363:
101:
145:
470:
216:
477:
75:
334:
401:
321:
660:
583:
202:
24:
copies of the body. For most bodies the value of the packing constant is unknown. The following is a list of bodies in
Euclidean spaces whose packing constant is known.
233:
605:
817:
Reinhardt, Karl (1934). "Über die dichteste gitterförmige
Lagerung kongruente Bereiche in der Ebene und eine besondere Art konvexer Kurven".
528:
914:
1046:
879:
Bezdek, András; Kuperberg, Włodzimierz (1990). "Maximum density space packing with congruent circular cylinders of infinite length".
162:
325:
965:
40:
packing constant. Therefore, any such body for which the lattice packing constant was previously known, such as any
1041:
25:
1036:
852:
Mount, David M.; Silverman, Ruth (1990). "Packing and covering the plane with translates of a convex polygon".
510:
433:
206:
33:
362:
21:
730:
Cohn, Henry; Kumar, Abhinav (2009). "Optimality and uniqueness of the Leech lattice among lattices".
495:
485:
690:
Bezdek, András; Kuperberg, Włodzimierz (2010). "Dense packing of space with various convex solids".
506:
429:
44:, consequently has a known packing constant. In addition to these bodies, the packing constants of
1012:
994:
943:
923:
834:
797:
776:
757:
739:
691:
775:
Chang, Hai-Chau; Wang, Lih-Chung (2010). "A Simple Proof of Thue's
Theorem on Circle Packing".
502:
392:
1004:
982:
933:
890:
861:
826:
749:
223:
152:
100:
63:
29:
17:
469:
316:{\displaystyle \eta _{so}={\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414}
144:
961:
476:
37:
215:
665:
588:
354:
108:
20:
of a geometric body is the largest average density achieved by packing arrangements of
1030:
1016:
865:
838:
947:
761:
796:
Hales, Thomas; Kusner, Wöden (2016). "Packings of regular pentagons in the plane".
82:
74:
753:
1008:
881:
350:
910:"Upper bounds on packing density for circular cylinders with high aspect ratio"
938:
909:
894:
333:
655:{\displaystyle {\frac {({\frac {\pi }{2}})^{12}}{12!}}\approx 0.000000471087}
136:
400:
45:
830:
112:
41:
578:{\displaystyle {\frac {({\frac {\pi }{2}})^{4}}{4!}}\approx 0.2536695}
744:
781:
696:
999:
802:
928:
711:
Fejes Tóth, László (1950). "Some packing and covering theorems".
94:
Shapes such that congruent copies can form a tiling of space
475:
468:
399:
361:
332:
214:
143:
99:
73:
197:{\displaystyle {\frac {5-{\sqrt {5}}}{3}}\approx 0.92131}
349:
Linear-time (in number of vertices) algorithm given by
985:(2016). "The sphere packing problem in dimension 8".
608:
531:
236:
165:
654:
577:
315:
196:
48:in 8 and 24 dimensions are almost exactly known.
32:body has a packing constant that is equal to its
488:whose inscribed sphere is contained in the shape
8:
966:"Sphere Packing Solved in Higher Dimensions"
998:
937:
927:
801:
780:
743:
695:
629:
615:
609:
607:
552:
538:
532:
530:
291:
281:
265:
253:
241:
235:
175:
166:
164:
50:
679:
685:
683:
915:Discrete & Computational Geometry
7:
341:All 2-fold symmetric convex polygons
14:
626:
612:
549:
535:
494:Fraction of the volume of the
1:
819:Abh. Math. Sem. Univ. Hamburg
866:10.1016/0196-6774(90)90010-C
754:10.4007/annals.2009.170.1003
28:proved that in the plane, a
1009:10.4007/annals.2017.185.3.7
1063:
484:All shapes contained in a
1047:Mathematics-related lists
939:10.1007/s00454-014-9593-6
895:10.1112/s0025579300012808
36:packing constant and its
511:rhombic enneacontahedron
908:Kusner, Wöden (2014).
