88:
is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its other edge lengths are 51, 52, 53, 80 and 84. 8064 is the smallest possible volume (and 6384 is the smallest possible surface area) of a perfect tetrahedron. The integral edge lengths of a
140:, tetrahedra in which all faces are congruent and each pair of opposite sides has equal lengths. In this case, there are only three edge lengths needed to describe the tetrahedron, rather than six, and the triples of lengths that define Heronian tetrahedra can be characterized using an
96:
with base 896 and sides 1073, and the other two faces are also isosceles with base 990 and the same sides. However, Starke made an error in reporting its volume which has become widely copied. The correct volume is
514:
368:
71:. The lengths of the edges on the path of axis-parallel edges are 153, 104, and 672, and the other three edge lengths are 185, 680, and 697, forming four right triangle faces described by the
587:
424:
281:
841:
along a common side. This definition has also been generalized to three dimensions, leading to a different class of tetrahedra that have also been called Heron tetrahedra.
129:(one with all faces being equilateral) cannot be a Heronian tetrahedron because, for regular tetrahedra whose edge lengths are integers, the face areas and volume are
1331:
1035:
733:
700:
813:
793:
773:
647:
627:
607:
209:
189:
169:
151:
There are also infinitely many
Heronian birectangular tetrahedra. One method for generating tetrahedra of this type derives the axis-parallel edge lengths
1088:
980:
430:
1210:
292:
875:
867:
144:. There are also infinitely many Heronian tetrahedra with a cycle of four equal edge lengths, in which all faces are
819:, a rectangular cuboid in which the sides, two of the three face diagonals, and the body diagonal are all integers.
526:
823:
110:
Sascha Kurz has used computer search algorithms to find all
Heronian tetrahedra with longest edge length at most
374:
1259:
1214:
926:
220:
64:
67:, a tetrahedron with a path of three edges parallel to the three coordinate axes and with all faces being
1040:
133:. For the same reason no Heronian tetrahedron can have an equilateral triangle as one of its faces.
1326:
126:
837:
An alternative definition of
Heronian triangles is that they can be formed by gluing together two
1276:
1183:
1129:
900:
838:
145:
93:
72:
44:
1299:
1112:
Starke, E. P. (June–July 1943), "E 544: A commensurable tetrahedron", Problems and solutions,
863:
130:
40:
1231:
1098:
1268:
1121:
1013:
995:
945:
935:
884:
1197:
1009:
959:
896:
1193:
1005:
955:
892:
136:
There are infinitely many
Heronian tetrahedra, and more strongly infinitely many Heronian
48:
1302:
888:
89:
Heronian tetrahedron with this volume and surface area are 25, 39, 56, 120, 153 and 160.
1149:
1044:
715:
682:
1084:
1058:
798:
778:
758:
632:
612:
592:
520:
212:
194:
174:
154:
141:
79:
68:
60:
1320:
1033:
Gardner, Martin (1983), "Chapter 2: Diophantine
Analysis and Fermat's Last Theorem",
904:
1093:
85:
1257:
Lin, C.-S. (November 2011), "95.66 The reciprocal volume of a Heron tetrahedron",
816:
32:
1272:
1000:
940:
137:
1307:
950:
924:
Chisholm, C.; MacDougall, J. A. (2006), "Rational and Heron tetrahedra",
1280:
92:
In 1943, E. P. Starke published another example, in which two faces are
1133:
36:
829:
A complete classification of all
Heronian tetrahedra remains unknown.
519:
For instance, the tetrahedron derived in this way from an identity of
16:
Tetrahedron whose edge lengths, face areas and volume are all integers
1125:
121:
Classification, infinite families, and special types of tetrahedron
1188:
78:
Eight examples of
Heronian tetrahedra were discovered in 1877 by
1174:
Kurz, Sascha (2008), "On the generation of
Heronian triangles",
75:(153,104,185), (104,672,680), (153,680,697), and (185,672,697).
745:, and the hypotenuse of the remaining two sides equal to
1236:
Sitzungsberichte der
Berliner Mathematische Gesellschaft
509:{\displaystyle c={\bigl |}(pr)^{2})-|(qs)^{2}{\bigr |}.}
826:
had been found and no one has proven that none exist.
