Talk:Honeycomb (geometry)

Merge

Shouldn't this be merged into Tessellation? --PeR 19:23, 28 March 2006 (UTC)

I would not suggest a merge, although I agree tessellation deserves a subsection mentioning higher dimensional tessellations, and linking here. Obviously this article is short and could use expansion as well. Tom Ruen 03:30, 29 March 2006 (UTC)

The way I see it, that's the same thing... While tesselation of space is currently so short it would fit into tesselation, it should be made a subarticle again once it grows large enough. --PeR 07:43, 29 March 2006 (UTC)

Well, I didn't create this article. In my expansions on the uniform "tessellations of space", I've been using the term honeycombs to mean polyhedron tessellations in 3-space and tilings for polygonal tessellations in 2-space. Although in mathematical usage, I've seen honeycomb used in "3 or higher dimensional" tessellations. Also check out list of regular polytopes for the full list of multidimensional "regular" tessellations of any dimension.

Anyway, if you want to move it I'll let you take the plunge. Easy enough to copy this text to a section of tessellation and add a redirect on this page to the main article. I don't know if Patrick (originated this article) has a watch here. Tom Ruen 08:22, 29 March 2006 (UTC)

definition of uniform

Congruency of vertices is insufficient, as the pseudorhombicuboctahedron (J37) illustrates for finite polytopes. Consider an elongated triangular prismatic tiling in which one layer of cubes is bisected and rotated, so that on one side of the cut-plane (call it the XY plane) the axis of the 3-prisms is parallel to the X axis, on the other side they are parallel to the Y axis. The symmetry group then has no Z component and therefore vertices at different Z are in different equivalence classes; this tiling is not uniform although each vertex figure is the same. —Tamfang 01:08, 11 June 2006 (UTC)

Merging Tessellation of space to Honeycomb (geometry)

I agree merge is useful. I suggest this name be kept and tessellation of space be deleted. I merged content as seemed helpful. It obviously needs some further expansion in either case. Tom Ruen 05:15, 20 July 2006 (UTC)

I relinked the articles from tessellation of space to this name (most of them added originally by me). Tom Ruen 05:31, 20 July 2006 (UTC)

Okay. Strictly speaking, you don't have to go to the trouble of relinking other articles, because ultimately we'll just redirect Tessellation of space here. Speaking of which, is it ready for that yet? Melchoir 01:04, 21 July 2006 (UTC)

I realized about redirects, but needed to change anyway, since I had a number of double links! And yes, I think it's ready for a merge/replacement, or whatever you call it. (I should have tried a "move this page" in the first place and it would have been easy!)

Oh, I see what you mean. Well, I'll put in the redirect and copy the talk page material. Melchoir 02:55, 21 July 2006 (UTC)

Category:Honeycombs (geometry)

I have created a new sub-Category:Honeycombs (geometry) page and re-categorised some of the individual honeycomb pages. Anybody feel like helping with the rest? To find them, browse to Category:Polytopes or Category:Tiling Steelpillow 17:04, 8 March 2007 (UTC)

I converted them all I think, changed to lower case (geometry). Tom Ruen 22:41, 8 March 2007 (UTC)

Why does someone think a honeycomb is only 3-dimensional???

The first two sentences read:

"In geometry, a honeycomb is a space filling or close packing of polyhedral cells, so that there are no gaps. It is a three-dimensional example of the more general mathematical tiling or tessellation in any number of dimensions. Honeycomb is also sometimes used for higher dimensional tessellations as well. For clarity, George Olshevsky advocates limiting the term honeycomb to 3-space tessellations and expanding a systematic terminology for higher dimensions: tetracomb as tessellations of 4-space, and pentacomb as tessellations of 5-space, and so on."

But Coxeter, who I believe invented the term, used it consistently for a tessellation of not only 3-dimensional space, but for any dimension.

With much due respect for George Olshevsky: I believe Coxeter's usage carries vastly greater weight, and that it is vastly inappropriate for an article to be based on what "George Olshevsky advocates", regardless of how nice his polytope website is.

(One reason: Searching on MathSciNet for all papers authored by H.S.M. Coxeter gives 254 results; searching for all papers authored by George Olshevsky; or Olshevsky, George; or G. Olshevsky; or Olshevsky, G. yields 0 results.)

