20:
47:. To be a valid common net, there shouldn't exist any non-overlapping sides and the resulting polyhedra must be connected through faces. The research of examples of this particular nets dates back to the end of the 20th century, despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search of common nets is usually made by either extensive search or the overlapping of nets that tile the plane.
245:
53:
There can be types of common nets, strict edge unfoldings and free unfoldings. Strict edge unfoldings refers to common nets where the different polyhedra that can be folded use the same folds, that is, to fold one polyhedra from the net of another there is no need to make new folds. Free unfoldings
704:
The first cases of common nets of polycubes found was the work by George Miller, with a later contribution of Donald Knuth, that culminated in the
Cubigami puzzle. It’s composed of a net that can fold to all 7 tree-like tetracubes. All possible common nets up to pentacubes were found. All the nets
252:
Common nets of cuboids have been deeply researched, mainly by Uehara and coworkers. To the moment, common nets of up to three cuboids have been found, It has, however, been proven that there exist infinitely many examples of nets that can be folded into more than one polyhedra.
965:
Araki, Y., Horiyama, T., Uehara, R. (2015). Common
Unfolding of Regular Tetrahedron and Johnson-Zalgaller Solid. In: Rahman, M.S., Tomita, E. (eds) WALCOM: Algorithms and Computation. WALCOM 2015. Lecture Notes in Computer Science, vol 8973. Springer, Cham.
1404:
1334:
Aloupis, Greg; Bose, Prosenjit K.; Collette, Sébastien; Demaine, Erik D.; Demaine, Martin L.; Douïeb, Karim; Dujmović, Vida; Iacono, John; Langerman, Stefan; Morin, Pat (2011).
74:"Can any Platonic solid be cut open and unfolded to a polygon that may be refolded to a different Platonic solid? For example, may a cube be so dissected to a tetrahedron?"
1397:
1390:
1355:
1023:
926:
1639:
1004:
Xu D., Horiyama T., Shirakawa T., Uehara R., Common developments of three incongruent boxes of area 30, Computational
Geometry, 64, 8 2017
1660:
79:
This problem has been partially solved by
Shirakawa et al. with a fractal net that is conjectured to fold to a tetrahedron and a cube.
1560:
944:
Toshihiro
Shirakawa, Takashi Horiyama, and Ryuhei Uehara, 27th European Workshop on Computational Geometry (EuroCG 2011), 2011, 47-50.
1039:
982:"Ryuuhei Uehara - Nonexistence of Common Edge Developments of Regular Tetrahedron and Other Platonic Solids - Papers - researchmap"
1701:
1771:
1467:
1498:
1300:
1646:
1413:
66:
54:
refer to the opposite case, when we can create as many folds as needed to enable the folding of different polyhedra.
1696:
50:
Demaine et al. proved that every convex polyhedron can be unfolded and refolded to a different convex polyhedron.
1437:
1731:
1667:
868:
Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin-ichi; Lubiw, Anna; Nara, Chie; OʼRourke, Joseph (2013-10-01).
1573:
1457:
1442:
981:
869:
1617:
1447:
1427:
1243:
1072:
1584:
1565:
1342:. Lecture Notes in Computer Science. Vol. 7033. Berlin, Heidelberg: Springer. pp. 44–54.
19:
1802:
1746:
1569:
1544:
1033:
1372:
1335:
57:
Multiplicity of common nets refers to the number of common nets for the same set of polyhedra.
1766:
1653:
1524:
1351:
1263:
1221:
1170:
Abel, Zachary; Demaine, Erik; Demaine, Martin; Matsui, Hiroaki; Rote, Günter; Uehara, Ryuhei.
1092:
1019:
922:
899:
1726:
1691:
1622:
1594:
1579:
1462:
1343:
1255:
1213:
1179:
1084:
889:
881:
40:
1706:
1534:
1519:
1493:
1776:
1751:
1741:
1736:
1721:
1716:
1599:
1432:
1796:
1529:
1781:
1711:
1483:
1144:
1200:
Xu, Dawei; Horiyama, Takashi; Shirakawa, Toshihiro; Uehara, Ryuhei (August 2017).
1217:
885:
705:
follow strict orthogonal folding despite still being considered free unfoldings.