363:{\displaystyle a={\bigl |}(pq)^{2}-(rs)^{2}{\bigr |},}
801:
781:
761:
718:
685:
635:
615:
595:
529:
433:
377:
295:
223:
197:
177:
157:
1245:
1072:
1061:(1877), "Über rationale Dreikante und Tetraeder",
807:
787:
767:
727:
694:
641:
621:
601:
581:
508:
418:
362:
275:
203:
183:
163:
51:so that its vertex coordinates are also integers.
35:whose edge lengths, face areas and volume are all
47:). Every Heronian tetrahedron can be arranged in
988:Bulletin of the Australian Mathematical Society
1284:(about a different concept with the same name)
1219:(3rd ed.), Dover, Table I(i), pp. 292–293
1036:Wheels, Life and Other Mathematical Amusements
582:{\displaystyle 59^{4}+158^{4}=133^{4}+134^{4}}
498:
442:
408:
386:
352:
304:
8:
1232:"Rationale Tetraeder mit kongruenten Seiten"
868:"Heronian tetrahedra are lattice tetrahedra"
419:{\displaystyle b={\bigl |}2pqrs{\bigr |},}
1187:
1028:
1026:
999:
949:
939:
800:
780:
760:
717:
684:
634:
614:
594:
573:
560:
547:
534:
528:
497:
496:
490:
472:
460:
441:
440:
432:
407:
406:
385:
384:
376:
351:
350:
344:
322:
303:
302:
294:
267:
254:
241:
228:
222:
196:
176:
156:
679:, with the hypotenuse of right triangle
850:
276:{\displaystyle p^{4}+s^{4}=q^{4}+r^{4}}
107:, twice the number reported by Starke.
1144:
1142:
919:
917:
915:
913:
858:
856:
854:
1332:Arithmetic problems of solid geometry
974:
972:
970:
968:
7:
889:10.4169/amer.math.monthly.120.02.140
712:, the hypotenuse of right triangle
39:. The faces must therefore all be
14:
1114:The American Mathematical Monthly
1039:, W. H. Freeman, pp. 10–19,
1246:Chisholm & MacDougall (2006)
1073:Chisholm & MacDougall (2006)
1063:Archiv der Mathematik und Physik
979:Buchholz, Ralph Heiner (1992),
866:; Perlis, Alexander R. (2013),
487:
477:
473:
466:
457:
447:
341:
331:
319:
309:
1:
1089:"Math Trek: Perfect Pyramids"
876:American Mathematical Monthly
1176:Serdica Journal of Computing
815:form the edge lengths of an
1048:; see in particular page 14
1348:
824:trirectangular tetrahedron
1273:10.1017/S0025557200003740
1001:10.1017/S0004972700030252
941:10.1016/j.jnt.2006.02.009
822:No example of a Heronian
65:birectangular tetrahedron
1260:The Mathematical Gazette
927:Journal of Number Theory
755:. For these tetrahedra,
839:integer right triangles
1303:"Heronian tetrahedron"
1163:(5): 162–166, May 1985
809:
789:
769:
729:
696:
643:
623:
603:
583:
510:
420:
364:
277:
205:
185:
165:
1230:Güntsche, R. (1907),
817:almost-perfect cuboid
810:
790:
770:
730:
697:
644:
624:
604:
584:
511:
421:
365:
278:
213:sums of fourth powers
206:
186:
166:
1101:on February 20, 2008
799:
779:
759:
716:
683:
633:
613:
593:
527:
431:
375:
293:
221:
195:
175:
155:
59:An example known to
21:Heronian tetrahedron
1157:Crux Mathematicorum
1045:1983wlom.book.....G
1019:on October 27, 2009
286:using the formulas
146:isosceles triangles
127:regular tetrahedron
94:isosceles triangles
73:Pythagorean triples
1300:Weisstein, Eric W.
981:"Perfect pyramids"
864:Marshall, Susan H.
805:
785:
765:
728:{\displaystyle bc}
725:
695:{\displaystyle ab}
692:
639:
619:
599:
579:
506:
416:
360:
273:
201:
181:
161:
131:irrational numbers
45:Hero of Alexandria
41:Heronian triangles
1216:Regular Polytopes
1211:Coxeter, H. S. M.
