Talk:Weaire–Phelan structure

Voronoi approximation
The structure appears to be approximated by a Voronoi partition whose nuclei are (0,0,0), (1,1,1), and all rotations of (0,1/2,1) and (0,3/2,1), with period 2 in each dimension. Is this correct? &mdash;Tamfang (talk) 21:18, 25 May 2008 (UTC)


 * Sounds quite likely, but I have not studied the Voronoi approach. The polyhedral structure also corresponds to what crystallographers call (or used to call?) a Frank-Kasper phase (ISTR that I originally wrote something about this, but it was deleted because I could not cite a reference - the book had gone back to the library and I lost my note of its title). Anyway, it's well enough known and its Voronoi derivation has surely been documented somewhere. Sorry I can't help more. -- Cheers, Steelpillow 09:43, 26 May 2008 (UTC)


 * A website by Guy Inchbald documents the Voronoi connection. Ferkel (talk) 16:46, 7 December 2022 (UTC)
 * You did notice the equality between domain name and the name here of the person you're replying to, right? Anyway, because the boundaries in the Weaire–Phelan structure are curved, they're definitely not actually Voronoi cells. —David Eppstein (talk) 18:21, 7 December 2022 (UTC)
 * Indeed. Although I am a peer-reviewed and published source on some polyhedral topics, the Weaire-Phelan packing is not one of them. Since writing the above in 2008, I have privately used arguments of symmetry to justify the claim to Voronoi characteristics for the associated polyhedral tiling, but I have not presented those arguments - or had them reviewed - anywhere, so I should not be cited here. &mdash; Cheers, Steelpillow (Talk) (aka Guy Inchbald) 18:36, 7 December 2022 (UTC)
 * The Symmetries of Things, which we are citing as a source already, states that Weaire and Phelan constructed this structure "by relaxing the Voronoi cells" of points at the centers of the cubical tetrastix approximation, in the same sense that the Voronoi cells of the bitruncated cubic honeycomb are relaxed to form Kelvin's foam. On the other hand, I doubt the Voronoi cells have equal volumes, so (unlike in the case for Kelvin) some adjustment would need to be made to get the volumes equal as well as to satisfy Plateau's laws. —David Eppstein (talk) 18:45, 7 December 2022 (UTC)
 * An intriguing question. When I calculated my polyhedral approximation I required constant volume. If you then construct the line between centroids of adjacent 12- and 14-hedra, will it be orthogonal to the joining face (which is a characteristic of Voronoi cells). I never checked that. &mdash; Cheers, Steelpillow (Talk) 09:13, 8 December 2022 (UTC)

and I have been corresponding about this off-wiki. They have now found a paper which describes the construction of the equal-volume polyhedral tiling as a weighted Voronoi partitioning. The classic Voronoi cells are not equal-volume, so the bubble pressures have to be weighted to even them up. Rob Kusner and John M. Sullivan; "Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin's Foam", Forma 11:3, 1996, pp 233-242. Reprinted in The Kelvin Problem, Taylor & Francis, 1996, pp 71-80. copy

Pyritohedron?
It is curiously similar. The Pyritohedron has tetrahedral symmetry. What's the symmetry of this irregular dodecahedron? Tom Ruen (talk) 19:38, 2 July 2008 (UTC)


 * Both of these irregular dodecahedra have tetrahedral symmetry and they do look closely similar. The way to judge whether they are the same polyhedron is to compare the precise face angles, since there is a whole continuum of such dodecahedra. In the absence of any published study demonstrating that the pyritohedron and the Weaire-Phelan dodecahedron have identical angles, we must assume that they do not. -- Cheers, Steelpillow (Talk) 15:14, 3 July 2008 (UTC)


 * Does "pyritohedron" mean the specific shape of pyrite crystals? I've always seen it used for any dodecapentagon with Th symmetry. —Tamfang (talk) 16:53, 10 July 2008 (UTC)


 * I have not seen it used in the context of any such dodecahedron, but then there is a lot that I have not read. Can you cite authoritative references, or does it only occur in less formal contexts (e.g. that have not been peer reviewed)? -- Cheers, Steelpillow (Talk) 19:58, 10 July 2008 (UTC)


 * I've only ever seen the word in relatively informal contexts. —Tamfang (talk) 18:41, 15 July 2008 (UTC)


 * I imagine the pyritohedron is defined in a way that generates a honeycomb (since it exists in a crystaline solid)! Is it cell-transitive? I can't find any nets to build it and see. Tom Ruen (talk) 20:31, 10 July 2008 (UTC)
 * No, it is not a spacefiller. It is the shape of the pyrites crystal. Crystals are not generally space fillers, they are the result of periodic atom packings: nothing in that says the crystal itself must be a space-filler (although an arbitrary few are). -- Cheers, Steelpillow (Talk) 21:24, 10 July 2008 (UTC)

Th or dihedral?

