656:
663:
649:
642:
635:
677:
591:
1150:
598:
584:
670:
619:
319:
628:
605:
1412:
563:
1423:
684:
612:
969:
44:
1401:
577:
570:
1456:
1445:
1434:
962:
791:
368:
757:
1201:
945:
1504:
1497:
1490:
1483:
1476:
1469:
1280:
278:
33:
263:
1266:
with its 6 square faces bisected on diagonals with pyritohedral symmetry. The icosahedra with pyritohedral symmetry constitute an infinite family of polyhedra which include the cuboctahedron, regular icosahedron, Jessen's icosahedron, and
1241:
452:
is the process of extending the faces or edges of a polyhedron until they meet to form a new polyhedron. It is done symmetrically so that the resulting figure retains the overall symmetry of the parent figure.
849:
1039:
has the symbol (332), , with order 12. These lower symmetries allow geometric distortions from 20 equilateral triangular faces, instead having 8 equilateral triangles and 12 congruent
1165:
of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form (2, 1, 0). These coordinates represent the
2110:
468:
Other stellations have more than one face in each plane or form compounds of simpler polyhedra. These are not strictly icosahedra, although they are often referred to as such.
2103:
2096:
1717:
554:
1416:
1604:
1427:
889:
414:
rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.
1113:
1102:
1092:
1082:
1074:
1064:
905:
784:
774:
764:
740:
730:
465:
Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them.
1405:
1257:
1133:
1123:
1054:
981:
750:
2015:
1128:
1118:
1097:
1087:
1069:
1059:
779:
769:
745:
735:
499:
292:
There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20
808:
1460:
1449:
1438:
422:
2512:
2309:
2250:
655:
534:
519:
1232:
to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
662:
648:
641:
634:
2517:
2339:
2299:
1905:
1592:
1229:
539:
529:
524:
514:
2334:
2329:
476:
458:
590:
1899:
1149:
676:
597:
583:
387:
300:. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a
2440:
2435:
2314:
2220:
2027:
1955:
1875:
1710:
1309:
1268:
1209:
489:
618:
2304:
2245:
2235:
2180:
1961:
1170:
1019:
549:
318:
180:
1252:, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.
2324:
2240:
2195:
1562:
1180:
in its regular icosahedron form, generated by the same operations carried out starting with the vector (
1162:
1032:
894:
669:
604:
221:
1411:
1422:
627:
2284:
2210:
2158:
1312:
by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not
1166:
1154:
1138:
1036:
1026:
1022:
910:
700:
350:
297:
293:
241:
968:
562:
43:
2450:
2319:
2294:
2279:
2215:
2163:
1400:
1301:
1295:
1283:
1107:
696:
683:
611:
576:
569:
509:
494:
313:
269:
233:
37:
2465:
2430:
2289:
2184:
2133:
2072:
1949:
1943:
1703:
1040:
798:
391:
339:
1455:
1444:
1433:
961:
410:}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a
2445:
2255:
2230:
2174:
2062:
1986:
1938:
1911:
1881:
1671:
1600:
1249:
1029:, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.
879:
544:
383:
362:
283:
225:
224:
shapes of icosahedra, some of them being more symmetrical than others. The best known is the (
2384:
2067:
2047:
1869:
1632:
1336:
949:
790:
323:
200:
131:
66:
1218:, the 12 isosceles faces are arranged differently so that the figure is non-convex and has
756:
2021:
1933:
1928:
1893:
1848:
1838:
1828:
1823:
1313:
1047:
921:
869:
855:
722:
418:
346:
980:
with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a
2205:
2128:
2077:
1980:
1843:
1833:
1271:
octahedron. Cyclical kinematic transformations among the members of this family exist.
1222:
1200:
944:
484:
372:
335:
237:
1637:
1620:
2506:
2410:
2266:
2200:
1998:
1992:
1887:
1818:
1808:
1370:
1357:
1263:
977:
1503:
1496:
1489:
1482:
1475:
1468:
1813:
1279:
1189:
999:
926:
367:
342:{3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
2143:
1787:
1777:
1767:
1762:
1686:
1219:
277:
2475:
2363:
2153:
2120:
2038:
1792:
1782:
1772:
1757:
1747:
1726:
1305:
1245:
936:
692:
449:
229:
176:
48:
1619:
Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01).
