85:(with one particle per repeating unit such that each particle has a common orientation). These packing constructions almost doubled the optimal Bravais-lattice-packing fraction 36.73% obtained by Hoylman. In 2007 and 2010, Chaikin and coworkers experimentally showed that tetrahedron-like dice can randomly pack in a finite container up to a packing fraction between 75% and 76%. In 2008, Chen was the first to propose a packing of hard, regular tetrahedra that packed more densely than spheres, demonstrating numerically a packing fraction of 77.86%. A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%.
67:
35:
99:
In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. These packings were also the basis of a slightly improved packing obtained by
Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, and by Chen, Engel,
84:
and
Torquato showed that a packing fraction about 72% can be obtained by constructing a non-Bravais lattice packing of tetrahedra (with multiple particles with generally different orientations per repeating unit), and thus they showed that the best tetrahedron packing cannot be a lattice packing
96:, which can be compressed to 83.24%. They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%.
100:
and
Glotzer in early 2010 with a packing fraction of 85.63%. The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra. Surprisingly, the square-triangle tiling packs denser than this
678:
Haji-Akbari, Amir; Engel, Michael; Keys, Aaron S.; Zheng, Xiaoyu; Petschek, Rolfe G.; Palffy-Muhoray, Peter; Glotzer, Sharon C. (2009). "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra".
353:
966:
414:
92:
simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal
912:
962:
157:
112:
of a tetrahedron and a sphere), making the 82-tetrahedron crystal the largest unit cell for a densest packing of identical particles to date.
744:
514:
266:
187:
884:
146:
1099:
789:
Torquato, S.; Jiao, Y. (2009). "Analytical
Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry".
1031:
905:
185:
Chen, Elizabeth R.; Engel, Michael; Glotzer, Sharon C. (2010). "Dense crystalline dimer packings of regular tetrahedra".
1120:
1026:
264:
Simon Gravel; Veit Elser; Yoav Kallus (2010). "Upper bound on the packing density of regular tetrahedra and octahedra".
132:
is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.
125:
237:
742:
Kallus, Yoav; Elser, Veit; Gravel, Simon (2010). "Dense
Periodic Packings of Tetrahedra with Small Repeating Units".
1016:
970:
958:
1125:
1006:
898:
1021:
459:
980:
1011:
66:
34:
823:
700:
633:
570:
468:
362:
105:
43:
333:
1130:
935:
812:"Evolution of the dense packings of spherotetrahedral particles: from ideal tetrahedra to spheres"
790:
771:
753:
724:
690:
657:
623:
594:
541:
523:
293:
275:
214:
196:
89:
81:
1078:
975:
851:
716:
649:
586:
494:
390:
921:
841:
831:
763:
708:
681:
641:
614:
578:
561:
533:
484:
476:
433:
423:
380:
370:
311:
285:
246:
206:
878:
250:
232:
161:
141:
51:
31:
throughout three-dimensional space so as to fill the maximum possible fraction of space.
872:
827:
704:
637:
612:
Torquato, S.; Jiao, Y. (2009). "Dense packings of the
Platonic and Archimedean solids".
574:
472:
366:
120:
Because the earliest lower bound known for packings of tetrahedra was less than that of
1057:
1036:
998:
985:
950:
846:
811:
385:
348:
129:
121:
101:
39:
315:
1114:
1094:
1041:
109:
775:
661:
545:
428:
409:
297:
218:
728:
598:
480:
93:
55:
58:
space, and an upper bound below 100% (namely, 1 − (2.6...)·10) has been reported.
453:
Jaoshvili, Alexander; Esakia, Andria; Porrati, Massimo; Chaikin, Paul M. (2010).
940:
28:
767:
537:
289:
210:
375:
150:
124:, it was suggested that the regular tetrahedra might be a counterexample to
74:
855:
720:
653:
590:
498:
394:
38:
The currently densest known packing structure for regular tetrahedra is a
20:
712:
645:
438:
489:
836:
582:
512:
Chen, Elizabeth R. (2008). "A Dense
Packing of Regular Tetrahedra".
