610:
603:
967:
44:
620:
582:
596:
589:
386:
bases are connected "back-to-back", so that the resulting object has no depth. The polygons must be congruent, but glued in such a way that one is the mirror image of the other. This applies only if the distance between the two faces is zero; for a distance larger than zero, the faces are infinite
406:. This characterization holds also for the distances on the surface of a dihedron, so the statement of Alexandrov's theorem requires that dihedra be considered as convex polyhedra.
1743:
398:, which characterizes the distances on the surface of any convex polyhedron as being locally Euclidean except at a finite number of points with positive
1736:
1063:
Gausmann, Evelise; Roland
Lehoucq; Jean-Pierre Luminet; Jean-Philippe Uzan; Jeffrey Weeks (2001). "Topological Lensing in Spherical Spaces".
1729:
1350:
1117:
395:
1292:
1255:
187:
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211:
159:
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835:
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803:
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764:
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179:
169:
1237:
830:
798:
769:
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682:
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174:
1648:
1202:
1167:
788:
759:
730:
164:
1942:
1883:
2150:
1972:
1932:
1538:
1280:
1967:
1962:
1532:
285:, a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a
2073:
2068:
1947:
1853:
1660:
1588:
1508:
1343:
278:
1937:
1878:
1868:
1813:
1594:
457:
388:
1957:
1873:
1828:
1917:
1843:
1791:
1220:
1185:
1082:
960:
915:
859:
566:
282:
107:
2083:
1952:
1927:
1912:
1848:
1796:
556:
2145:
2098:
2063:
1922:
1817:
1766:
1705:
1582:
1576:
1336:
1210:
1175:
1098:
1072:
528:
418:
239:
1284:
1274:
627:
437:
137:
1247:
1241:
2078:
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1807:
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1571:
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1514:
1311:
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1251:
542:
95:
2017:
1700:
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1090:
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998:
535:
410:
375:
313:
62:
1140:
1136:
1654:
1566:
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1526:
1481:
1471:
1461:
1456:
1132:
657:
549:
433:
340:
274:
234:
151:
83:
69:
1224:
1189:
1086:
1838:
1761:
1710:
1613:
1476:
1466:
1270:
1034:
399:
123:
1094:
2139:
2043:
1899:
1833:
1631:
1625:
1520:
1451:
1441:
1102:
1446:
609:
602:
484:
465:
329:
966:
619:
581:
43:
1776:
1486:
1420:
1410:
1400:
1395:
595:
588:
2108:
1996:
1786:
1753:
1671:
1425:
1415:
1405:
1390:
1380:
1359:
1314:
1039:
491:
352:
286:
270:
259:
246:
58:
409:
Some dihedra can arise as lower limit members of other polyhedra families: a
2103:
2093:
2038:
2022:
1858:
1685:
1319:
880:
1044:
871:
391:'s digon faces, having a width larger than zero, are infinite stripes).
2113:
2088:
1690:
847:
383:
263:
17:
1077:
479:
is made of two regular spherical polygons which share the same set of
865:
469:
325:
1721:
1215:
1180:
853:
414:
359:
1781:
1725:
1332:
1118:"On the volume of unbounded polyhedra in the hyperbolic space"
1328:
985:-dimensional analogue of a dihedron, with Schläfli symbol {
490:
The regular polyhedron {2,2} is self-dual, and is both a
468:
equator; each polygon of a spherical dihedron fills a
1207:
On flat polyhedra deriving from
Alexandrov's theorem
324:-sided faces covering the sphere, each face being a
2056:
2031:
2006:
1981:
1897:
1805:
1760:
1670:
1641:
1606:
1554:
1495:
1434:
1373:
502:22 symmetry mutations of regular dihedral tilings:
233:
210:
186:
150:
136:
122:
106:
94:
82:
68:
53:
32:
421:with a digon base would be a triangular dihedron.
281:if its faces are flat, while in three-dimensional
1246:(3rd ed.), Dover Publications Inc., p.
