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415:. These prisms surround cubical voids which form one fourth of the cells of the cubical tiling; the remaining three fourths of the cells fill the prisms, offset by half a unit from the integer grid aligned with the prism walls. Similarly, in the Weaire–Phelan structure itself, which has the same symmetries as the tetrastix structure, 1/4 of the cells are dodecahedra and 3/4 are tetrakaidecahedra.
400:
410:
The tetrakaidecahedron cells, linked up in face-to-face chains of cells along their hexagonal faces, form chains in three perpendicular directions. A combinatorially equivalent structure to the Weaire–Phelan structure can be made as a tiling of space by unit cubes, lined up face-to-face into infinite
312:
Since the discovery of the Weaire–Phelan structure, other counterexamples to the Kelvin conjecture have been found, but the Weaire–Phelan structure continues to have the smallest known surface area per cell of these counterexamples. Although numerical experiments suggest that the Weaire–Phelan
422:
associated with the Weaire–Phelan structure (obtained by flattening the faces and straightening the edges) is also referred to loosely as the Weaire–Phelan structure. It was known well before the Weaire–Phelan structure was discovered, but the application to the Kelvin problem was overlooked.
395:
symmetry. Like the hexagons in the Kelvin structure, the pentagons in both types of cells are slightly curved. The surface area of the Weaire–Phelan structure is 0.3% less than that of the Kelvin structure.
234:
126:
asked the corresponding question for three-dimensional space: how can space be partitioned into cells of equal volume with the least area of surface between them? Or, in short, what was the most efficient
1170:
Cros, Christian; Pouchard, Michel; Hagenmuller, Paul (December 1970), "Sur une nouvelle famille de clathrates minéraux isotypes des hydrates de gaz et de liquides, interprétation des résultats obtenus",
359:
The Weaire–Phelan structure differs from Kelvin's in that it uses two kinds of cells, although they have equal volume. Like the cells in Kelvin's structure, these cells are combinatorially equivalent to
297:
Although Kelvin did not state it explicitly as a conjecture, the idea that the foam of the bitruncated cubic honeycomb is the most efficient foam, and solves Kelvin's problem, became known as the
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and his student Robert Phelan discovered the Weaire–Phelan structure through computer simulations of foam, and showed that it was more efficient, disproving the Kelvin conjecture.
288:
186:
261:
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Frank, F. C.; Kasper, J. S. (1959), "Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures",
317:. The minimality of the sphere as a surface enclosing a single volume was not proven until the 19th century, and the next simplest such problem, the
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for its faces, obeying
Plateau's laws and reducing the area of the structure by 0.2% compared with the corresponding polyhedral structure.
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structure is optimal, this remains unproven. In general, it has been very difficult to prove the optimality of structures involving
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of tiling space by equal volume cells of minimum surface area than the previous best-known solution, the Kelvin structure.
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In two dimensions, the subdivision of the plane into cells of equal area with minimum average perimeter is given by the
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Kasper, J. S.; Hagenmuller, P.; Pouchard, M.; Cros, C. (December 1965), "Clathrate structure of silicon Na
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Gabbrielli, Ruggero (1 August 2009), "A new counter-example to Kelvin's conjecture on minimal surfaces",
142:
95:
731:, NATO Advanced Science Institutes Series E: Applied Sciences, vol. 354, Kluwer, pp. 379–402,
301:. It was widely believed, and no counter-example was known for more than 100 years. Finally, in 1993,
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on hexagonal faces. Therefore, Kelvin's proposed structure uses curvilinear edges and slightly warped
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with two hexagonal and twelve pentagonal faces, in this case only possessing two mirror planes and a
161:, formulated by Joseph Plateau in the 19th century, according to which minimal foam surfaces meet at
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Where the components of the crystal lie at the corners of the polyhedra, it is known as the "Type I
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236:. The angles of the polyhedral structure are different; for instance, its edges meet at angles of
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463:. Where the components of the crystal lie at the centres of the polyhedra it forms one of the
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827:"Approximation of partitions of least perimeter by Γ-convergence: around Kelvin's conjecture"
406:, modeling the face-to-face chains of tetrakaidecahedron cells in the Weaire–Phelan structure
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formed by methane, propane and carbon dioxide at low temperatures have a structure in which
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1049:"Complex alloy structures regarded as sphere packings. I. Definitions and basic principles"
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also form this structure, with silicon or germanium at nodes, and alkali metals in cages.