890:
454:
795:
758:
695:
628:
528:
334:"Jeffrey Lagarias and Chuanming Zong to receive 2015 Conant Prize"
280:
201:
65:
33:
894:
559:
Cohn, Henry (2009). "Mathematical physics: A tight squeeze".
50:
Currently, the best lower bound achieved on the optimal
455:"Experiments on the Random Packing of Tetrahedral Dice"
810:
Jin, Weiwei; Lu, Peng; Li, Shuixiang (December 2015).
873:
77:
claimed that tetrahedra could fill space completely.
1087:
1066:
1050:
997:
949:
928:
54:of regular tetrahedra is 85.63%. Tetrahedra do not
673:
671:
354:Proceedings of the National Academy of Sciences
349:"Packing, tiling, and covering with tetrahedra"
27:is the problem of arranging identical regular
906:
415:Bulletin of the American Mathematical Society
235:(1925). "Het probleem 'De Impletione Loci'".
88:In mid-2009 Haji-Akbari et al. showed, using
8:
410:"The densest lattice packing of tetrahedra"
153:packing of irregular tetrahedra in 3-space.
913:
899:
891:
180:
178:
108:when tetrahedra are slightly rounded (the
845:
835:
794:
757:
694:
627:
527:
488:
437:
427:
384:
374:
316:"Mysteries in Packing Regular Tetrahedra"
279:
200:
885:Pyramids are the best shape for packing
174:
158:triakis truncated tetrahedral honeycomb
116:Relationship to other packing problems
745:Discrete & Computational Geometry
515:Discrete & Computational Geometry
267:Discrete & Computational Geometry
188:Discrete & Computational Geometry
16:Concept in three-dimensional geometry
7:
164:and based on a regular tetrahedron.
14:
314:and Chuanming Zong (2012-12-04).
147:Disphenoid tetrahedral honeycomb
429:10.1090/S0002-9904-1970-12400-4
1042:Sphere-packing (Hamming) bound
481:10.1103/PhysRevLett.104.185501
1:
128:that the optimal density for
408:Hoylman, Douglas J. (1970).
332:News Release (2014-12-03).
238:Nieuw Archief voor Wiskunde
1147:
768:10.1007/s00454-010-9254-3
538:10.1007/s00454-008-9101-y
290:10.1007/s00454-010-9304-x
211:10.1007/s00454-010-9273-0
130:packing congruent spheres
46:and fills 85.63% of space
967:isosceles right triangle
460:Physical Review Letters
376:10.1073/pnas.0601389103
981:Circle packing theorem
347:Conway, J. H. (2006).
71:
47:
106:triangular bipyramids
70:Tetrahedral packaging
69:
44:triangular bipyramids
37:
963:equilateral triangle
1121:History of geometry
1100:Slothouber–Graatsma
828:2015NatSR...515640J
713:10.1038/nature08641
705:2009Natur.462..773H
646:10.1038/nature08239
638:2009Natur.460..876T
575:2009Natur.460..801C
473:2010PhRvL.104r5501J
367:2006PNAS..10310612C
361:(28): 10612–10617.
25:tetrahedron packing
816:Scientific Reports
72:
62:Historical results
48:
1108:
1107:
1067:Other 3-D packing
1051:Other 2-D packing
976:Apollonian gasket
837:10.1038/srep15640
689:(7274): 773–777.
622:(7257): 876–879.
569:(7257): 801–802.