 * One is an irregular dodecahedron with pentagonal faces, possessing dihedral symmetry

More precisely, what? D2h, say? —Tamfang (talk) 03:49, 11 July 2010 (UTC)


 * Just noticed, the symmetries of both cells are all wrong. The dodecahedron has tetrahedral symmetry, the tetrakaidecahedron has a lower symmetry like a tetrahedron stretched along a twofold axis. The pentagonal faces have the symmetry of an isosceles triangle, whatever that is. &mdash; Cheers, Steelpillow (Talk) 20:52, 11 July 2010 (UTC)


 * Okay, Th and D2d. —Tamfang (talk) 03:38, 14 July 2010 (UTC)

Hyphen or en dash in name
Traditionally, a "double-barrelled" name is hyphenated. An en dash indicates some form of "from–to" grammatical construction. Weaire and Phelan were co-discoverers of their bubble. Therefore the correct form of the name uses a hyphen and not an en dash. wrt the Manual of Style, this corresponds to the use of a hyphen "to link related terms in compound adjectives". To clarify one possible confusion, the MoS also says, "When naming an article, a hyphen is not used as a substitute for an en dash that properly belongs in the title". Note that in the present article, the en dash does not properly belong in the title, and so this condition does not apply. -- Cheers, Steelpillow (Talk) 16:25, 2 May 2009 (UTC)
 * Please have a look at MOS:DASH:
 * "As a substitute for some uses of and, to, or versus for marking a relationship involving independent elements in certain compound expressions (Canada–US border, blood–brain barrier, time–altitude graph, 4–3 win in the opening game, male–female ratio, 3–2 majority verdict, Lincoln–Douglas debate, diode–transistor logic; but a hyphen is used in Sino-Japanese trade, in which Sino-, being a prefix, lacks lexical independence.)"
 * So, there should be an en-dash in the article name, as a substitute for "and". The move automatically generates the required redirect with the hyphen in the name.
 * Traditionally, there is an en-dash in these constructs, please have a look at scientific textbook classics from e.g. Cambridge University Press. -- Crowsnest (talk) 16:49, 2 May 2009 (UTC)
 * Good point (though most of my books are currently in storage, so I take your assertion on trust). Meanwhile the Dash article notes that:
 * The Chicago Manual of Style uses a hyphen in this context.
 * For relationships and connections, "a 'simple' compound used as an adjective is written with a hyphen; at least one authority considers name pairs, as in the Taft-Hartley Act to be 'simple',[5] while most consider an en dash appropriate there[citation needed] to represent the parallel relationship, as in the McCain–Feingold bill or Bose–Einstein statistics."
 * When that missing citation arrives, to justify the assertion that "most" consider and en dash appropriate, I will be willing to agree that The Chicago Manual of Style and I are both out of touch with reality.
 * Meanwhile, the WP:MOS Talk page has a long and ongoing discussion on this very point. You may wish to add your CUP references to the pot, as most examples given to date seem to be pointing my way. -- Cheers, Steelpillow (Talk) 20:41, 2 May 2009 (UTC)
 * What counts for me is the Wikipedia Manual of Style. The current style is obviously an en dash, as in the MoS, and it has been since quite some time, see e.g. this 2006 version Did not read well, there have been changes. Crowsnest (talk) 22:50, 2 May 2009 (UTC) . But let us wait on what happens on WT:MOS. -- Crowsnest (talk) 22:35, 2 May 2009 (UTC)
 * Apart from established style rules, ease of typing, and the fact that hyphens are universally used for compound names (in Wikipedia and elsewhere), there is a practical reason for using a plain hyphen: most readers and editors will not notice the difference and may be quite confused on noticing that "Weaire-Phelan structure" is a redirect to "Weaire–Phelan structure" (!). It means, for example, that links to the former (which is how most people would type the name) must be eventually fixed to point to the second. Please, folks, be reasonable: the en-dash here has no advantage whatsoever over a hyphen, not even aesthetics, and several obvious disadvantages. Please move it back to the hypenated form.  All the best, --Jorge Stolfi (talk) 16:15, 28 June 2009 (UTC)
 * Unfortunately, the MOS has been taken over by endash fundamentalists who seem able to blank The Chicago Manual of Style from their consciousness. I did actually move the article back once, even briefly tried to reason with them, silly me. -- Cheers, Steelpillow (Talk) 18:07, 28 June 2009 (UTC)

