28:
17:
497:
718:
317:
114:
422:
194:
174:
154:
134:
85:
379:
229:
427:
344:
276:
249:
563:
The case when the problem is restricted to a square grid was solved in 1989 by
Jaigyoung Choe who proved that the optimal figure is an irregular hexagon.
553:, who mentions in his work that there is reason to believe that the conjecture may have been present in the minds of mathematicians before Varro.
795:
759:
176:
has additional components that are unbounded or whose area is not one; allowing these additional components cannot shorten
608:
690:
Fejes, László (1943). "Über das kürzeste
Kurvennetz, das eine Kugeloberfläche in flächengleiche konvexe Teile zerlegt".
560:
of the plane, in which every circle is tangent to six other circles, which fill just over 90% of the area of the plane.
790:
577:
136:) all of which are bounded and have unit area. Then, averaged over large disks in the plane, the average length of
499:
The value on the right hand side of the inequality is the limiting length per unit area of the hexagonal tiling.
55:
of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician
156:
per unit area is at least as large as for the hexagon tiling. The theorem applies even if the complement of
572:
281:
785:
90:
557:
384:
542:
603:
545:
published a proof for a special case of the conjecture, in which each cell is required to be a
508:
179:
159:
139:
119:
70:
349:
199:
703:
639:
512:
322:
254:
8:
665:
643:
617:
234:
740:
647:
730:
627:
27:
764:
719:"On the existence and regularity of fundamental domains with least boundary area"
699:
635:
550:
381:
covered by bounded unit-area components. (If these are the only components, then
56:
116:, subdividing the plane into regions (connected components of the complement of
546:
44:
32:
21:
492:{\displaystyle \limsup _{r\to \infty }{\frac {L_{r}}{A_{r}}}\geq {\sqrt{12}}.}
779:
744:
735:
527:
631:
669:
622:
538:
535:
52:
48:
507:
The first record of the conjecture dates back to 36 BC, from
16:
760:"Mathematicians Complete Quest to Build 'Spherical Cubes'"
549:. The full conjecture was proven in 1999 by mathematician
531:
430:
387:
352:
325:
284:
257:
237:
202:
182:
162:
142:
122:
93:
73:
491:
416:
373:
338:
311:
270:
243:
223:
188:
168:
148:
128:
108:
79:
777:
432:
582:on the solution of the similar problem in 3D.
606:(January 2001). "The Honeycomb Conjecture".
734:
621:
96:
35:, with circles centered on each hexagon.
26:
15:
692:Math. Naturwiss. Anz. Ungar. Akad. Wiss
778:
757:
598:
596:
689:
663:
602:
716:
659:
657:
530:used a similar theorem to argue why
609:Discrete and Computational Geometry
593:
13:
556:It is also related to the densest
442:
312:{\displaystyle \Gamma \cap B(0,r)}
285:
183:
163:
143:
123:
87:be any system of smooth curves in
74:
14:
807:
796:Conjectures that have been proved
654:
723:Journal of Differential Geometry
424:.) Then the theorem states that
109:{\displaystyle \mathbb {R} ^{2}}
417:{\displaystyle A_{r}=\pi r^{2}}
751:
717:Choe, Jaigyoung (1989-01-01).
710:
683:
439:
368:
356:
306:
294:
218:
206:
1:
586:
516:
511:, but is often attributed to
251:centered at the origin, let
7:
575:, a counter-example to the
566:
278:denote the total length of
10:
812:
502:
231:denote the disk of radius
62:
346:denote the total area of
791:Euclidean plane geometry
526:). In the 17th century,
573:Weaire–Phelan structure
189:{\displaystyle \Gamma }
169:{\displaystyle \Gamma }
149:{\displaystyle \Gamma }
129:{\displaystyle \Gamma }
80:{\displaystyle \Gamma }
31:This honeycomb forms a
736:10.4310/jdg/1214443065
666:"Honeycomb Conjecture"
509:Marcus Terentius Varro
493:
418:
375:
374:{\displaystyle B(0,r)}
340:
313:
272:
245:
225:
224:{\displaystyle B(0,r)}
190:
170:
150:
130:
110:
81:
43:states that a regular
36:
24:
758:Cepelewicz, Jordana.
632:10.1007/s004540010071
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419:
376:
341:
339:{\displaystyle A_{r}}
314:
273:
271:{\displaystyle L_{r}}
246:
226:
191:
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131:
111:
82:
30:
19:
513:Pappus of Alexandria
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200:
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91:
71:
51:has the least total
41:honeycomb conjecture
664:Weisstein, Eric W.
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126:
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37:
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786:Discrete geometry
543:László Fejes Tóth
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244:{\displaystyle r}
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604:Hales, Thomas C.
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196:. Formally, let
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811:
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776:
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765:Quanta Magazine
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551:Thomas C. Hales
523:
519:
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100:
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88:
72:
69:
68:
65:
57:Thomas C. Hales
12:
11:
5:
809:
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798:
793:
788:
772:
771:
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709:
682:
653:
591:
590:
588:
585:
584:
583:
568:
565:
558:circle packing
547:convex polygon
504:
501:
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433:lim sup
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45:hexagonal grid
33:circle packing
22:hexagonal grid
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673:. Retrieved
623:math/9906042
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607:
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66:
40:
38:
698:: 349–354.
616:(1): 1–22.
541:. In 1943,
780:Categories
587:References
580:conjecture
539:honeycombs
528:Jan Brożek
524: 350
522: – c.
520: 290
319:, and let
20:A regular
745:0022-040X
670:MathWorld
536:hexagonal
472:≥
443:∞
440:→
402:π
289:∩
286:Γ
184:Γ
164:Γ
144:Γ
124:Γ
75:Γ
53:perimeter
49:honeycomb
648:14849112
567:See also
704:0024155
640:1797293
534:create
503:History
63:Theorem
743:
702:
675:27 Dec
646:
638:
578:Kelvin
729:(3).
644:S2CID
618:arXiv
741:ISSN
677:2010
532:bees
67:Let
39:The
731:doi
628:doi
47:or
782::
762:.
739:.
727:29
725:.
721:.
700:MR
696:62
694:.
668:.
656:^
642:.
636:MR
634:.
626:.
614:25
612:.
595:^
517:c.
477:12
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768:.
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733::
706:.
679:.
650:.
630::
620::
515:(
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481:4
465:r
461:A
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363:,
360:0
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328:A
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304:r
301:,
298:0
295:(
292:B
264:r
260:L
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216:r
213:,
210:0
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204:B
102:2
97:R
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