2470:
2460:
2405:
2389:
2225:
2052:
1656:
1350:
1343:
462:, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.
411:
376:
17:
2356:
1752:
1567:
56:
262:
32:
2480:
2455:
2057:
1572:
1329:
691:
The stellation process on the icosahedron creates a number of related
441:, 3}, having three regular star pentagonal faces around each vertex.
1240:
330:
The convex regular icosahedron is usually referred to simply as the
2088:
1278:
1239:
1199:
1148:
713:
366:
317:
296:
faces with five meeting at each of its twelve vertices. Both have
42:
31:
353:{5, 3} having three regular pentagonal faces around each vertex.
2148:
2092:
1699:
982:
kinematic transformation between cuboctahedron and icosahedron
167:
158:
102:
93:
1324:
Common icosahedra with pyramid and prism symmetries include:
1695:
1308:
made up of 20 congruent rhombs. It can be derived from the
209:
149:
134:
120:
111:
84:
72:
1595:(2003) , Peter Roach; James Hartmann; Jane Setter (eds.),
1106:
respectively, each representing the lower symmetry to the
143:
78:
1673:
Connections: The
Geometric Bridge Between Art and Science
1262:
A regular icosahedron is topologically identical to a
976:
A regular icosahedron is topologically identical to a
820:
811:
197: 'seat'. The plural can be either "icosahedra" (
146:
140:
137:
117:
108:
81:
75:
69:
1375:
470:
206:
203:
164:
161:
99:
96:
2423:
2398:
2373:
2348:
2264:
2172:
2127:
2037:
2008:
1973:
1921:
1862:
1801:
1740:
844:{\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}
155:
152:
90:
87:
843:
326:inscribed in a convex regular icosahedron
1676:(2nd ed.). World Scientific. p. 475.
2104:
1711:
1625:Sultan Qaboos University Journal for Science
8:
956:
1621:"Symmetry of the Pyritohedron and Lattices"
716:
2377:
2111:
2097:
2089:
1718:
1704:
1696:
1214:In Jessen's icosahedron, sometimes called
1636:
1599:, Cambridge: Cambridge University Press,
1509:
1397:
998:can be distorted or marked up as a lower
815:
810:
717:Pyritohedral and tetrahedral symmetries
1584:
1035:has the symbol (3*2), , with order 24.
179:with 20 faces. The name comes from
1650:
1648:
7:
1153:Construction from the vertices of a
1735:Listed by number of faces and type
1417:Elongated triangular orthobicupola
1346:(2 sets of 9 sides + 2 ends = 20).
25:
1638:10.24200/squjs.vol21iss2pp139-149
1428:Elongated triangular gyrobicupola
1655:John Baez (September 11, 2011).
1502:
1495:
1488:
1481:
1474:
1467:
1454:
1443:
1432:
1421:
1410:
1399:
1131:
1126:
1121:
1116:
1111:
1100:
1095:
1090:
1085:
1080:
1072:
1067:
1062:
1057:
1052:
967:
960:
943:
789:
782:
777:
772:
767:
762:
755:
748:
743:
738:
733:
728:
682:
675:
668:
661:
654:
647:
640:
633:
626:
617:
610:
603:
596:
589:
582:
575:
568:
561:
386:is one of the four regular star
276:
261:
199:
130:
65:
1406:Gyroelongated triangular cupola
1258:Kinematics of the cuboctahedron
1216:Jessen's orthogonal icosahedron
932:
920:
904:
888:
878:
868:
854:
797:
721:
255:Two kinds of regular icosahedra
1597:English Pronouncing Dictionary
1176:This construction is called a
1157:, showing internal rectangles.
477:stellations of the icosahedron
220:There are infinitely many non-
1:
1946:(two infinite groups and 75)
2491:Degenerate polyhedra are in
1964:(two infinite groups and 50)
1461:Triangular hebesphenorotunda
1450:Metabiaugmented dodecahedron
1439:Parabiaugmented dodecahedron
1320:Pyramid and prism symmetries
423:great stellated dodecahedron
338:, and is represented by its
2310:pentagonal icositetrahedron
2251:truncated icosidodecahedron
1195:
1025:. If all the triangles are
535:Compound of five tetrahedra
520:Medial triambic icosahedron
190: 'twenty' and
2534:
2340:pentagonal hexecontahedron
2300:deltoidal icositetrahedron
1360:(2 sets of 10 sides = 20).