713:Acta Sci. Math. Szeged
656:
579:
480:
473:
442:Half-infinite cylinder
404:
366:
337:
317:
219:
198:
148:
104:
78:
987:Annals of Mathematics
854:Journal of Algorithms
732:Annals of Mathematics
657:
580:
505:. Examples pictured:
479:
472:
403:
365:
336:
318:
218:
199:
147:
103:
77:
606:
529:
496:rhombic dodecahedron
486:rhombic dodecahedron
408:Bi-infinite cylinder
234:
163:
135:Proof attributed to
666:Hypersphere packing
589:Hypersphere packing
507:rhombicuboctahedron
498:filled by the shape
964:(March 30, 2016),
831:10.1007/bf02940676
652:
575:
481:
474:
405:
367:
338:
313:
220:
194:
149:
105:
79:
1042:Discrete geometry
983:Viazovska, Maryna
671:
670:
644:
623:
567:
546:
503:Kepler conjecture
393:Kepler conjecture
305:
296:
270:
209:and Wöden Kusner
186:
180:
1054:
1037:Packing problems
1021:
1020:
1002:
979:
973:
972:
962:Klarreich, Erica
958:
952:
951:
941:
931:
905:
899:
898:
876:
870:
869:
849:
843:
842:
814:
808:
807:
805:
793:
787:
786:
784:
772:
766:
765:
747:
738:(3): 1003–1050.
727:
721:
720:
708:
702:
701:
699:
687:
661:
659:
658:
653:
645:
643:
635:
634:
633:
624:
616:
610:
584:
582:
581:
576:
568:
566:
558:
557:
556:
547:
539:
533:
460:
458:
457:
426:
424:
423:
388:
386:
385:
322:
320:
319:
314:
306:
304:
297:
292:
286:
285:
271:
266:
254:
249:
248:
224:Smoothed octagon
203:
201:
200:
195:
187:
182:
181:
176:
167:
153:Regular pentagon
132:
130:
129:
64:Packing constant
51:
18:packing constant
1062:
1061:
1057:
1056:
1055:
1053:
1052:
1051:
1027:
1026:
1025:
1024:
993:(3): 991–1015.
981:
980:
976:
970:Quanta Magazine
960:
959:
955:
907:
906:
902:
878:
877:
873:
851:
850:
846:
816:
815:
811:
795:
794:
790:
774:
773:
769:
729:
728:
724:
710:
709:
705:
689:
688:
681:
676:
636:
625:
611:
604:
603:
559:
548:
534:
527:
526:
455:
453:
448:
421:
419:
414:
383:
381:
376:
287:
255:
237:
232:
231:
168:
161:
160:
127:
125:
120:
30:point symmetric
12:
11:
5:
1060:
1058:
1050:
1049:
1044:
1039:
1029:
1028:
1023:
1022:
974:
953:
922:(4): 964–978.
900:
871:
860:(4): 564–580.
844:
809:
788:
767:
722:
703:
678:
677:
675:
672:
669:
668:
662:
651:
650:0.000000471087
648:
642:
639:
632:
628:
622:
619:
614:
601:
598:
595:
592:
591:
585:
574:
571:
565:
562:
555:
551:
545:
542:
537:
524:
521:
518:
515:
514:
499:
492:
489:
482:
465:
464:
461:
446:
443:
440:
437:
436:
427:
412:
409:
406:
396:
395:
389:
374:
371:
368:
358:
357:
355:Ruth Silverman
347:
345:
342:
339:
329:
328:
323:
312:
309:
303:
300:
295:
290:
284:
280:
277:
274:
269:
264:
261:
258:
252:
247:
244:
240:
229:
226:
221:
211:
210:
204:
193:
190:
185:
179:
174:
171:
158:
155:
150:
140:
139:
133:
118:
115:
106:
96:
95:
92:
89:
86:
80:
70:
69:
66:
61:
58:
55:
13:
10:
9:
6:
4:
3:
2:
1059:
1048:
1045:
1043:
1040:
1038:
1035:
1034:
1032:
1018:
1014:
1010:
1006:
1001:
996:
992:
988:
984:
978:
975:
971:
967:
963:
957:
954:
949:
945:
940:
935:
930:
925:
921:
917:
916:
911:
904:
901:
896:
892:
888:
884:
883:
875:
872:
867:
863:
859:
855:
848:
845:
840:
836:
832:
828:
824:
820:
813:
810:
804:
799:
792:
789:
783:
778:
771:
768:
763:
759:
755:
751:
746:
741:
737:
733:
726:
723:
718:
714:
707:
704:
698:
693:
686:
684:
680:
673:
667:
663:
649:
646:
640:
637:
630:
620:
617:
602:
599:
596:
594:
593:
590:
586:
572:
569:
563:
560:
553:
543:
540:
525:
522:
519:
517:
516:
512:
508:
504:
501:Corollary of
500:
497:
493:
490:
487:
483:
478:
471:
467:
466:
463:Wöden Kusner
462:
451:
447:
444:
441:
439:
438:
435:
431:
428:
417:
413:
410:
407:
402:
398:
397:
394:
390:
379:
375:
372:
369:
364:
360:
359:
356:
352:
348:
346:
343:
340:
335:
331:
330:
327:
324:
310:
307:
301:
298:
293:
288:
282:
278:
275:
272:
267:
262:
259:
256:
250:
245:
242:
238:
230:
227:
225:
222:
217:
213:
212:
208:
205:
191:
188:
183:
177:
172:
169:
159:
156:
154:
151:
146:
142:
141:
138:
134:
123:
119:
116:
114:
110:
107:
102:
98:
97:
93:
90:
87:
84:
81:
76:
72:
71:
67:
65:
62:
59:
56:
53:
52:
49:
47:
43:
39:
35:
31:
27:
23:
19:
990:
986:
977:
969:
956:
919:
913:
903:
886:
880:
874:
857:
853:
847:
822:
818:
812:
791:
770:
745:math/0403263
735:
731:
725:
716:
712:
706:
449:
415:
377:
207:Thomas Hales
121:
46:hyperspheres
15:
882:Mathematika
825:: 216–230.
782:1009.4322v1
697:1008.2398v1
597:Hypersphere
520:Hypersphere
387:≈ 0.7404805
57:Description
34:translative
1031:Categories
1000:1603.04246
803:1602.07220
674:References
459:≈ 0.906900
425:≈ 0.906900
131:≈ 0.906900
85:prototiles
83:Monohedral
26:Fejes Tóth
1017:119286185
929:1309.6996
889:: 74–80.
839:120336230
647:≈
618:π
573:0.2536695
570:≈
541:π
434:Kuperberg
326:Reinhardt
308:≈
299:−
279:
273:−
260:−
239:η
189:≈
173:−
68:Comments
60:Dimension
22:congruent
948:38234737
762:10696627
311:0.902414
454:√
420:√
382:√
192:0.92131
126:√
113:Ellipse
42:ellipse
38:lattice
1015:
946:
837:
760:
450:π
430:Bezdek
416:π
378:π
370:Sphere
122:π
109:Circle
1013:S2CID
995:arXiv
944:S2CID
924:arXiv
835:S2CID
798:arXiv
777:arXiv
758:S2CID
740:arXiv
692:arXiv
351:Mount
54:Image
664:See
587:See
509:and
432:and
391:See
353:and
137:Thue
16:The
1005:doi
991:185
934:doi
891:doi
862:doi
827:doi
750:doi
736:170
88:all
1033::
1011:.
1003:.
989:.
968:,
942:.
932:.
920:51
918:.
912:.
887:37
885:.
858:11
856:.
833:.
823:10
821:.
756:.
748:.
734:.
717:12
715:.
682:^
638:12
631:12
600:24
513:.
456:12
422:12
384:18
276:ln
128:12
111:,
1019:.
1007::
997::
950:.
936::
926::
897:.
893::
868:.
864::
841:.
829::
806:.
800::
785:.
779::
764:.
752::
742::
719:.
700:.
694::
641:!
627:)
621:2
613:(
564:!
561:4
554:4
550:)
544:2
536:(
523:8
491:3
452:/
445:3
418:/
411:3
380:/
373:3
344:2
302:1
294:2
289:2
283:2
268:2
263:4
257:8
251:=
246:o
243:s
228:2
184:3
178:5
170:5
157:2
124:/
117:2
91:1
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