645:
652:
638:
631:
624:
666:
580:
1139:
587:
573:
659:
608:
308:
617:
594:
1401:
552:
1412:
673:
601:
958:
33:
1390:
566:
559:
1445:
1434:
1423:
951:
780:
357:
746:
1190:
934:
1493:
1486:
1479:
1472:
1465:
1458:
1269:
267:
22:
252:
1255:
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
1230:
441:
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.
838:
1028:
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
1154:
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
2099:
457:
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.
2092:
2085:
1706:
543:
1405:
1593:
1416:
878:
403:
rather than a pentagon, leading to geometrically intersecting faces. The intersections of the triangles do not represent new edges.
1102:
1091:
1081:
1071:
1063:
1053:
894:
773:
763:
753:
729:
719:
454:
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.
1394:
1246:
1122:
1112:
1043:
970:
739:
2004:
1117:
1107:
1086:
1076:
1058:
1048:
768:
758:
734:
724:
488:
281:
There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20
797:
1449:
1438:
1427:
411:
2501:
2298:
2239:
644:
523:
508:
1221:
to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube.
651:
637:
630:
623:
2506:
2328:
2288:
1894:
1581:
1218:
528:
518:
513:
503:
2323:
2318:
465:
447:
579:
1888:
1138:
665:
586:
572:
376:
289:. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a
2429:
2424:
2303:
2209:
2016:
1944:
1864:
1699:
1298:
1257:
1198:
478:
607:
2293:
2234:
2224:
2169:
1950:
1159:
1008:
538:
307:
169:
1241:, and cuboctahedron. The cuboctahedron can flex this way even if its edges (but not its faces) are rigid.
2313:
2229:
2184:
1551:
1169:
in its regular icosahedron form, generated by the same operations carried out starting with the vector (
1151:
1021:
883:
658:
593:
210:
1400:
1411:
616:
2273:
2199:
2147:
1301:
by removing 10 middle faces. Even though all the faces are congruent, the rhombic icosahedron is not
1155:
1143:
1127:
1025:
1015:
1011:
899:
689:
339:
286:
282:
230:
957:
551:
32:
2439:
2308:
2283:
2268:
2204:
2152:
1389:
1290:
1284:
1272:
1096:
685:
672:
600:
565:
558:
498:
483:
302:
258:
222:
26:
2454:
2419:
2278:
2173:
2122:
2061:
1938:
1932:
1692:
1029:
787:
380:
328:
1444:
1433:
1422:
950:
399:}. Like the convex form, it also has 20 equilateral triangle faces, but its vertex figure is a
2434:
2244:
2219:
2163:
2051:
1975:
1927:
1900:
1870:
1660:
1589:
1238:
1018:, the symmetry can also be distinguished by colouring the 8 and 12 triangle sets differently.
868:
533:
372:
351:
272:
214:
213:
shapes of icosahedra, some of them being more symmetrical than others. The best known is the (
2373:
2056:
2036:
1858:
1621:
1325:
938:
779:
312:
189:
120:
55:
1207:, the 12 isosceles faces are arranged differently so that the figure is non-convex and has
745:
2010:
1922:
1917:
1882:
1837:
1827:
1817:
1812:
1302:
1036:
910:
858:
844:
711:
407:
335:
969:
with its 6 square faces bisected on diagonals with pyritohedral symmetry. There exists a
2194:
2117:
2066:
1969:
1832:
1822:
1260:
octahedron. Cyclical kinematic transformations among the members of this family exist.
1211:
1189:
933:
473:
361:
324:
226:
1626:
1609:
2495:
2399:
2255:
2189:
1987:
1981:
1876:
1807:
1797:
1359:
1346:
1252:
966:
1492:
1485:
1478:
1471:
1464:
1457:
1802:
1268:
1178:
988:
915:
356:
331:{3, 5}, containing 20 triangular faces, with 5 faces meeting around each vertex.
2132:
1776:
1766:
1756:
1751:
1675:
1208:
266:
2464:
2352:
2142:
2109:
2027:
1781:
1771:
1761:
1746:
1736:
1715:
1294:
1234:
925:
681:
438:
218:
165:
37:
1608:
Koca, Nazife; Al-Mukhaini, Aida; Koca, Mehmet; Al Qanobi, Amal (2016-12-01).
2459:
2449:
2394:
2378:
2214:
2041:
1645:
1339:
1332:
451:, Coxeter et al. enumerated 58 such stellations of the regular icosahedron.
400:
365:
2345:
1741:
1556:
45:
251:
21:
2469:
2444:
2046:
1561:
1318:
680:
The stellation process on the icosahedron creates a number of related
430:, 3}, having three regular star pentagonal faces around each vertex.
1229:
319:
The convex regular icosahedron is usually referred to simply as the
2077:
1267:
1228:
1188:
1137:
702:
355:
306:
285:
faces with five meeting at each of its twelve vertices. Both have
31:
20:
342:{5, 3} having three regular pentagonal faces around each vertex.