1353:(2 sets of 10 sides = 20).
1293:
1255:
1207:
1002:symmetry, and is called a
690:
540:Compound of ten tetrahedra
530:Compound of five octahedra
525:Great triambic icosahedron
515:Small triambic icosahedron
473:
360:
334:, one of the five regular
311:
308:Convex regular icosahedron
2489:
2380:
2335:disdyakis triacontahedron
2330:deltoidal hexecontahedron
2016:Kepler–Poinsot polyhedron
1733:
1018:. This can be seen as an
975:
955:
942:
639:
574:
503:
493:
488:
459:The Fifty-Nine Icosahedra
1244:Progressions between an
388:Kepler-Poinsot polyhedra
29:Polyhedron with 20 faces
2441:gyroelongated bipyramid
2315:rhombic triacontahedron
2221:truncated cuboctahedron
2028:Uniform star polyhedron
1956:quasiregular polyhedron
1310:rhombic triacontahedron
1046:These symmetries offer
874:30 (6 short + 24 long)
2436:truncated trapezohedra
2305:disdyakis dodecahedron
2271:(duals of Archimedean)
2246:rhombicosidodecahedron
2236:truncated dodecahedron
1962:semiregular polyhedron
1670:Kappraff, Jay (1991).
1286:
1253:
1205:
1158:
845:
550:Excavated dodecahedron
379:
327:
52:
40:
2325:pentakis dodecahedron
2241:truncated icosahedron
2196:truncated tetrahedron
2009:non-convex polyhedron
1563:Truncated icosahedron
1282:
1243:
1203:
1163:Cartesian coordinates
1152:
1145:Cartesian coordinates
1033:Pyritohedral symmetry
1008:snub tetratetrahedron
846:
710:Pyritohedral symmetry
370:
321:
242:equilateral triangles
217:) or "icosahedrons".
46:
35:
2285:rhombic dodecahedron
2211:truncated octahedron
1210:Jessen's icosahedron
1204:Jessen's icosahedron
1196:Jessen's icosahedron
1167:truncated octahedron
1155:truncated octahedron
1139:icosahedral symmetry
1037:Tetrahedral symmetry
1023:truncated octahedron
916:, , (332), order 12
900:, , (3*2), order 24
809:
701:icosahedral symmetry
510:(Convex) icosahedron
445:Stellated icosahedra
351:regular dodecahedron
298:icosahedral symmetry
294:equilateral triangle
240:—whose faces are 20
2320:triakis icosahedron
2295:tetrakis hexahedron
2280:triakis tetrahedron
2216:rhombicuboctahedron
1339:(plus 2 ends = 20).
1332:(plus 1 base = 20).
1302:rhombic icosahedron
1296:Rhombic icosahedron
1290:Rhombic icosahedron
1284:Rhombic icosahedron
1108:regular icosahedron
1041:isosceles triangles
996:regular icosahedron
332:regular icosahedron
322:Three interlocking
314:Regular icosahedron
270:regular icosahedron
234:regular icosahedron
38:regular icosahedron
2513:Geodesic polyhedra
2290:triakis octahedron
2175:Archimedean solids
1950:regular polyhedron
1944:uniform polyhedron
1906:Hectotriadiohedron
1287:
1254:
1230:scissors congruent
1206:
1173:vertices deleted.
