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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)
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)
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 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)
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)
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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)
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)
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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)
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)
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 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)
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 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)
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)