This strongly suggests this article should be merged with that of Tessellation.Daqu (talk) 00:35, 29 August 2008 (UTC)

I agree that the title and content are a poor match. Although Olskevsky is at the forefront of current activity in polytope discovery and enumeration, he seldom publishes his results formally. He is also fond of coining his own jargon. So overall I think you are right about Coxeter vs. Olshevsky too. Also, the geometric tilings/tessellations/honeycombs articles are all in a mess. Not, of course, helped by the fact that all three terms are mostly interchangeable. But I think merging this article with Tessellation would be wrong, because this focuses on 3D while the other focuses on 2D. No doubt somewhere these are articles on 4D too. Also, very likely, there are articles which muddle tilings generally with regular and uniform varieties. IMHO we need a single "home" article (I would suggest Tiling (geometry) as the plainest and clearest) giving an overall summary, with Tessellation and Honeycomb (geometry) as redirects, and build up a set of sub-topics from there. Meanwhile, some pruning of the "Olshevsky advocates" stuff would be welcome. -- Cheers, Steelpillow (Talk) 08:49, 29 August 2008 (UTC)

Space-filling polyhedra

It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. Note that if we interpret each brick face as a hexagon having two interior angles of 180 degrees, we can now accept the pattern as a proper tiling. However, not all geometers accept such hexagons.

I added a list of articles by Michael Goldberg which give many space-filling polyhedra. Most of the articles point out that a space-filling polyhedron can be divided into halfs, quarters, etc as smaller polyhedra which are also space-filling in the same way. But Goldberg doesn't clearly differentiate between topological honeycombs (those connected face-to-face) and those that whose volume fit but not face-to-face, which can happen when a space-filler is divided in certain ways. Non-matching faces can be divided further, allowing for coplanar faces. Anyway, the issue to me suggests this article should point out these distinctions. OTOH, like pentagonal tiling also includes non-edge-to-edge connectivity without comment. The value I see in the face-to-face forms is that duality can be defined between two honeycombs. As best I can tell, we can only call them face-to-face honeycombs, or not. Tom Ruen (talk) 04:54, 4 May 2017 (UTC)

Actually I see it is addressed with an example image (copied here), talking about a proper honeycomb, but the word proper is not used elsewhere in this article. Tom Ruen (talk) 05:33, 4 May 2017 (UTC)

Grunbaum and Shephard, in Tilings and Patterns defines a proper tiling: (Chapter 9, Tilings by polygons, p.472) "A normal tiling by polygons is called proper if the intersection of any two tiles is contained in a side (edge) of each tile." This definition still seems confusing, still seems to include cases that are not edge-to-edge, like this one with isohedral triangles: P3-3, and rectangles in a brick pattern P4-22 is also included. Tom Ruen (talk) 05:48, 4 May 2017 (UTC)

This seems to be one of those things that nobody ever clears up. Every discussion I have ever seen at best describes the difference before stating which kind it is interested in, then promptly forgets that the other field exists. For example nobody interested in classifying and enumerating the many pentagonal tilings would say that they are "not proper" just because there are corners of two tiles meeting part way along an edge of a third. We just need to say what we are talking about and make sure that every statement is clear from the context what it means. That may limit a single article's ability to cover both kinds, and frankly I think that is all to the good: the more we mix them, the more confusion we create. — Cheers, Steelpillow (Talk) 19:21, 4 May 2017 (UTC)

Some n-dimensional terminology

Norman Johnson has some terminology, but all seems to apply specifically to face-to-face honeycombs. Tom Ruen (talk) 05:30, 4 May 2017 (UTC)

• (From Geometries and Transformations, section 11.1 Polytopes and Honeycombs): "An n-honeycomb, or polytopal space-filling, is abstractly the same as an (n+1)-polytope but has a scope (all or part of n-space) instead of a body. The only 0-honeycomb is an antipodation, consisting of the two points of S⁰. A 1-honeycomb is a partition of a line or a circle, and its 1-faces are called parts. A 2-honeycomb is a tessellation of a plane or a sphere, and a 3-honeycomb is a cellulation of 3-space or a 3-sphere. The n-faces (“hypercells”) of an n-honeycomb are its cellules, and the (n-1)-faces separating adjacent cellules are walls."