1761:
1452:
1347:
244:
1382:
1201:
1756:
1514:
1315:
1285:
1259:
1088:
44:
24:
1267:
1225:
1096:
903:
1674:
1488:
1057:
32:
894:
1183:
1539:
1171:
1248:
International
Journal of Computational Geometry & Applications
1077:
International
Journal of Computational Geometry & Applications
1386:
1338:. In Akiyama, Jin; Bo, Jiang; Kano, Mikio; Tan, Xuehou (eds.).
1244:"Common Developments of Three Incongruent Orthogonal Boxes"
1202:"Common developments of three incongruent boxes of area 30"
1172:"Common Developments of Several Different Orthogonal Boxes"
1073:"Common Developments of Three Incongruent Orthogonal Boxes"
1016:
Geometric
Folding Algorithms: Linkages, Origami, Polyhedra
919:
Geometric folding algorithms: linkages, origami, polyhedra
1145:"Polygons Folding to Plural Incongruent Orthogonal Boxes"
1373:"The four common nets of the five 7-vertex deltahedra"
1242:
Shirakawa, Toshihiro; Uehara, Ryuhei (February 2013).
1071:
Shirakawa, Toshihiro; Uehara, Ryuhei (February 2013).
1176:
The 23rd
Canadian Conference on Computational Geometr
953:
Koichi Hirata, Personal communication, December 2000
1684:
1631:
1610:
1553:
1507:
1476:
1420:
16:Edge-joined polygon with multiple principle shapes
67:Geometric Folding Algorithm by Rourke and Demaine
1336:"Common Unfoldings of Polyominoes and Polycubes"
1340:Computational Geometry, Graphs and Applications
1398:
1152:Canadian Conference on Computational Geometry
8:
966:https://doi.org/10.1007/978-3-319-15612-5_26
917:Demaine, Erik D.; O'Rourke, Joseph (2007).
1405:
1391:
1383:
921:. Cambridge: Cambridge university press.
893:
809:
707:
255:
243:
81:
18:
860:
1329:
1327:
1325:
1138:
1136:
1134:
1132:
1130:
1128:
1126:
1031:
248:Common net of a 1x1x5 and 1x2x3 cuboid
1279:
1277:
1237:
1235:
1195:
1193:
1165:
1163:
1161:
1124:
1122:
1120:
1118:
1116:
1114:
1112:
1110:
1108:
1106:
1014:Demaine, Erik; O'Rourke (July 2007).
870:"Refold rigidity of convex polyhedra"
7:
1640:Geometric Exercises in Paper Folding
1143:Mitani, Jun; Uehara, Ryuhei (2008).
1051:
1049:
976:
974:
972:
961:
959:
940:
938:
1661:A History of Folding in Mathematics
1301:"Ambiguous unfoldings of polycubes"
14:
43:that can be folded onto several
1561:Alexandrov's uniqueness theorem
1284:Miller, George; Knuth, Donald.
1018:. Cambridge University Press.
1:
1499:Regular paperfolding sequence
1038:: CS1 maint: date and year (
1647:Geometric Folding Algorithms
1414:Mathematics of paper folding
1218:10.1016/j.comgeo.2017.03.001
886:10.1016/j.comgeo.2013.05.002
1348:10.1007/978-3-642-24983-9_5
1819:
1697:Margherita Piazzola Beloch
1468:Yoshizawa–Randlett system
1260:10.1142/S0218195913500040
1089:10.1142/S0218195913500040
756:All tree-like tetracubes
178:Octahedron (non-Regular)
1668:Origami Polyhedra Design
781:22 tree-like pentacubes
695:*Non-orthogonal foldings
166:1x1x7 and 1x3x3 Cuboids
833:Both 8 face deltahedra
807:3D Simplicial polytope
1458:Napkin folding problem
1206:Computational Geometry
874:Computational Geometry
794:Non-planar pentacubes
249:
77:
65:Open problem 25.31 in
28:
23:Common net for both a
247:
235:Non-regular polyhedra
71:
27:and a Tritetrahedron.