1172:Flat zipper-unfolding pairs for Platonic solids
1737:
1344:
8:
2010:
1744:
1730:
1722:
1351:
1337:
1329:
320:can exist as nondegenerate form, with two
42:
1214:
1179:
1076:
496:
417:bases would be a square dihedron, and a
1055:
48:Example hexagonal dihedron on a sphere
29:
1162:
1160:
7:
336:if the vertices are equally spaced.
1368:Listed by number of faces and type
1125:Beiträge zur Algebra und Geometrie
266:faces which share the same set of
25:
1273:; Schulte, Egon (December 2002),
297:). Dihedra have also been called
965:
963:as a 2-dimensional tessellation:
833:
828:
823:
818:
813:
801:
796:
791:
786:
781:
772:
767:
762:
757:
752:
743:
738:
733:
728:
723:
714:
709:
704:
699:
694:
685:
680:
675:
670:
665:
618:
608:
601:
594:
587:
580:
177:
172:
167:
162:
157:
396:Alexandrov's uniqueness theorem
374:can be considered a degenerate
483:vertices, equally spaced on a
1:
1579:(two infinite groups and 75)
1065:Classical and Quantum Gravity
498:Family of regular dihedra ¡ *
2124:Degenerate polyhedra are in
1597:(two infinite groups and 50)
460:which share the same set of
1943:pentagonal icositetrahedron
1884:truncated icosidodecahedron
1095:10.1088/0264-9381/18/23/311
959:-gonal dihedron becomes an
2167:
1973:pentagonal hexecontahedron
1933:deltoidal icositetrahedron
1281:Cambridge University Press
1276:Abstract Regular Polytopes
477:regular spherical dihedron
366:As a flat-faced polyhedron
362:faces share two vertices.
2122:
2013:
1968:disdyakis triacontahedron
1963:deltoidal hexecontahedron
1649:KeplerâPoinsot polyhedron
1366:
512:
448:As a tiling of the sphere
387:polygons (a bit like the
41:
428:, with Schläfli symbol {
2074:gyroelongated bipyramid
1948:rhombic triacontahedron
1854:truncated cuboctahedron
1661:Uniform star polyhedron
1589:quasiregular polyhedron
394:Dihedra can arise from
307:doubly covered polygons
273:. In three-dimensional
27:Polyhedron with 2 faces
2069:truncated trapezohedra
1938:disdyakis dodecahedron
1904:(duals of Archimedean)
1879:rhombicosidodecahedron
1869:truncated dodecahedron
1595:semiregular polyhedron
955:tends to infinity, an
389:apeirogonal hosohedron
347:-gonal dihedron is an
1958:pentakis dodecahedron
1874:truncated icosahedron
1829:truncated tetrahedron
1642:non-convex polyhedron
1918:rhombic dodecahedron
1844:truncated octahedron
961:apeirogonal dihedron
947:Apeirogonal dihedron
432:,2}, is made of two
328:, and vertices on a
108:Vertex configuration
1953:triakis icosahedron
1928:tetrakis hexahedron
1913:triakis tetrahedron
1849:rhombicuboctahedron
1225:2010arXiv1007.2016O
1190:2010arXiv1010.2450O
1116:KĂĄntor, S. (2003),
1087:2001CQGra..18.5155G
1013:}, which share all
506:
378:whose two (planar)
1923:triakis octahedron
1808:Archimedean solids
1583:regular polyhedron
1577:uniform polyhedron
1539:Hectotriadiohedron
1312:Weisstein, Eric W.
497:
458:spherical polygons
454:spherical dihedron
2151:Regular polyhedra
2133:
2132:
2052:
2051:
1889:snub dodecahedron
1864:icosidodecahedron
1719:
1718:
1620:Archimedean solid
1607:convex polyhedron
1515:Icosidodecahedron
1243:Regular Polytopes
1238:Coxeter, H. S. M.