126:Ulam's conjecture
1138:
1126:Packing problems
989:
929:Abstract packing
922:Packing problems
915:
908:
901:
892:
879:Efficient shapes
860:
859:
849:
839:
807:
801:
800:
798:
786:
780:
779:
761:
739:
733:
732:
698:
675:
666:
665:
631:
609:
603:
602:
556:
550:
549:
531:
509:
503:
502:
492:
450:
444:
443:
441:
431:
405:
399:
398:
388:
378:
344:
338:
337:
329:
323:
322:
320:
312:Jeffrey Lagarias
308:
302:
301:
283:
261:
255:
254:
229:
223:
222:
204:
182:
52:packing fraction
1146:
1145:
1141:
1140:
1139:
1137:
1136:
1135:
1111:
1110:
1109:
1104:
1083:
1062:
1046:
993:
987:
986:Tammes problem
945:
924:
919:
887:, New Scientist
881:, The Economist
869:
864:
863:
809:
808:
804:
788:
787:
783:
741:
740:
736:
677:
676:
669:
611:
610:
606:
583:10.1038/460801a
558:
557:
553:
511:
510:
506:
452:
451:
447:
407:
406:
402:
346:
345:
341:
331:
330:
326:
318:
310:
309:
305:
263:
262:
258:
231:
230:
226:
184:
183:
176:
171:
162:cell-transitive
142:Packing problem
138:
118:
64:
17:
12:
11:
5:
1144:
1142:
1134:
1133:
1128:
1123:
1113:
1112:
1106:
1105:
1103:
1102:
1097:
1091:
1089:
1085:
1084:
1082:
1081:
1076:
1070:
1068:
1064:
1063:
1061:
1060:
1058:Square packing
1054:
1052:
1048:
1047:
1045:
1044:
1039:
1037:Kissing number
1034:
1029:
1024:
1019:
1014:
1009:
1003:
1001:
999:Sphere packing
995:
994:
992:
991:
983:
978:
973:
955:
953:
951:Circle packing
947:
946:
944:
943:
938:
932:
930:
926:
925:
920:
918:
917:
910:
903:
895:
889:
888:
882:
876:
868:
867:External links
865:
862:
861:
802:
781:
752:(2): 245–252.
734:
667:
604:
551:
522:(2): 214–240.
504:
467:(18): 185501.
445:
400:
339:
324:
303:
274:(4): 799–818.
256:
224:
195:(2): 253–280.
173:
172:
170:
167:
166:
165:
154:
144:
137:
134:
117:
114:
102:double lattice
63:
60:
40:double lattice
15:
13:
10:
9:
6:
4:
3:
2:
1143:
1132:
1129:
1127:
1124:
1122:
1119:
1118:
1116:
1101:
1098:
1096:
1093:
1092:
1090:
1086:
1080:
1077:
1075:
1072:
1071:
1069:
1065:
1059:
1056:
1055:
1053:
1049:
1043:
1040:
1038:
1035:
1033:
1032:Close-packing
1030:
1028:
1027:In a cylinder
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1004:
1002:
1000:
996:
990:
984:
982:
979:
977:
974:
972:
968:
964:
960:
957:
956:
954:
952:
948:
942:
939:
937:
934:
933:
931:
927:
923:
916:
911:
909:
904:
902:
897:
896:
893:
886:
883:
880:
877:
874:
871:
870:
866:
857:
853:
848:
843:
838:
833:
829:
825:
821:
817:
813:
806:
803:
797:
792:
785:
782:
777:
773:
769:
765:
760:
755:
751:
747:
746:
738:
735:
730:
726:
722:
718:
714:
710:
706:
702:
697:
692:
688:
684:
683:
674:
672:
668:
663:
659:
655:
651:
647:
643:
639:
635:
630:
625:
621:
617:
616:
608:
605:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
563:
555:
552:
547:
543:
539:
535:
530:
525:
521:
517:
516:
508:
505:
500:
496:
491:
486:
482:
478:
474:
470:
466:
462:
461:
456:
449:
446:
440:
435:
430:
425:
421:
417:
416:
411:
404:
401:
396:
392:
387:
382:
377:
372:
368:
364:
360:
356:
355:
350:
343:
340:
335:
328:
325:
317:
313:
307:
304:
299:
295:
291:
287:
282:
277:
273:
269:
268:
260:
257:
252:
248:
244:
240:
239:
234:
233:Struik, D. J.
