystal arrangement? -- strauss at uark.edu

I added an enumerations of the 28 tessellations, using the added online links provided. I grouped them as best I could for a first pass. Obviously some pictures need to be added.

The dynamic VRML models (from the first link) were very effective in showing the arrangements of polyhedra at each vertex. For single-view images, transparent faces might be more helpful. Maybe I can manipulate the settings a bit to get some good pictures.

Tom Ruen 12:08, 23 October 2005 (UTC)

alternated cubic

Does "alternated cubic" mean the face-centered cubic tiling by rhombic dodecahedra? If so, it's interesting that a nonuniform pattern can become uniform by truncation; I wonder whether there are any analogous finite examples. --Anton Sherwood 01:35, 10 January 2006 (UTC)

So far, names just come from description at [1], and so I can't say more. Sorry. Tom Ruen 04:34, 10 January 2006 (UTC)

I see now: the "alternated cubic" tilings come from treating alternate cells differently, so that the vertex figure has tetrahedral but not cubic symmetry. —Tamfang 18:47, 16 March 2006 (UTC)

Can you draw the fundamental domain for them? The cubic forms represent 1/48 of a cube - connecting a tetrahedron of vertex, mid-edge, mid-face, and cube center AND one generating point. What does the fundamental domain for an alternated cubic look like? Tom Ruen 00:28, 17 March 2006 (UTC)

The only difference is that the plane {0,1,2} is no longer a mirror, so take that simplex and add its reflection in that plane. —Tamfang 07:24, 17 March 2006 (UTC)

...or equivalently in the plane {1,2,3}. (Obvious, given duality, but it took me awhile to see it) —Tamfang 20:15, 21 March 2006 (UTC)

Interesting, so there's 4 points: 0,1,3a,3b. The face-center point (2) disappears, and we have two cell-centers. I'll have to think what effect this has! Same thing can be done on a square tiling, but resulting triangle is same 45-45-90 triangle, while a tetrahedron "doubled" generates a different symmetry. If a cubic honeycomb has a symmetry group [4,3,3], I wonder what is the "alternated cubic" symmetry group(s) is/are called? I've got Coexter's notation on sphere symmetry page at least: List of spherical symmetry groups, but not seen any tables for higher [a,b,c] types. Tom Ruen 07:44, 17 March 2006 (UTC)

you mean [4,3,4] not [4,3,3] ;)

—Tamfang 05:52, 19 April 2006 (UTC)

In terms of crystal symmetry, perhaps it is Tetragonal versus Cubic? Not that I understand much at all about these Crystal structure groups either, but just guessing blindly from the choices and taking away one symmetry plane??? Tom Ruen 07:57, 17 March 2006 (UTC)

No. —Tamfang 05:52, 19 April 2006 (UTC)

The "bitruncated alternated cubic" is not made by the usual kaleidoscope method with the cell I just described! (It has the right symmetries, but its edges do not meet the mirrors at right angles.) For that you need to double the cell again: reflect the original cell in both of its isosceles triangles, yielding a cell whose corners may be called (±1,0,0), (0,1,±1) — the origin being "1" or "2". This cell can also produce the "octet", truncated alternated cubic, rectified cubic and bitruncated cubic, all of which can also be made with the doubled cell, and of which the last two can be made with the single cell.

The Coxeter-Dynkin graph of the original cubic cell is

o--4--o--3--o--4--o

That of the doubled cell is

o--3--o--3--o
      |
      4
      |
      o

And that of the quadrupled cell is

o--3--o
|     |
3     3
|     |
o--3--o

—Tamfang 05:52, 19 April 2006 (UTC)

and after laboriously working out these graphs I opened Coxeter's Regular Polytopes and found them staring at me, as it were, at 5.62 (page 85). —Tamfang 04:10, 20 April 2006 (UTC)

skeleton images

See User:Tamfang/Tilings: skeleton images of bitruncated, cantellated, cantitruncated, runcitruncated and omnitruncated cubic tilings. More to come! —Tamfang 07:45, 24 February 2006 (UTC)

Very cool! For fun I linked a thumb image for first in the article. Tom Ruen 09:21, 24 February 2006 (UTC)

the prismatics

The duals of hexagonal-prismatic and triangular-prismatic are not uniform: the edges are not equal. —Tamfang 07:40, 16 April 2006 (UTC)

arrangement by symmetry

User:Tamfang/Tilings now contains a proposed rewrite of Andreini tessellation. —Tamfang 06:11, 21 April 2006 (UTC)

I'm glad for your work. My only concern is I tend towards wanting a compact table over wordy descriptions. I'd put the descriptions in individual articles. I don't need a specific arrangement, but I like some statistical information like my [a/b] notation for #-cells/vertex and #types of cells. In general I'd try to support different naming systems, like Guy Inchbald uses codes like O+T even if not unique. [2]. Well, and I don't remember what Grunbaum had in his paper, but there was a number of different indices at least to identify them between papers.

For me, I'd USE this article as a reference, like "Oh, I have this space-filling tessellation, what the HECK is it, and look at a table to identify it quickly." I accept my table approach could get ugly anyway and could as well go in List of Andreini tessellations. Since I have no time to spare, it's your call what and when you want to replace it. Tom Ruen 07:41, 21 April 2006 (UTC)

Something like this?

... and so on ... —Tamfang 21:42, 21 April 2006 (UTC)

Something like that. I'm now STUPIDLY thinking of backtracking again, specifically the 2D parallel of uniform tilings, currently at least grouped in given in Tiling by regular polygons or List of uniform planar tilings. Basically whatever sort of table I'd support ought to exist in a parallel article uniform tiling.