1159:
1016:pseudo-icosahedron
841:
835:
490:Uniform duals
380:
328:
248:Regular icosahedra
53:
41:
2518:Individual graphs
2500:
2499:
2419:
2418:
2256:snub dodecahedron
2231:icosidodecahedron
2086:
2085:
1987:Archimedean solid
1974:convex polyhedron
1882:Icosidodecahedron
1554:
1553:
1250:pseudoicosahedron
992:
991:
988:
987:
707:
706:
545:Great icosahedron
495:Regular compounds
384:great icosahedron
363:Great icosahedron
357:Great icosahedron
324:golden rectangles
302:great icosahedron
284:Great icosahedron
16:(Redirected from
2525:
2378:
2374:Dihedral uniform
2349:Dihedral regular
2272:
2188:
2137:
2113:
2106:
2099:
2090:
1922:elemental things
1900:Enneacontahedron
1870:Icositetrahedron
1720:
1713:
1706:
1697:
1690:
1684:
1678:
1677:
1667:
1661:
1660:
1652:
1643:
1642:
1640:
1616:
1610:
1609:
1589:
1506:
1499:
1492:
1485:
1478:
1471:
1458:
1447:
1436:
1425:
1414:
1403:
1376:
1373:are icosahedra:
1275:Other icosahedra
1178:snub tetrahedron
1136:
1135:
1134:
1130:
1129:
1125:
1124:
1120:
1119:
1115:
1114:
1105:
1104:
1103:
1099:
1098:
1094:
1093:
1089:
1088:
1084:
1083:
1077:
1076:
1075:
1071:
1070:
1066:
1065:
1061:
1060:
1056:
1055:
1048:Coxeter diagrams
1012:snub tetrahedron
971:
964:
957:
947:
850:
848:
847:
842:
840:
839:
793:
787:
786:
785:
781:
780:
776:
775:
771:
770:
766:
765:
759:
753:
752:
751:
747:
746:
742:
741:
737:
736:
732:
731:
723:Coxeter diagrams
714:
686:
679:
672:
665:
658:
651:
644:
637:
630:
621:
614:
607:
600:
593:
586:
579:
572:
565:
555:Final stellation
471:
440:
438:
437:
434:
431:
409:
407:
406:
403:
400:
280:
265:
216:
215:
212:
211:
208:
205:
174:
173:
170:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
136:
127:
126:
123:
122:
119:
114:
113:
110:
105:
104:
101:
98:
95:
92:
89:
86:
83:
80:
77:
74:
71:
21:
2533:
2532:
2528:
2527:
2526:
2524:
2523:
2522:
2503:
2502:
2501:
2496:
2485:
2424:Dihedral others
2415:
2394:
2369:
2344:
2273:
2270:
2269:
2260:
2189:
2178:
2177:
2168:
2131:
2129:Platonic solids
2123:
2117:
2087:
2082:
2033:
2022:Star polyhedron
2004:
1969:
1917:
1894:Hexecontahedron
1876:Triacontahedron
1858:
1849:Enneadecahedron
1839:Heptadecahedron
1829:Pentadecahedron
1824:Tetradecahedron
1797:
1736:
1729:
1724:
1694:
1693:
1685:
1681:
1669:
1668:
1664:
1654:
1653:
1646:
1618:
1617:
1613:
1607:
1591:
1590:
1586:
1581:
1559:
1549:
1547:
1545:
1540:
1538:
1533:
1531:
1526:
1521:
1516:
1514:
1512:
1459:
1448:
1437:
1426:
1415:
1404:
1367:
1322:
1314:face-transitive
1298:
1292:
1277:
1260:
1238:
1223:dihedral angles
1212:
1198:
1184:, 1, 0), where
1147:
1132:
1127:
1122:
1117:
1112:
1110:
1101:
1096:
1091:
1086:
1081:
1079:
1073:
1068:
1063:
1058:
1053:
1051:
1004:snub octahedron
948:
922:Dual polyhedron
914:
898:
863:
861:
834:
833:
827:
826:
816:
807:
806:
804:
799:Schläfli symbol
783:
778:
773:
768:
763:
761:
760:
754:(pyritohedral)
749:
744:
739:
734:
729:
727:
712:
447:
435:
432:
429:
428:
426:
419:dual polyhedron
404:
401:
398:
397:
395:
392:Schläfli symbol
365:
359:
347:dual polyhedron
340:Schläfli symbol
336:Platonic solids
316:
310:
290:
289:
288:
287:
286:
281:
273:
272:
266:
257:
256:
250:
238:Platonic solids
202:
198:
133:
129:
116:
107:
68:
64:
30:
23:
22:
15:
12:
11:
5:
2531:
2529:
2521:
2520:
2515:
2505:
2504:
2498:
2497:
2490:
2487:
2486:
2484:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2427:
2425:
2421:
2420:
2417:
2416:
2414:
2413:
2408:
2402:
2400:
2396:
2395:
2393:
2392:
2387:
2381:
2375:
2371:
2370:
2368:
2367:
2360:
2352:
2350:
2346:
2345:
2343:
2342:
2337:
2332:
2327:
2322:
2317:
2312:
2307:
2302:
2297:
2292:
2287:
2282:
2276:
2274:
2267:Catalan solids
2265:
2262:
2261:
2259:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2218:
2213:
2208:
2206:truncated cube
2203:
2198:
2192:
2190:
2173:
2170:
2169:
2167:
2166:
2161:
2156:
2151:
2146:
2140:
2138:
2125:
2124:
2118:
2116:
2115:
2108:
2101:
2093:
2084:
2083:
2081:
2080:
2078:parallelepiped
2075:
2070:
2065:
2060:
2055:
2050:
2044:
2042:
2035:
2034:
2032:
2031:
2025:
2019:
2012:
2010:
2006:
2005:
2003:
2002:
1996:
1990:
1984:
1981:Platonic solid
1977:
1975:
1971:
1970:
1968:
1967:
1966:
1965:
1959:
1953:
1941:
1936:
1931:
1925:
1923:
1919:
1918:
1916:
1915:
1909:
1903:
1897:
1891:
1885:
1879:
1873:
1866:
1864:
1860:
1859:
1857:
1856:
1851:
1846:
1844:Octadecahedron
1841:
1836:
1834:Hexadecahedron
1831:
1826:
1821:
1816:
1811:
1805:
1803:
1799:
1798:
1796:
1795:
1790:
1785:
1780:
1775:
1770:
1765:
1760:
1755:
1750:
1744:
1742:
1738:
1737:
1734:
1731:
1730:
1725:
1723:
1722:
1715:
1708:
1700:
1692:
1691:
1679:
1662:
1644:
1611:
1605:
1583:
1582:
1580:
1577:
1576:
1575:
1570:
1565:
1558:
1555:
1552:
1551:
1542:
1535:
1528:
1523:
1518:
1508:
1507:
1500:
1493:
1486:
1479:
1472:
1464:
1463:
1452:
1441:
1430:
1419:
1408:
1396:
1395:
1392:
1389:
1386:
1383:
1380:
1371:Johnson solids
1366:
1365:Johnson solids
1363:
1362:
1361:
1354:
1347:
1340:
1333:
1321:
1318:
1294:Main article:
1291:
1288:
1276:
1273:
1256:Main article:
1237:
1234:
1208:Main article:
1197:
1194:
1146:
1143:
1141:of order 120.
990:
989:
986:
985:
973:
972:
965:
953:
952:
940:
939:
934:
930:
929:
924:
918:
917:
912:
908:
906:Rotation group
902:
901:
896:
892:
890:Symmetry group
886:
885:
882:
876:
875:
872:
866:
865:
858:
852:
851:
838:
832:
829:
828:
825:
822:
821:
819:
814:
801:
795:
794:
788:(tetrahedral)
725:
719:
718:
711:
708:
705:
704:
688:
687:
680:
673:
666:
659:
652:
645:
638:
631:
623:
622:
615:
608:
601:
594:
587:
580:
573:
566:
558:
557:
552:
547:
542:
537:
532:
527:
522:
517:
512:
506:
505:
502:
497:
492:
487:
481:
480:
456:In their book
446:
443:
361:Main article:
358:
355:
312:Main article:
309:
306:
282:
275:
274:
267:
260:
259:
258:
254:
253:
252:
251:
249:
246:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2530:
2519:
2516:
2514:
2511:
2510:
2508:
2494:
2488:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2428:
2426:
2422:
2412:
2409:
2407:
2404:
2403:
2401:
2397:
2391:
2388:
2386:
2383:
2382:
2379:
2376:
2372:
2366:
2365:
2361:
2359:
2358:
2354:
2353:
2351:
2347:
2341:
2338:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2277:
2275:
2268:
2263:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2201:cuboctahedron
2199:
2197:
2194:
2193:
2191:
2186:
2182:
2176:
2171:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2141:
2139:
2135:
2130:
2126:
2122:
2114:
2109:
2107:
2102:
2100:
2095:
2094:
2091:
2079:
2076:
2074:
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2045:
2043:
2040:
2036:
2029:
2026:
2023:
2020:
2017:
2014:
2013:
2011:
2007:
2000:
1999:Johnson solid
1997:
1994:
1993:Catalan solid
1991:
1988:
1985:
1982:
1979:
1978:
1976:
1972:
1963:
1960:
1957:
1954:
1951:
1948:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1926:
1924:
1920:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1888:Hexoctahedron
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1867:
1865:
1861:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1819:Tridecahedron
1817:
1815:
1812:
1810:
1809:Hendecahedron
1807:
1806:
1804:
1800:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1771:
1769:
1766:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1745:
1743:
1739:
1732:
1728:
1721:
1716:
1714:
1709:
1707:
1702:
1701:
1698:
1689:on Mathworld.