808:{\displaystyle c}
788:{\displaystyle b}
768:{\displaystyle a}
642:{\displaystyle c}
622:{\displaystyle b}
602:{\displaystyle a}
204:{\displaystyle c}
184:{\displaystyle b}
164:{\displaystyle a}
25:Heron tetrahedron
1339:
1313:
1312:
1285:
1283:
1267:(534): 542–545,
1254:
1248:
1243:
1227:
1221:
1220:
1207:
1201:
1200:
1191:
1171:
1165:
1164:
1154:
1146:
1137:
1136:
1109:
1103:
1102:
1097:, archived from
1081:
1075:
1070:
1055:
1049:
1047:
1030:
1021:
1020:
1018:
1012:, archived from
1003:
985:
976:
963:
962:
953:
943:
921:
908:
907:
872:
860:
814:
812:
811:
806:
794:
792:
791:
786:
774:
772:
771:
766:
754:
753:
750:
744:
743:
740:
734:
732:
731:
726:
711:
710:
707:
701:
699:
698:
693:
678:
677:
674:
668:
667:
664:
658:
657:
654:
648:
646:
645:
640:
628:
626:
625:
620:
608:
606:
605:
600:
588:
586:
585:
580:
578:
577:
565:
564:
552:
551:
539:
538:
515:
513:
512:
507:
502:
501:
495:
494:
476:
465:
464:
446:
445:
425:
423:
422:
417:
412:
411:
390:
389:
369:
367:
366:
361:
356:
355:
349:
348:
327:
326:
308:
307:
282:
280:
279:
274:
272:
271:
259:
258:
246:
245:
233:
232:
210:
208:
207:
202:
190:
188:
187:
182:
170:
168:
167:
162:
116:
115:
106:
105:
102:
1347:
1346:
1342:
1341:
1340:
1338:
1337:
1336:
1317:
1316:
1298:
1297:
1294:
1289:
1288:
1256:
1255:
1251:
1229:
1228:
1224:
1209:
1208:
1204:
1173:
1172:
1168:
1152:
1148:
1147:
1140:
1126:10.2307/2303724
1111:
1110:
1106:
1085:Peterson, Ivars
1083:
1082:
1078:
1057:
1056:
1052:
1032:
1031:
1024:
1016:
983:
978:
977:
966:
923:
922:
911:
870:
862:
861:
852:
847:
835:
797:
796:
777:
776:
757:
756:
751:
748:
746:
741:
738:
736:
714:
713:
708:
705:
703:
681:
680:
675:
672:
670:
665:
662:
660:
655:
652:
650:
631:
630:
611:
610:
591:
590:
569:
556:
543:
530:
525:
524:
486:
456:
429:
428:
373:
372:
340:
318:
291:
290:
263:
250:
237:
224:
219:
218:
211:from two equal
193:
192:
173:
172:
153:
152:
123:
113:
111:
103:
100:
98:
69:right triangles
57:
49:Euclidean space
29:perfect pyramid
23:(also called a
17:
12:
11:
5:
1345:
1343:
1335:
1334:
1329:
1319:
1318:
1315:
1314:
1293:
1292:External links
1290:
1287:
1286:
1249:
1244:, as cited by
1222:
1202:
1182:(2): 181–196,
1166:
1138:
1104:
1076:
1071:, as cited by
1050:
1022:
994:(3): 353–368,
964:
934:(1): 153–185,
909:
883:(2): 140–149,
849:
848:
846:
843:
834:
833:Related shapes
831:
804:
784:
764:
724:
721:
691:
688:
638:
618:
598:
576:
572:
568:
563:
559:
555:
550:
546:
542:
537:
533:
521:Leonhard Euler
517:
516:
505:
500:
493:
489:
485:
482:
479:
475:
471:
468:
463:
459:
455:
452:
449:
444:
439:
436:
426:
415:
410:
405:
402:
399:
396:
393:
388:
383:
380:
370:
359:
354:
347:
343:
339:
336:
333:
330:
325:
321:
317:
314:
311:
306:
301:
298:
284:
283:
270:
266:
262:
257:
253:
249:
244:
240:
236:
231:
227:
200:
180:
160:
142:elliptic curve
122:
119:
80:Reinhold Hoppe
63:is a Heronian
61:Leonhard Euler
56:
53:
15:
13:
10:
9:
6:
4:
3:
2:
1344:
1333:
1330:
1328:
1325:
1324:
1322:
1310:
1309:
1304:
1301:
1296:
1295:
1291:
1282:
1278:
1274:
1270:
1266:
1262:
1261:
1253:
1250:
1247:
1241:
1237:
1233:
1226:
1223:
1218:
1217:
1212:
1206:
1203:
1199:
1195:
1190:
1185:
1181:
1177:
1170:
1167:
1162:
1158:
1155:, Solutions,
1151:
1150:"Problem 930"
1145:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1108:
1105:
1100:
1096:
1095:
1090:
1087:(July 2003),
1086:
1080:
1077:
1074:
1068:
1064:
1060:
1054:
1051:
1046:
1042:
1038:
1037:
1029:
1027:
1023:
1015:
1011:
1007:
1002:
997:
993:
989:
982:
975:
973:
971:
969:
965:
961:
957:
952:
951:1959.13/26739
947:
942:
937:
933:
929:
928:
920:
918:
916:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
877:
869:
865:
859:
857:
855:
851:
844:
842:
840:
832:
830:
827:
825:
820:
818:
802:
782:
762:
722:
719:
689:
686:
636:
616:
596:
574:
570:
566:
561:
557:
553:
548:
544:
540:
535:
531:
522:
503:
491:
483:
480:
469:
461:
453:
450:
437:
434:
427:
413:
403:
400:
397:
394:
391:
381:
378:
371:
357:
345:
337:
334:
328:
323:
315:
312:
299:
296:
289:
288:
287:
268:
264:
260:
255:
251:
247:
242:
238:
234:
229:
225:
217:
216:
215:
214:
198:
178:
158:
149:
147:
143:
139:
134:
132:
128:
120:
118:
108:
95:
90:
87:
83:
81:
76:
74:
70:
66:
62:
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1306:
1264:
1258:
1252:
1239:
1235:
1225:
1215:
1205:
1179:
1175:
1169:
1160:
1156:
1117:
1113:
1107:
1099:the original
1094:Science News
1092:
1079:
1066:
1062:
1053:
1034:
1014:the original
991:
987:
931:
925:
880:
874:
836:
828:
821:
518:
285:
150:
135:
124:
109:
91:
84:
77:
58:
28:
24:
20:
18:
138:disphenoids
43:(named for
33:tetrahedron
1327:Tetrahedra
1321:Categories
1120:(6): 390,
845:References
1308:MathWorld
1189:1401.6150
1059:Hoppe, R.
735:equal to
702:equal to
649:equal to
470:−
329:−
1281:23248533
1213:(1973),
905:15888158
55:Examples
37:integers
1242:: 38–53
1198:2473583
1134:2303724
1069:: 86–98
1041:Bibcode
1010:1165142
960:2268761
897:3029939
31:) is a
1279:
1196:
1132:
1008:
958:
903:
895:
795:, and
669:, and
629:, and
589:, has
191:, and
1277:JSTOR
1184:arXiv
1153:(PDF)
1130:JSTOR
1017:(PDF)
984:(PDF)
901:S2CID
871:(PDF)
1269:doi
1122:doi
996:doi
946:hdl
936:doi
932:121
885:doi
881:120
752:657
749:318
747:635
742:032
739:093
737:504
709:993
706:828
704:509
676:360
673:083
671:379
666:368
663:273
661:332
656:175
653:678
651:386
571:134
558:133
545:158
114:000
112:600
104:600
101:185
99:124
86:117
27:or
1323::
1305:.
1275:,
1265:95
1263:,
1238:,
1234:,
1194:MR
1192:,
1178:,
1161:11
1159:,
1141:^
1128:,
1118:50
1116:,
1091:,
1067:61
1065:,
1025:^
1006:MR
1004:,
992:45
990:,
986:,
967:^
956:MR
954:,
944:,
930:,
912:^
899:,
893:MR
891:,
879:,
873:,
853:^
775:,
659:,
609:,
532:59
523:,
171:,
148:.
125:A
117:.
82:.
19:A
1311:.
1271::
1240:6
1186::
1180:2
1124::
1043::
998::
948::
938::
887::
803:c
783:b
763:a
723:c
720:b
690:b
687:a
637:c
617:b
597:a
575:4
567:+
562:4
554:=
549:4
541:+
536:4
504:.
499:|
492:2
488:)
484:s
481:q
478:(
474:|
467:)
462:2
458:)
454:r
451:p
448:(
443:|
438:=
435:c
414:,
409:|
404:s
401:r
398:q
395:p
392:2
387:|
382:=
379:b
358:,
353:|
346:2
342:)
338:s
335:r
332:(
324:2
320:)
316:q
313:p
310:(
305:|
300:=
297:a
269:4
265:r
261:+
256:4
252:q
248:=
243:4
239:s
235:+
230:4
226:p
199:c
179:b
159:a
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.