22:
1618:Fold-and-cut theorem
1574:Steffen's polyhedron
1438:Huzita–Hatori axioms
1428:Big-little-big lemma
846:7-vertex deltahedra
1566:Flexible polyhedron
116:Cuboid (1x1x1.232)
1747:Toshikazu Kawasaki
1570:Bricard octahedron
1545:Yoshimura buckling
1443:Kawasaki's theorem
250:
142:Jonhson Solid J84
129:Jonhson Solid J17
29:
1790:
1789:
1654:Geometric Origami
1525:Paper bag problem
1448:Maekawa's theorem
1357:978-3-642-24983-9
1056:Weisstein, Eric.
1025:978-0-521-85757-4
928:978-0-521-85757-4
852:
851:
800:
799:
696:
692:
691:
232:
231:
61:Regular polyhedra
1810:
1727:David A. Huffman
1692:Roger C. Alperin
1595:Source unfolding
1463:Pureland origami
1407:
1400:
1393:
1384:
1377:
1376:
1368:
1362:
1361:
1331:
1320:
1319:
1314:Miller, George.
1311:
1305:
1304:
1296:
1290:
1289:
1281:
1272:
1271:
1239:
1230:
1229:
1197:
1188:
1187:
1167:
1156:
1155:
1149:
1140:
1101:
1100:
1068:
1062:
1061:
1053:
1044:
1043:
1037:
1029:
1011:
1005:
1002:
996:
995:
993:
992:
978:
967:
963:
954:
951:
945:
942:
933:
932:
914:
908:
907:
897:
865:
810:
708:
694:
256:
226:Tetramonohedron
202:tetramonohedron
190:Tetramonohedron
154:Tetramonohedron
82:
1818:
1817:
1813:
1812:
1811:
1809:
1808:
1807:
1793:
1792:
1791:
1786:
1772:Joseph O'Rourke
1707:Robert Connelly
1680:
1627:
1606:
1549:
1535:Schwarz lantern
1520:Modular origami
1503:
1472:
1416:
1411:
1381:
1380:
1370:
1369:
1365:
1358:
1333:
1332:
1323:
1313:
1312:
1308:
1298:
1297:
1293:
1283:
1282:
1275:
1241:
1240:
1233:
1199:
1198:
1191:
1169:
1168:
1159:
1147:
1142:
1141:
1104:
1070:
1069:
1065:
1055:
1054:
1047:
1030:
1026:
1013:
1012:
1008:
1003:
999:
990:
988:
980:
979:
970:
964:
957:
952:
948:
943:
936:
929:
916:
915:
911:
867:
866:
862:
857:
805:
702:
242:
237:
214:Tritetrahedron
63:
17:
12:
11:
5:
1816:
1814:
1806:
1805:
1795:
1794:
1788:
1787:
1785:
1784:
1779:
1777:Tomohiro Tachi
1774:
1769:
1764:
1759:
1754:
1752:Robert J. Lang
1749:
1744:
1742:Humiaki Huzita
1739:
1734:
1729:
1724:
1722:Rona Gurkewitz
1719:
1717:Martin Demaine
1714:
1709:
1704:
1699:
1694:
1688:
1686:
1682:
1681:
1679:
1678:
1671:
1664:
1657:
1650:
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1635:
1633:
1629:
1628:
1626:
1625:
1620:
1614:
1612:
1608:
1607:
1605:
1604:
1603:
1602:
1600:Star unfolding
1597:
1592:
1587:
1577:
1563:
1557:
1555:
1551:
1550:
1548:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1511:
1509:
1505:
1504:
1502:
1501:
1496:
1491:
1486:
1480:
1478:
1474:
1473:
1471:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1433:Crease pattern
1430:
1424:
1422:
1418:
1417:
1412:
1410:
1409:
1402:
1395:
1387:
1379:
1378:
1363:
1356:
1321:
1306:
1291:
1273:
1231:
1189:
1157:
1102:
1063:
1045:
1024:
1006:
997:
986:researchmap.jp
968:
955:
946:
934:
927:
909:
880:(8): 979–989.