1688:
1683:
1680:
1675:
1674:
1666:
1663:
1658:
1657:"Fool's Gold"
1651:
1649:
1645:
1639:
1634:
1630:
1626:
1622:
1615:
1612:
1608:
1606:3-12-539683-2
1602:
1598:
1594:
1593:Jones, Daniel
1588:
1585:
1578:
1574:
1571:
1569:
1566:
1564:
1561:
1560:
1556:
1543:
1541:10 pentagons
1536:
1534:10 pentagons
1529:
1524:
1519:
1510:
1505:
1501:
1498:
1494:
1491:
1487:
1484:
1480:
1477:
1473:
1470:
1466:
1465:
1462:
1457:
1453:
1451:
1446:
1442:
1440:
1435:
1431:
1429:
1424:
1420:
1418:
1413:
1409:
1407:
1402:
1398:
1393:
1390:
1387:
1384:
1381:
1378:
1377:
1374:
1372:
1364:
1359:
1358:trapezohedron
1355:
1352:
1348:
1345:
1341:
1338:
1334:
1331:
1327:
1326:
1325:
1319:
1317:
1315:
1311:
1307:
1303:
1297:
1289:
1285:
1281:
1274:
1272:
1270:
1265:
1264:cuboctahedron
1259:
1251:
1247:
1242:
1236:Cuboctahedron
1235:
1233:
1231:
1226:
1224:
1221:
1217:
1211:
1202:
1193:
1191:
1187:
1183:
1179:
1174:
1172:
1168:
1164:
1156:
1151:
1144:
1142:
1140:
1109:
1049:
1044:
1042:
1038:
1034:
1030:
1028:
1024:
1021:
1017:
1013:
1009:
1005:
1001:
997:
983:
979:
978:cuboctahedron
974:
970:
966:
963:
959:
958:
954:
951:
946:
941:
938:
935:
931:
928:
925:
923:
919:
915:
909:
907:
903:
899:
893:
891:
887:
883:
881:
877:
873:
871:
867:
864:12 isosceles
862:8 equilateral
860:20 triangles:
859:
857:
853:
836:
830:
823:
817:
812:
802:
800:
796:
792:
758:
726:
724:
720:
715:
709:
702:
698:
694:
689:
685:
681:
678:
674:
671:
667:
664:
660:
657:
653:
650:
646:
643:
636:
632:
629:
625:
624:
620:
616:
613:
609:
606:
602:
599:
595:
592:
588:
585:
581:
578:
571:
567:
564:
560:
559:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
531:
528:
526:
523:
521:
518:
516:
513:
511:
508:
507:
501:
498:
496:
491:
486:
483:
482:
479:
478:
472:
469:
466:
463:
461:
460:
454:
451:
444:
442:
424:
420:
415:
413:
393:
389:
385:
378:
374:
369:
364:
356:
354:
352:
348:
343:
341:
337:
333:
325:
320:
315:
307:
305:
303:
299:
295:
285:
279:
271:
264:
247:
245:
243:
239:
235:
231:
227:
223:
218:
214:
196:
192:
189:
185:
182:
181:Ancient Greek
178:
172:
125:
62:
58:
50:
45:
39:
34:
27:
19:
2492:
2411:trapezohedra
2362:
2355:
2159:dodecahedron
1912:Apeirohedron
1863:>20 faces
1853:
1814:Dodecahedron
1682:
1672:
1665:
1628:
1624:
1614:
1596:
1587:
1544:13 triangles
1537:10 triangles
1530:10 triangles
1511:16 triangles
1368:
1323:
1299:
1269:double cover
1261:
1227:
1215:
1213:
1190:golden ratio
1185:
1181:
1177:
1175:
1160:
1045:
1031:
1015:
1011:
1007:
1003:
1000:pyritohedral
995:
993:
927:Pyritohedron
500:Regular star
474:
467:
464:
457:
455:
448:
416:
381:
375:monument in
371:A detail of
344:
331:
329:
301:
291:
236:—one of the
219:
194:
191:
187:
184:
60:
54:
26:
2181:semiregular
2164:icosahedron
2144:tetrahedron
1854:Icosahedron
1802:11–20 faces
1788:Enneahedron
1778:Heptahedron
1768:Pentahedron
1763:Tetrahedron
1687:Icosahedron
1548:3 pentagons
1527:12 squares
1525:8 triangles
1522:12 squares
1520:8 triangles
1137:, (*532),
1027:equilateral
805:sr{3,3} or
61:icosahedron
51:icosahedron
18:Icosahedral
2507:Categories
2476:prismatoid
2406:bipyramids
2390:antiprisms
2364:hosohedron
2154:octahedron
2039:prismatoid
2024:(infinite)
1793:Decahedron
1783:Octahedron
1773:Hexahedron
1748:Monohedron
1741:1–10 faces
1631:(2): 139.