Applications section
There's another application, vaguely related to finite element mesh generation, in my paper "Tiling space and slabs with acute tetrahedra" (cs.CG/0302027, 10.1016/j.comgeo.2003.11.003): if one places a vertex at the center of each cell of the Weaire-Phelan structure, one can find a tiling of space by tetrahedra all of whose dihedral angles are acute. I won't add this myself because it would be too self-serving, but I won't mind if someone else reviews it and thinks it relevant enough to mention. —David Eppstein (talk) 03:38, 4 May 2009 (UTC)
 * I found another neat tiling with integer vertex coordinates, by dissecting the dodecahedron into twelve congruent pentagonal pyramids with apices at its centre, and sticking the pyramids onto the incident faces of the tetrakaidecahedron. Facets line up such that each cell has 8 vertices, 2 rhombic faces and 8 obtuse isosceles triangles. I think these are more like "related honeycombs" than applications. &mdash; Cheers, Steelpillow (Talk) 20:59, 11 July 2010 (UTC)

One suggestion
I am no expert on the subject matter of the article, but one thing I would really like to see is this:

A concise and readily identifiable description of the precise mathematical problem (in mathematical terms) that the Kelvin solution and the Weaire-Phelan solution are trying to solve. (E.g., Are the surfaces supposed to be piecewise differentiable? Is the number of distinct cells limited to be finite? Are we assuming a lattice arrangement, and if so, are we minimizing average surface area per fundamental domain among all lattices in R3, or something else?)

When I say "readily identifiable", I mean a definition that one does have to hunt through the text to find. By "concise" I mean that one should not have to read a lot of text in order to extract this most basic information.

And of course I am aware that anyone can click on the highlighted text "Kelvin Problem". But it's always a good idea to include at least a brief definition of what is the problem that this world record holder holds the world record in instead of expecting readers to piece together text from two articles often written with different notation and even terminology. 2601:200:C000:1A0:9D37:4459:C381:5539 (talk) 21:32, 27 March 2022 (UTC)
 * I think that having a concise and readily identifiable form of the precise mathematical assumptions of the problem is likely to be an impossible task, in part because figuring out the right definition is a big part of the problem, and in part because solutions to similar problems in the past have involved abstruse concepts in geometric measure theory like currents rather than anything easily explained. It is obviously not required to be a lattice arrangement: the cells of the Weaire–Phelan structure do not form a lattice. I think the article already does define the problem adequately if a little vaguely (with good reason for the vagueness): a partition of space into infinitely many cells of equal volume, minimizing the surface area per cell as averaged over large volumes of space. —David Eppstein (talk) 06:02, 28 March 2022 (UTC)
 * I assume our OP means Kelvin and not Kepler. However equal volume is not necessarily assumed. Investigators tend to just take it as a starting point to make life easier (I seem to recall that Weaire and Phelan were explicit about this, but I do not have any of their studies to hand). One approach which I suggest elsewhere (not peer reviewed) is; "How would bubbles at a given density pack together, to give the least possible area of surface film between them?" where bubble density is the number of bubbles per unit volume. But it might be original research to make any such statement here. &mdash; Cheers, Steelpillow (Talk) 09:36, 28 March 2022 (UTC)
 * You definitely need equal volume to make the problem interesting. Otherwise a repeating pattern of one large bubble together with many very tiny bubbles would have much less surface area than the patterns described in this article. —David Eppstein (talk) 16:05, 28 March 2022 (UTC)
 * Good point. Probably the most useful parameter would be related to the mean surface energy, or whatever governs the stability of the foam. &mdash; Cheers, Steelpillow (Talk) 16:42, 28 March 2022 (UTC)