859:
858:
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850:
849:
847:
844:
841:
837:
836:
834:
831:
828:
824:
823:
820:
817:
814:
804:
801:
798:
797:
795:
792:
789:
785:
784:
782:
779:
776:
772:
771:
769:
768:23 pentacubes
766:
764:
760:
759:
757:
754:
751:
747:
746:
744:
741:
739:
735:
734:
732:
729:
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722:
721:
718:
715:
712:
701:
698:
690:
689:
687:
684:
681:
678:
676:
672:
671:
669:
666:
663:
660:
658:
654:
653:
651:
649:
648:√10x2√10x2√10
646:
643:
641:
637:
636:
634:
632:
629:
626:
623:
619:
618:
616:
614:
611:
608:
605:
601:
600:
598:
596:
593:
590:
587:
583:
582:
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572:
569:
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547:
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389:
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108:
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99:
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92:
89:
86:
62:
59:
15:
13:
10:
9:
6:
4:
3:
2:
1815:
1804:
1801:
1800:
1798:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
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1740:
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1735:
1733:
1730:
1728:
1725:
1723:
1720:
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1713:
1710:
1708:
1705:
1703:
1700:
1698:
1695:
1693:
1690:
1689:
1687:
1683:
1677:
1676:
1672:
1670:
1669:
1665:
1663:
1662:
1658:
1656:
1655:
1651:
1649:
1648:
1644:
1642:
1641:
1637:
1636:
1634:
1630:
1624:
1623:Lill's method
1621:
1619:
1616:
1615:
1613:
1611:Miscellaneous
1609:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1582:
1581:
1578:
1575:
1571:
1567:
1564:
1562:
1559:
1558:
1556:
1552:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1530:Rigid origami
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1512:
1510:
1508:3d structures
1506:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1481:
1479:
1477:Strip folding
1475:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1425:
1423:
1419:
1415:
1408:
1403:
1401:
1396:
1394:
1389:
1388:
1385:
1374:
1371:Mabry, Rick.
1367:
1364:
1359:
1353:
1349:
1345:
1341:
1337:
1330:
1328:
1326:
1322:
1317:
1310:
1307:
1302:
1299:Mabry, Rick.
1295:
1292:
1287:
1280:
1278:
1274:
1269:
1265:
1261:
1257:
1253:
1249:
1245:
1238:
1236:
1232:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1196:
1194:
1190:
1185:
1181:
1177:
1173:
1166:
1164:
1162:
1158:
1153:
1146:
1139:
1137:
1135:
1133:
1131:
1129:
1127:
1125:
1123:
1121:
1119:
1117:
1115:
1113:
1111:
1109:
1107:
1103:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1067:
1064:
1059:
1052:
1050:
1046:
1041:
1035:
1027:
1021:
1017:
1010:
1007:
1001:
998:
987:
983:
977:
975:
973:
969:
962:
960:
956:
950:
947:
941:
939:
935:
930:
924:
920:
913:
910:
905:
901:
896:
891:
887:
883:
879:
875:
871:
864:
861:
854:
848:
845:
842:
839:
838:
835:
832:
829:
826:
825:
821:
818:
816:Multiplicity
815:
812:
811:
808:
802:
796:
793:
790:
787:
786:
783:
780:
777:
774:
773:
770:
767:
765:
762:
761:
758:
755:
752:
749:
748:
745:
743:All tricubes
742:
740:
737:
736:
733:
731:All tricubes
730:
727:
724:
723:
719:
716:
714:Multiplicity
713:
710:
709:
706:
699:
697:
688:
685:
682:
679:
677:
674:
673:
670:
667:
664:
661:
659:
656:
655:
652:
650:
647:
644:
642:
639:
638:
635:
633:
630:
627:
624:
621:
620:
617:
615:
612:
609:
606:
603:
602:
599:
597:
594:
591:
588:
585:
584:
581:
579:
576:
573:
570:
567:
566:
563:
561:
558:
555:
552:
549:
548:
545:
543:
540:
537:
534:
531:
530:
527:
525:
522:
519:
516:
513:
512:
509:
507:
504:
501:
498:
495:
494:
491:
489:
486:
483:
480:
477:
476:
473:
471:
468:
465:
462:
459:
458:
455:
453:
450:
447:
444:
441:
440:
437:
435:
432:
429:
426:
423:
422:
419:
417:
414:
411:
408:
405:
404:
401:
399:
396:
393:
390:
387:
386:
383:
381:
378:
375:
372:
369:
368:
365:
363:
360:
357:
354:
351:
350:
347:
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262:Multiplicity
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85:Multiplicity
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34:
26:
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1782:Eve Torrence
1712:Erik Demaine
1673:
1666:
1659:
1652:
1645:
1638:
1632:Publications
1589:
1494:Möbius strip
1484:Dragon curve
1421:Flat folding
1366:
1339:
1309:
1294:
1254:(1): 65–71.