1071:(23): 5155â5186.
944:
943:
252:
251:
16:(Redirected from
2158:
2011:
2007:Dihedral uniform
1982:Dihedral regular
1905:
1821:
1770:
1746:
1739:
1732:
1723:
1555:elemental things
1533:Enneacontahedron
1503:Icositetrahedron
1353:
1346:
1339:
1330:
1325:
1324:
1298:
1297:
1279:(1st ed.),
1267:
1261:
1260:
1240:(January 1973),
1234:
1228:
1227:
1218:
1203:O'Rourke, Joseph
1199:
1193:
1192:
1183:
1168:O'Rourke, Joseph
1164:
1155:
1153:
1152:
1151:
1145:
1139:, archived from
1122:
1113:
1107:
1106:
1080:
1060:
997:,2}. It has two
969:
838:
837:
836:
832:
831:
827:
826:
822:
821:
817:
816:
806:
805:
804:
800:
799:
795:
794:
790:
789:
785:
784:
777:
776:
775:
771:
770:
766:
765:
761:
760:
756:
755:
748:
747:
746:
742:
741:
737:
736:
732:
731:
727:
726:
719:
718:
717:
713:
712:
708:
707:
703:
702:
698:
697:
690:
689:
688:
684:
683:
679:
678:
674:
673:
669:
668:
622:
612:
605:
598:
591:
584:
507:
494:and a dihedron.
434:regular polygons
426:regular dihedron
405:
314:spherical tiling
182:
181:
180:
176:
175:
171:
170:
166:
165:
161:
160:
63:spherical tiling
46:
30:
21:
2166:
2165:
2161:
2160:
2159:
2157:
2156:
2155:
2136:
2135:
2134:
2129:
2118:
2057:Dihedral others
2048:
2027:
2002:
1977:
1906:
1903:
1902:
1893:
1822:
1811:
1810:
1801:
1764:
1762:Platonic solids
1756:
1750:
1720:
1715:
1666:
1655:Star polyhedron
1637:
1602:
1550:
1527:Hexecontahedron
1509:Triacontahedron
1491:
1482:Enneadecahedron
1472:Heptadecahedron
1462:Pentadecahedron
1457:Tetradecahedron
1430:
1369:
1362:
1357:
1310:
1309:
1306:
1301:
1295:
1271:McMullen, Peter
1269:
1268:
1264:
1258:
1236:
1235:
1231:
1201:
1200:
1196:
1166:
1165:
1158:
1149:
1147:
1143:
1120:
1115:
1114:
1110:
1062:
1061:
1057:
1053:
1031:
975:
964:
949:
917:
888:
834:
829:
824:
819:
814:
812:
802:
797:
792:
787:
782:
780:
773:
768:
763:
758:
753:
751:
744:
739:
734:
729:
724:
722:
715:
710:
705:
700:
695:
693:
686:
681:
676:
671:
666:
664:
659:
629:
576:
568:
558:
551:
544:
537:
530:
524:
464:vertices, on a
456:is made of two
450:
438:Schläfli symbol
403:
368:
283:spherical space
275:Euclidean space
235:Dual polyhedron
221:
198:
178:
173:
168:
163:
158:
156:
152:Coxeter diagram
138:Schläfli symbol
49:
33:Set of regular
28:
23:
22:
15:
12:
11:
5:
2164:
2162:
2154:
2153:
2148:
2138:
2137:
2131:
2130:
2123:
2120:
2119:
2117:
2116:
2111:
2106:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2060:
2058:
2054:
2053:
2050:
2049:
2047:
2046:
2041:
2035:
2033:
2029:
2028:
2026:
2025:
2020:
2014:
2008:
2004:
2003:
2001:
2000:
1993:
1985:
1983:
1979:
1978:
1976:
1975:
1970:
1965:
1960:
1955:
1950:
1945:
1940:
1935:
1930:
1925:
1920:
1915:
1909:
1907:
1900:Catalan solids
1898:
1895:
1894:
1892:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1839:truncated cube
1836:
1831:
1825:
1823:
1806:
1803:
1802:
1800:
1799:
1794:
1789:
1784:
1779:
1773:
1771:
1758:
1757:
1751:
1749:
1748:
1741:
1734:
1726:
1717:
1716:
1714:
1713:
1711:parallelepiped
1708:
1703:
1698:
1693:
1688:
1683:
1677:
1675:
1668:
1667:
1665:
1664:
1658:
1652:
1645:
1643:
1639:
1638:
1636:
1635:
1629:
1623:
1617:
1614:Platonic solid
1610:
1608:
1604:
1603:
1601:
1600:
1599:
1598:
1592:
1586:
1574:
1569:
1564:
1558:
1556:
1552:
1551:
1549:
1548:
1542:
1536:
1530:
1524:
1518:
1512:
1506:
1499:
1497:
1493:
1492:
1490:
1489:
1484:
1479:
1477:Octadecahedron
1474:
1469:
1467:Hexadecahedron
1464:
1459:
1454:
1449:
1444:
1438:
1436:
1432:
1431:
1429:
1428:
1423:
1418:
1413:
1408:
1403:
1398:
1393:
1388:
1383:
1377:
1375:
1371:
1370:
1367:
1364:
1363:
1358:
1356:
1355:
1348:
1341:
1333:
1327:
1326:
1305:
1304:External links
1302:
1300:
1299:
1293:
1262:
1256:
1229:
1194:
1156:
1131:(1): 145â154,
1108:
1054:
1052:
1049:
1048:
1047:
1042:
1037:
1035:Dihedral group
1030:
1027:
974:
971:
948:
945:
942:
941:
938:
935:
932:
929:
926:
923:
920:
912:
911:
908:
905:
902:
899:
896:
893:
890:
884:
883:
877:
874:
868:
862:
856:
850:
844:
840:
839:
810:
807:
778:
749:
720:
691:
662:
654:
653:
650:
647:
644:
641:
638:
635:
632:
624:
623:
616:
613:
606:
599:
592:
585:
578:
572:
571:
564:
561:
554:
547:
540:
533:
526:
520:
519:
516:
514:
511:
449:
446:
400:angular defect
367:
364:
303:flat polyhedra
262:, made of two
250:
249:
237:
231:
230:
217:
214:
212:Rotation group
208:
207:
193:
190:
188:Symmetry group
184:
183:
154:
148:
147:
140:
134:
133:
126:
124:Wythoff symbol
120:
119:
110:
104:
103:
98:
92:
91:
86:
80:
79:
72:
66:
65:
55:
51:
50:
47:
39:
38:
37:-gonal dihedra
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2163:
2152:
2149:
2147:
2144:
2143:
2141:
2127:
2121:
2115:
2112:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2055:
2045:
2042:
2040:
2037:
2036:
2034:
2030:
2024:
2021:
2019:
2016:
2015:
2012:
2009:
2005:
1999:
1998:
1994:
1992:
1991:
1987:
1986:
1984:
1980:
1974:
1971:
1969:
1966:
1964:
1961:
1959:
1956:
1954:
1951:
1949:
1946:
1944:
1941:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1914:
1911:
1910:
1908:
1901:
1896:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1834:cuboctahedron
1832:
1830:
1827:
1826:
1824:
1819:
1815:
1809:
1804:
1798:
1795:
1793:
1790:
1788:
1785:
1783:
1780:
1778:
1775:
1774:
1772:
1768:
1763:
1759:
1755:
1747:
1742:
1740:
1735:
1733:
1728:
1727:
1724:
1712:
1709:
1707:
1704:
1702:
1699:
1697:
1694:
1692:
1689:
1687:
1684:
1682:
1679:
1678:
1676:
1673:
1669:
1662:
1659:
1656:
1653:
1650:
1647:
1646:
1644:
1640:
1633:
1632:Johnson solid
1630:
1627:
1626:Catalan solid
1624:
1621:
1618:
1615:
1612:
1611:
1609:
1605:
1596:
1593:
1590:
1587:
1584:
1581:
1580:
1578:
1575:
1573:
1570:
1568:
1565:
1563:
1560:
1559:
1557:
1553:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1521:Hexoctahedron
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1500:
1498:
1494:
1488:
1485:
1483:
1480:
1478:
1475:
1473:
1470:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1452:Tridecahedron
1450:
1448:
1445:
1443:
1442:Hendecahedron
1440:
1439:
1437:
1433:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1382:
1379:
1378:
1376:
1372:
1365:
1361:
1354:
1349:
1347:
1342:
1340:
1335:
1334:
1331:
1322:
1321:
1316:
1313:
1308:
1307:
1303:
1296:
1294:0-521-81496-0
1290:
1286:
1282:
1278:
1277:
1272:
1266:
1263:
1259:
1257:0-486-61480-8
1253:
1249:
1245:
1244:
1239:
1233:
1230:
1226:
1222:
1217:
1212:
1208:
1204:
1198:
1195:
1191:
1187:
1182:
1177:
1173:
1169:
1163:
1161:
1157:
1146:on 2017-02-15
1142:
1138:
1134:
1130:
1126:
1119:
1112:
1109:
1104:
1100:
1096:
1092:
1088:
1084:
1079:
1078:gr-qc/0106033
1074:
1070:
1066:
1059:
1056:
1050:
1046:
1043:
1041:
1038:
1036:
1033:
1032:
1028:
1026:
1025:} in common.
1024:
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1016:
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1008:
1004:
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996:
992:
988:
984:
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972:
970:
968:
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288:
284:
280:
276:
272:
269:
265:
261:
258:is a type of
257:
248:
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215:
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202:
196:
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155:
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149:
145:
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105:
102:
99:
97:
93:
90:
87:
85:
81:
77:
73:
71:
67:
64:
60:
56:
52:
45:
40:
36:
31:
19:
2125:
2044:trapezohedra
1995:
1989:
1988:
1792:dodecahedron
1545:Apeirohedron
1496:>20 faces
1447:Dodecahedron
1385:
1318:
1275:
1265:
1242:
1232:
1206:
1197:
1171:
1148:, retrieved
1141:the original
1128:
1124:
1111:
1068:
1064:
1058:
1022:
1018:
1010:
1006:
1002:
994:
990:
986:
982:
978:
976:
956:
952:
950:
567:Apeirogonal
503:
499:
489:
485:great circle
480:
476:
474:
466:great circle
461:
453:
451:
441:
436:, each with
429:
425:
423:
408:
402:summing to 4
393:
379:
371:
369:
356:
348:
344:
338:
333:
330:great circle
321:
317:
311:
306:
302:
298:
294:
290:
267:
255:
253:
242:
227:
223:
218:
204:
200:
194:
143:
129:
116:
112:
100:
88:
75:
34:
1814:semiregular
1797:icosahedron
1777:tetrahedron
1487:Icosahedron
1435:11â20 faces
1421:Enneahedron
1411:Heptahedron
1401:Pentahedron
1396:Tetrahedron
557:Pentagonal
2140:Categories
2109:prismatoid
2039:bipyramids
2023:antiprisms
1997:hosohedron
1787:octahedron
1672:prismatoid
1657:(infinite)
1426:Decahedron
1416:Octahedron
1406:Hexahedron
1381:Monohedron
1374:1â10 faces
1315:"Dihedron"
1283:, p.
1150:2017-02-14
1051:References
1040:Polyhedron
977:A regular
887:Edges and
529:Monogonal
518:Euclidean
492:hosohedron
470:hemisphere
353:hosohedron
326:hemisphere
287:lens space
279:degenerate
260:polyhedron
247:hosohedron
226:), order 2
203:), order 4
59:polyhedron
2146:Polyhedra
2104:birotunda
2094:bifrustum
1859:snub cube
1754:polyhedra
1686:antiprism
1391:Trihedron
1360:Polyhedra
1320:MathWorld
1216:1007.2016
1181:1010.2450
889:vertices
628:Schläfli
543:Trigonal
513:Spherical
487:equator.