1579:References
1550:1 hexagon
1517:1 hexagon
1306:zonohedron
1246:octahedron
1171:alternated
1020:alternated
933:Properties
450:Stellation
177:polyhedron
49:tensegrity
2471:birotunda
2461:bifrustum
2226:snub cube
2121:polyhedra
2053:antiprism
1758:Trihedron
1727:Polyhedra
1546:3 squares
1513:3 squares
1356:10-sided
1351:bipyramid
1349:10-sided
1344:antiprism
1335:18-sided
1328:19-sided
697:compounds
693:polyhedra
412:pentagram
377:Amsterdam
230:stellated
2451:bicupola
2431:pyramids
2357:dihedron
1753:Dihedron
1568:600-cell
1557:See also
1369:Several
1342:9-sided
880:Vertices
475:Notable
188:(eíkosi)
57:geometry
2493:italics
2481:scutoid
2466:rotunda
2456:frustum
2185:uniform
2134:regular
2119:Convex
2073:pyramid
2058:frustum
1573:Icosoku
1330:pyramid
1188:is the
504:Others
485:Regular
439:
427:
421:is the
408:
396:
394:is {3,
373:Spinoza
349:is the
268:Convex
222:similar
195:(hédra)
175:) is a
36:Convex
2446:cupola
2399:duals:
2385:prisms
2063:cupola
1939:vertex
1603:
1539:
1532:
1515:
1228:It is
1014:, and
937:convex
803:s{3,4}
390:. Its
228:, non-
226:convex
186:εἴκοσι
2068:wedge
2048:prism
1908:(132)
1337:prism
1304:is a
1220:right
1169:with
870:Edges
856:Faces
699:with
183:
59:, an
2149:cube
2030:(57)
2001:(92)
1995:(13)
1989:(13)
1958:(16)
1934:edge
1929:face
1902:(90)
1896:(60)
1890:(48)
1884:(32)
1878:(30)
1872:(24)
1601:ISBN
1394:J92
1391:J60
1388:J59
1385:J36
1382:J35
1379:J22
1300:The
1161:The
1078:and
695:and
417:Its
382:The
345:Its
193:ἕδρα
2183:or
2018:(4)
1983:(5)
1952:(9)
1914:(∞)
1633:doi
950:Net
884:12
128:or
115:-,-
55:In
2509::
2041:s
1647:^
1629:21
1627:.
1623:.
1316:.
1248:,
1225:.
1192:.
1050::
1043:.
1010:,
1006:,
994:A
984:.
703:.
304:.
244:.
232:)
201:/-
168:ən
159:iː
135:aɪ
124:-/
121:oʊ
106:,-
103:ən
94:iː
73:aɪ
47:A
2495:.
2187:)
2179:(
2136:)
2132:(
2112:e
2105:t
2098:v
1719:e
1712:t
1705:v
1659:.
1641:.
1635::
1186:ϕ
1182:ϕ
913:d
911:T
897:h
895:T
837:}
831:3
824:3
818:{
813:s
436:2
433:/
430:5
425:{
405:2
402:/
399:5
213:/
210:ə
207:r
204:d
171:/
165:r
162:d
156:h
153:ˈ
150:ə
147:s
144:ɒ
141:k
138:ˌ
132:/
118:k
112:ə
109:k
100:r
97:d
91:h
88:ˈ
85:ə
82:s
79:ɒ
76:k
70:ˌ
67:/
63:(
20:)
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