There's good parallels to consider, like Elongated triangular tiling similar special case like [3] and [4].

Well, also nice to have a table like Archimedean solid except can't have nice VEFC counts.

I'll be quiet since I'm indecisive as well as no time! Sorry. :( Tom Ruen 07:12, 22 April 2006 (UTC)

(VRML clients for MacOS don't work well, alas.)

In List of uniform polyhedra, I counted the VEFC per period. —Tamfang 15:53, 22 April 2006 (UTC)

It's shaping up, go have a look! (User:Tamfang/Tilings) Perhaps instead of the vertex figure it should show the dual cell? —Tamfang 04:49, 5 May 2006 (UTC)

I don't think the wire frames of the andreini tesselations are particularly enlightening. I'd much prefer to see the tesselations in solid form. I've rendered most of the 'interesting' tesselations if images are needed. http://xaviergisz.googlepages.com/andreinitesselations —xaviergisz

I see both are useful. If you can help add nice solid images, please do! Tom Ruen 04:02, 13 May 2006 (UTC)

I moved the added solid images into the table. I also noticed the NEW table is missing the 10 "prismatic" forms (semiregular tilings extruded into prism slabs). These ought to be (RE)added here, even if in a split table. Tom Ruen 06:41, 14 May 2006 (UTC)

Okay, I added back the 11 prismatic forms in a brief table. Tom Ruen 07:17, 14 May 2006 (UTC)

illustration

I agree that some of the edge views are not helpful, particularly those in which the edges continue through the vertices. At least they were easy to make. ;) It might help to make the edges parti-colored so that e.g. triangles have white edges, squares have yellow edges and so on. I'm unsure how to code that idea. —Tamfang 16:19, 15 May 2006 (UTC)

Solid views are sometimes hard to understand because of the hidden parts! I have two ideas to get around that: an "exploded" view, where each cell is appropriately centred but half its proper size; and a "composition" view, where the central cluster showing all the cells together is surrounded by groups of only one cell type. (A picture, when I get it ready, will better show what I mean. To do this well, I have to re-learn Povray texture and lighting technique, which I haven't used in ages.) —Tamfang 16:19, 15 May 2006 (UTC)

Someone improved the solid views while I wasn't looking. Kudos! —66.52.133.106 22:16, 6 July 2006 (UTC)

Stub articles

I completed stub articles for all the nonprismatic forms. Just dropped in a sentence and nearly empty table. I figured it was worth the start for completeness since we had images. Lots of table fields to fill. All my time is done for now! Tom Ruen 00:13, 13 July 2006 (UTC)

Get some sleep ;) —Tamfang 06:19, 13 July 2006 (UTC)

Conway's take on things

for what it's worth, Conway has a whole batch of names for the vertex-uniform, non isotropic tilings of space by archimedean polyhedra. (By non-isotropic, I just mean to rule out the tilings by "slabs"-- that is, the tilings with one of the cubic symmetries) The enumeration is of course the same as the Andreini tesselations.

I've put an excerpt from our forthcoming book that includes pix at comp.uark.edu/~strauss/downloads/archilles.pdf

Conway calls these tilings architectronic and their duals catotropic (you can read the rationale in the file)

strauss at uark.edu

Reference indexing

I used Grumbaum's paper to cross reference the honeycombs from varied sources. I also reordered them to match Johnson's indexing, since the truncation operation approach (for cubic at least) matches his derviations. For the cubics I also added four cell columns, as done with the uniform polychoron article. I've not seen Johnson's 1991 paper, but it looks like he had a systematic nonseequential approach for indexing that skipped numbers and grouped them into 10's. Tom Ruen 18:03, 22 July 2006 (UTC)

List of figures

The list of figures i use is of my device. George Olshevsky quotes me as discovering two in the 143 convex tilings in four dimensions. 1 occurs in 1dt, 1-4 in 2dt, 1-6 in 3dt. This list arranges all of the prism-layers at the beginning. So 1 designates the square, cubic, tesseractic, etc. The 8 at #2 include hexagon-prisms, etc. Because we count the cubic at #1, it is removed from later contention. This is why 434 gives only 7, and 434 only 14.

1. The comb products on the horogon (square, cubic, tesseractic, etc)
-. in one dimension: 1
2. Ten non-comb tilings in 3d: 44 = 2, 36 = 6.
3. The snubs on 44 and 36
4 The laminate tiling LC1P
-. In two dimensions, 11
5. Wythoff Mirror edge on these groups 434 (7), 43A (4) and 3333: 1
6. Laminate tilings: LC2, LC2P, LB2, LA2P, LB2P
-. In 3 dimensions. 28
7. 55 products of #2 * #2.
8 Wythoff's Mirror edge on 3343 (28), 4334 (14), 433A (4), E33A (0), 33333: (7) = 53
9 The snub tiling s3433
10 The laminate tilings LB3, LC3, LA3P, LB3P, LC3P, LC1A2, LC1B2, LC1C2 = 8
-. In four dimensions, 145

This list consists of 145, not 143 that George gives. I am not shure which two he suppresses. --Wendy.krieger 10:42, 23 September 2007 (UTC)

Thanks Wendy. I'll get through George's list sometime soon and compare myself. Tom Ruen (talk) 07:10, 26 June 2008 (UTC)

Noncompact and hyperbolic honeycombs

I added two sections for noncompact honeycombs in Euclidean 3-space, and compact honeycombs in hyperbolic 3-space. Just a start, just included Coxeter groups so far. Probably I'll move them to the end and expand the intro when I get the enumeration included. Tom Ruen (talk) 07:10, 26 June 2008 (UTC)