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angles at their edges, with these edges meeting each other in sets of four with angles of
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1014:(2011), "Scientists make the 'perfect' foam: Theoretical low-energy foam made for real",
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963:"An experimental realization of the Weaire-Phelan structure in monodisperse liquid foam"
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on enclosing two volumes, remained open for over 100 years until being proven in 2002.
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page with illustrations and freely downloadable 'nets' for printing and making models.
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578:; Phelan, R. (1994), "A counter-example to Kelvin's conjecture on minimal surfaces",
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with 6 square faces and 8 hexagonal faces. However, this honeycomb does not satisfy
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write that it is "implicit rather than directly stated in Kelvin's original papers"
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together, and the larger gas molecules are trapped in the polyhedral cages. Some
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square prisms in the same way to form a structure of interlocking prisms called
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Gabbrielli, R.; Meagher, A.J.; Weaire, D.; Brakke, K.A.; Hutzler, S. (2012),
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The associated polyhedral honeycomb is found in two related geometries of
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and Robert Phelan found that this structure was a better solution of the
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A close-up of the mold used for the growth of ordered liquid foams.
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molecules lie at the nodes of the Weaire–Phelan structure and are
666:
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The Weaire–Phelan structure is the inspiration for the design by
727:
Sullivan, John M. (1999), "The geometry of bubbles and foams",
229:{\displaystyle \arccos {\tfrac {1}{3}}\approx 109.47^{\circ }}
74:
of equal-sized bubbles, with two different shapes. In 1993,
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891:
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71:
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is a three-dimensional structure representing an idealised
1229:
3D models of the Weaire–Phelan, Kelvin and P42a structures
23:
876:(2009), "Chapter 14. Proof of Double Bubble Conjecture",
669:"On the Division of Space with Minimum Partitional Area"
102:
cells are deformed slightly to form the Kelvin structure
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202:
905:, Wellesley, Massachusetts: A K Peters, p. 351,
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foam? This problem has since been referred to as the
947:(3rd ed.), Cornell University Press, p.
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118:(116-27 BCE), it was not proven until the work of
1242:, Alexandru Pintea, 2017, Individual First Prize
895:; Burgiel, Heidi; Goodman-Strauss, Chaim (2008),
85:
1250:
1207:"A Problem of Bubbles Frames an Olympic Design"
548:, a book by Weaire on this and related problems
1240:"Weaire-Phelan Smart Modular Space Settlement"
878:Geometric Measure Theory: A Beginner's Guide
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444:Experiments have shown that, with favorable
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799:Freiberger, Marianne (24 September 2009),
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619:Discrete & Computational Geometry
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448:, equal-volume bubbles spontaneously
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1047:Frank, F. C.; Kasper, J. S. (1958),
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616:(2001), "The honeycomb conjecture",
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1244:NASA Ames Space Settlement Contest:
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729:Foams and emulsions (Cargèse, 1997)
13:
1205:Fountain, Henry (August 5, 2008),
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452:into the Weaire–Phelan structure.