2084:bicupola
2064:pyramids
1990:dihedron
1386:Dihedron
1205:(2010),
1170:(2010),
1103:34259877
1045:Polytope
1029:See also
658:Coxeter
569:dihedron
559:dihedron
552:dihedron
545:dihedron
538:dihedron
536:Digonal
531:dihedron
372:dihedron
355:, where
332:. It is
318:dihedron
277:, it is
256:dihedron
199:, , (*22
96:Vertices
57:regular
2126:italics
2114:scutoid
2099:rotunda
2089:frustum
1818:uniform
1767:regular
1752:Convex
1706:pyramid
1691:frustum
1221:Bibcode
1186:Bibcode
1137:1990989
1083:Bibcode
973:Ditopes
918:config.
916:Vertex
660:diagram
575:Tiling
550:Square
523:Tiling
419:pyramid
384:polygon
382:-sided
351:-gonal
334:regular
299:bihedra
264:polygon
245:-gonal
240:regular
222:, , (22
18:Dihedra
2079:cupola
2032:duals:
2018:prisms
1696:cupola
1572:vertex
1291:
1254:
1135:
1101:
1015:ridges
999:facets
981:is an
979:ditope
843:Faces
652:{â,2}
630:symbol
577:image
510:Space
343:of an
1701:wedge
1681:prism
1541:(132)
1211:arXiv
1176:arXiv
1144:(PDF)
1121:(PDF)
1099:S2CID
1073:arXiv
1021:,...,
1005:,...,
989:,...,
646:{5,2}
643:{4,2}
640:{3,2}
637:{2,2}
634:{1,2}
525:name
415:digon
413:with
411:prism
376:prism
360:digon
312:As a
305:, or
271:edges
84:Edges
78:-gons
70:Faces
1782:cube
1663:(57)
1634:(92)
1628:(13)
1622:(13)
1591:(16)
1567:edge
1562:face
1535:(90)
1529:(60)
1523:(48)
1517:(32)
1511:(30)
1505:(24)
1289:ISBN
1252:ISBN
940:â.â
809:...
615:...
563:...
341:dual
339:The
316:, a
128:2 |
54:Type
1816:or
1651:(4)
1616:(5)
1585:(9)
1547:(â)
1285:158
1091:doi
1017:, {
1001:, {
951:As
937:...
934:5.5
931:4.4
928:3.3
925:2.2
922:1.1
907:...
881:{â}
876:...
872:{5}
866:{4}
860:{3}
854:{2}
848:{1}
649:...
444:}.
146:,2}
61:or
2142::
1674:âs
1317:.
1287:,
1250:,
1248:12
1219:,
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1184:,
1174:,
1159:^
1133:MR
1129:44
1127:,
1123:,
1097:.
1089:.
1081:.
1069:18
1067:.
910:â
879:2
870:2
864:2
858:2
852:2
846:2
504:nn
475:A
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452:A
424:A
370:A
309:.
301:,
289:L(
254:A
74:2
2128:.
1820:)
1812:(
1769:)
1765:(
1745:e
1738:t
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1223::
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1188::
1178::
1154:.
1105:.
1093::
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1075::
1023:q
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1007:q
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993:,
991:q
987:p
983:n
957:n
953:n
904:5
901:4
898:3
895:2
892:1
500:n
481:n
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430:n
404:Ď
380:n
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349:n
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322:n
295:q
293:,
291:p
268:n
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219:n
216:D
205:n
201:n
197:h
195:n
192:D
144:n
142:{
132:2
130:n
117:n
115:.
113:n
101:n
89:n
76:n
35:n
20:)
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