1251:
1247:
1209:
1205:
1175:
1151:
1083:(1): 65–71.
1080:
1076:
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1000:
989:. Retrieved
985:
949:
918:
912:
895:1721.1/99989
877:
873:
863:
806:
703:
693:
251:
223:Icosahedron
139:Tetrahedron
126:Tetrahedron
113:Tetrahedron
101:Tetrahedron
91:Polyhedra 2
88:Polyhedra 1
78:
73:
72:
64:
56:
52:
49:
36:
30:
1767:Kōryō Miura
1762:Jun Maekawa
1737:Kôdi Husimi
1453:Map folding
1184:10119/10308
211:Octahedron
199:Octahedron
187:Octahedron
1757:Anna Lubiw
1590:Common net
1515:Miura fold
1316:"Cubigami"
1286:"Cubigami"
991:2024-08-01
855:References
822:Reference
819:Polyhedra
803:Deltahedra
720:Reference
717:Polyhedra
324:√2x√2x3√2
274:Reference
94:Reference
37:common net
25:octahedron
1803:Polyhedra
1675:Origamics
1554:Polyhedra
1268:0218-1959
1226:0925-7721
1178:: 77–82.
1097:0218-1959
1034:cite book
904:0925-7721
700:Polycubes
345:√5x√5x√5
271:Cuboid 3
268:Cuboid 2
265:Cuboid 1
45:polyhedra
1797:Category
1732:Tom Hull
1702:Yan Chen
1585:Blooming
1489:Flexagon
1212:: 1–12.
686:2x13x58
683:7x14x38
668:2x13x16
33:geometry
680:7x8x56
665:2x4x43
662:7x8x14
628:2x2x10
610:2x2x10
592:1x2x11
574:1x1x17
541:1x2x10
502:1x1x14
466:1x1x13
448:1x1x13
412:1x1x11
309:0x1x11
240:Cuboids
1685:People
1540:Sonobe
1354:
1266:
1224:
1095:
1022:
925:
902:
728:29026
645:4x4x8
631:2x4x6
613:1x4x8
595:1x3x8
577:1x5x5
559:2x4x4
556:2x2x7
538:2x2x7
523:2x3x5
520:1x3x7
505:1x4x5
487:3x3x3
484:1x3x6
469:1x3x6
451:3x3x3
433:1x3x5
430:1x2x7
415:1x3x5
397:1x3x4
394:1x1x9
379:1x2x5
376:1x1x8
373:11291
361:1x3x3
358:1x1x7
342:1x3x3
339:1x1x7
321:1x2x4
306:1x2x3
303:1x1x5
288:1x2x3
285:1x1x5
69:reads:
1148:(PDF)
1058:"Net"
813:Area
711:Area
675:1792
463:1806
445:1735
391:2334
355:1080
282:6495
259:Area
175:Cube
163:Cube
151:Cube
104:Cube
39:is a
1352:ISBN
1264:ISSN
1222:ISSN
1093:ISSN
1040:link
1020:ISBN
923:ISBN
900:ISSN
657:532
640:160
607:218
481:387
409:568
35:, a
1580:Net
1344:doi
1256:doi
1214:doi
1180:hdl
1085:doi
890:hdl
882:doi
840:10
788:22
775:22
763:22
753:68
750:18
738:14
725:14
625:86
622:88
604:88
589:11
586:70
568:70
550:64
535:50
532:64
514:62
499:37
496:58
478:54
460:54
442:54
427:92
424:46
406:46
388:38
370:34
352:30
336:30
333:30
316:28
297:22
279:22
136:37
123:87
41:net
31:In
1799::
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892::
884::
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