372:with pentagonal faces, possessing
137:Kelvin proposed a foam called the
14:
1285:
1222:
897:"Understanding the Irish Bubbles"
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385:truncated hexagonal trapezohedron
1173:Journal of Solid State Chemistry
528:Beijing National Aquatics Centre
517:Beijing National Aquatics Centre
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24:
943:The Nature of the Chemical Bond
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880:(4th ed.), Academic Press
751:Philosophical Magazine Letters
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545:The Pursuit of Perfect Packing
324:
86:History and the Kelvin problem
1:
1142:10.1126/science.150.3704.1713
801:"Kelvin's bubble burst again"
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1193:10.1016/0022-4596(70)90053-8
990:10.1080/09500839.2011.645898
845:10.1080/10586458.2011.565233
530:, the 'Water Cube', for the
283:{\displaystyle 120^{\circ }}
181:{\displaystyle 120^{\circ }}
7:
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383:). The second is a form of
256:{\displaystyle 90^{\circ }}
143:bitruncated cubic honeycomb
141:. His foam is based on the
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96:bitruncated cubic honeycomb
10:
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715:Weaire & Phelan (1994)
1090:10.1107/s0365110x59001499
1069:10.1107/s0365110x58000487
807:, University of Cambridge
771:10.1080/09500830903022651
688:10.1080/14786448708628135
600:10.1080/09500839408241577
1026:10.1038/nature.2011.9504
902:The Symmetries of Things
832:Experimental Mathematics
319:double bubble conjecture
147:convex uniform honeycomb
19:Weaire–Phelan structure
825:Oudet, Édouard (2011),
68:Weaire–Phelan structure
676:Philosophical Magazine
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340:Irregular dodecahedron
303:Trinity College Dublin
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1234:Weaire–Phelan Bubbles
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532:2008 Summer Olympics
374:tetrahedral symmetry
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263:on square faces, or
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151:truncated octahedron
112:honeycomb conjecture
100:truncated octahedron
1185:1970JSSCh...2..570C
1134:1965Sci...150.1713K
1128:(3704): 1713–1714,
982:2012PMagL..92....1G
763:2009PMagL..89..483G
592:1994PMagL..69..107W
465:Frank–Kasper phases
446:boundary conditions
432:In physical systems
1274:1994 introductions
520:
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389:tetrakaidecahedron
352:Tetrakaidecahedron
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153:, a space-filling
122:in 1999. In 1887,
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38:Fibrifold notation
1077:Acta Crystallogr.
1056:Acta Crystallogr.
912:978-1-56881-220-5
457:crystal structure
299:Kelvin conjecture
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155:convex polyhedron
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614:Hales, T. C.
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492:alkali metal
480:Gas hydrates
478:structure".
473:
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427:Applications
417:
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370:dodecahedron
366:pyritohedron
358:
311:
307:Denis Weaire
298:
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132:
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76:Denis Weaire
67:
61:
626:(1): 1–22,
364:. One is a
325:Description
129:soap bubble
124:Lord Kelvin
34:Space group
1253:Categories
976:(1): 1–6,
701:2012-06-15
576:Weaire, D.
553:References
502:germanides
305:physicist
1264:Polyhedra
1034:136626668
787:137653272
779:0950-0839
498:silicides
476:clathrate
469:A15 phase
461:chemistry
420:honeycomb
413:tetrastix
404:Tetrastix
276:∘
249:∘
222:∘
214:≈
199:
174:∘
1158:21291705
1150:17768869
998:25427974
939:(1960),
538:See also
495:hydrides
64:geometry
1181:Bibcode
1130:Bibcode
1122:Science
978:Bibcode
921:2410150
861:2945749
853:2836251
759:Bibcode
737:1688327
652:1797293
588:Bibcode
526:of the
51:n (223)
1156:
1148:
1112:and Na
1032:
1017:Nature
996:
919:
909:
859:
851:
811:4 July
785:
777:
735:
650:
467:, the
218:109.47
196:arccos
66:, the
1154:S2CID
1052:(PDF)
1030:S2CID
994:S2CID
966:(PDF)
857:S2CID
783:S2CID
695:(PDF)
672:(PDF)
484:water
1146:PMID
907:ISBN
813:2017
775:ISSN
500:and
145:, a
94:The
72:foam
1189:doi
1138:doi
1126:150
1118:136
1085:doi
1064:doi
1022:doi
986:doi
949:471
841:doi
767:doi
684:doi
638:hdl
628:doi
596:doi
459:in
272:120
170:120
62:In
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471:.
245:90
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47:Pm
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1183::
1177:2
1140::
1132::
1114:x
1106:8
1094:.
1087::
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1037:.
1024::
1001:.
988::
980::
882:.
843::
769::
761::
705:.
686::
640::
630::
603:.
598::
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380:h
378:T
376:(
208:3
205:1
54:2
49:3
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