228:
225:
220:
216:
212:
208:
203:
198:
194:
190:
189:
181:
179:
175:
168:
163:
159:
155:
152:
148:
145:
143:
140:
139:
135:
133:
131:
127:
123:
115:
113:
111:
110:Minkowski sum
107:
103:
97:
95:
91:
86:
83:
78:
76:
68:
61:
59:
57:
53:
45:
41:
36:
32:
30:
26:
22:
1073:
969: /
965: /
961: /
822:(1): 15640.
819:
815:
805:
784:
749:
743:
737:
686:
680:
619:
613:
607:
566:
560:
554:
519:
513:
507:
464:
458:
448:
439:10150/288016
419:
413:
403:
358:
352:
342:
327:
306:
271:
265:
259:
242:
236:
227:
192:
186:
119:
98:
94:quasicrystal
87:
79:
73:
49:
24:
18:
1074:Tetrahedron
1017:In a sphere
988:(on sphere)
959:In a circle
490:10919/24495
422:: 135–138.
245:: 121–134.
241:. 2nd ser.
1131:Tetrahedra
1115:Categories
1007:Apollonian
251:52.0002.04
169:References
29:tetrahedra
1079:Ellipsoid
1022:In a cube
875:, NYTimes
796:0912.4210
759:0910.5226
696:1012.5138
629:0908.4107
529:0908.1884
281:1008.2830
202:1001.0586
151:isohedral
80:In 2006,
75:Aristotle
856:26490670
776:13385357
721:20010683
662:52819935
654:19675649
591:19675632
546:32166668
499:20482187
395:16818891
298:18908213
219:18523116
136:See also
21:geometry
1088:Puzzles
847:4614866
824:Bibcode
729:4412674
701:Bibcode
634:Bibcode
599:5157975
571:Bibcode
469:Bibcode
386:1502280
363:Bibcode
122:spheres
1095:Conway
1012:Finite
971:square
854:
844:
774:
727:
719:
682:Nature
660:
652:
615:Nature
597:
589:
562:Nature
544:
497:
393:
383:
296:
249:
217:
82:Conway
791:arXiv
772:S2CID
754:arXiv
725:S2CID
691:arXiv
658:S2CID
624:arXiv
595:S2CID
542:S2CID
524:arXiv
319:(PDF)
294:S2CID
276:arXiv
215:S2CID
197:arXiv
149:- an
852:PMID
717:PMID
650:PMID
587:PMID
495:PMID
391:PMID
156:The
56:tile
941:Set
936:Bin
842:PMC
832:doi
764:doi
709:doi
687:462
642:doi
620:460
579:doi
567:460
534:doi
485:hdl
477:doi
465:104
434:hdl
424:doi
381:PMC
371:doi
359:103
286:doi
247:JFM
207:doi
160:is
104:of
42:of
19:In
1117::
850:.
840:.
830:.
818:.
814:.
770:.
762:.
750:44
748:.
723:.
715:.
707:.
699:.
685:.
670:^
656:.
648:.
640:.
632:.
618:.
593:.
585:.
577:.
565:.
540:.
532:.
520:40
518:.
493:.
483:.
475:.
463:.
457:.
432:.
420:76
418:.
412:.
389:.
379:.
369:.
357:.
351:.
292:.
284:.
272:46
270:.
243:15
213:.
205:.
193:44
191:.
177:^
90:MC
23:,
914:e
907:t
900:v
858:.
834::
826::
820:5
799:.
793::
778:.
766::
756::
731:.
711::
703::
693::
664:.
644::
636::
626::
601:.
581::
573::
548:.
536::
526::
501:.
487::
479::
471::
442:.
436::
426::
397:.
373::
365::
336:.
321:.
300:.
288::
278::
253:.
221:.
209::
199::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