2137:
2081:
1688:
971:
kinematic transformation between cuboctahedron and icosahedron
156:
147:
91:
82:
1313:
Common icosahedra with pyramid and prism symmetries include:
1684:
1297:
made up of 20 congruent rhombs. It can be derived from the
198:
138:
123:
109:
100:
73:
61:
1584:(2003) , Peter Roach; James Hartmann; Jane Setter (eds.),
1095:
respectively, each representing the lower symmetry to the
132:
67:
1662:
Connections: The
Geometric Bridge Between Art and Science
1251:
A regular icosahedron is topologically identical to a
965:
A regular icosahedron is topologically identical to a
809:
800:
186: 'seat'. The plural can be either "icosahedra" (
135:
129:
126:
106:
97:
70:
64:
58:
1364:
459:
195:
192:
153:
150:
88:
85:
2412:
2387:
2362:
2337:
2253:
2161:
2116:
2026:
1997:
1962:
1910:
1851:
1790:
1729:
833:{\displaystyle s{\begin{Bmatrix}3\\3\end{Bmatrix}}}
144:
141:
79:
76:
832:
315:inscribed in a convex regular icosahedron
1665:(2nd ed.). World Scientific. p. 475.
2093:
1700:
1614:Sultan Qaboos University Journal for Science
8:
945:
1610:"Symmetry of the Pyritohedron and Lattices"
705:
2366:
2100:
2086:
2078:
1707:
1693:
1685:
1203:In Jessen's icosahedron, sometimes called
1625:
1588:, Cambridge: Cambridge University Press,
1498:
1386:
987:can be distorted or marked up as a lower
804:
799:
706:Pyritohedral and tetrahedral symmetries
1573:
1024:has the symbol (3*2), , with order 24.
168:with 20 faces. The name comes from
1639:
1637:
7:
1142:Construction from the vertices of a
1724:Listed by number of faces and type
1406:Elongated triangular orthobicupola
1335:(2 sets of 9 sides + 2 ends = 20).
14:
1627:10.24200/squjs.vol21iss2pp139-149
1417:Elongated triangular gyrobicupola
1644:John Baez (September 11, 2011).
1491:
1484:
1477:
1470:
1463:
1456:
1443:
1432:
1421:
1410:
1399:
1388:
1120:
1115:
1110:
1105:
1100:
1089:
1084:
1079:
1074:
1069:
1061:
1056:
1051:
1046:
1041:
956:
949:
932:
778:
771:
766:
761:
756:
751:
744:
737:
732:
727:
722:
717:
671:
664:
657:
650:
643:
636:
629:
622:
615:
606:
599:
592:
585:
578:
571:
564:
557:
550:
375:is one of the four regular star
265:
250:
188:
119:
54:
1395:Gyroelongated triangular cupola
1247:Kinematics of the cuboctahedron
1205:Jessen's orthogonal icosahedron
921:
909:
893:
877:
867:
857:
843:
786:
710:
244:Two kinds of regular icosahedra
1586:English Pronouncing Dictionary
1165:This construction is called a
1146:, showing internal rectangles.
466:stellations of the icosahedron
209:There are infinitely many non-
1:
1935:(two infinite groups and 75)
2480:Degenerate polyhedra are in
1953:(two infinite groups and 50)
1450:Triangular hebesphenorotunda
1439:Metabiaugmented dodecahedron
1428:Parabiaugmented dodecahedron
1309:Pyramid and prism symmetries
412:great stellated dodecahedron
327:, and is represented by its
2299:pentagonal icositetrahedron
2240:truncated icosidodecahedron
1184:
1014:. If all the triangles are
524:Compound of five tetrahedra
509:Medial triambic icosahedron
179: 'twenty' and
2525:
2329:pentagonal hexecontahedron
2289:deltoidal icositetrahedron
1349:(2 sets of 10 sides = 20).
1342:(2 sets of 10 sides = 20).
1282:
1244:
1196:
991:symmetry, and is called a
679:
529:Compound of ten tetrahedra
519:Compound of five octahedra
514:Great triambic icosahedron
504:Small triambic icosahedron
462:
349:
323:, one of the five regular
300:
297:Convex regular icosahedron
2478:
2369:
2324:disdyakis triacontahedron
2319:deltoidal hexecontahedron
2005:Kepler–Poinsot polyhedron
1722:
1007:. This can be seen as an
964:
944:
931:
628:
563:
492:
482:
477:
448:The Fifty-Nine Icosahedra
1233:Progressions between an
377:Kepler-Poinsot polyhedra
18:Polyhedron with 20 faces
2430:gyroelongated bipyramid
2304:rhombic triacontahedron
2210:truncated cuboctahedron
2017:Uniform star polyhedron
1945:quasiregular polyhedron
1299:rhombic triacontahedron
1035:These symmetries offer
863:30 (6 short + 24 long)
2425:truncated trapezohedra
2294:disdyakis dodecahedron
2260:(duals of Archimedean)
2235:rhombicosidodecahedron
2225:truncated dodecahedron
1951:semiregular polyhedron
1659:Kappraff, Jay (1991).
1275:
1242:
1194:
1147:
834:
539:Excavated dodecahedron
368:
316:
41:
29:
2314:pentakis dodecahedron
2230:truncated icosahedron
2185:truncated tetrahedron
1998:non-convex polyhedron
1552:Truncated icosahedron
1271:
1232:
1192:
1152:Cartesian coordinates
1141:
1134:Cartesian coordinates
1022:Pyritohedral symmetry
997:snub tetratetrahedron
835:
699:Pyritohedral symmetry
359:
310:
231:equilateral triangles
206:) or "icosahedrons".
35:
24:
2274:rhombic dodecahedron
2200:truncated octahedron
1199:Jessen's icosahedron
1193:Jessen's icosahedron
1185:Jessen's icosahedron
1156:truncated octahedron
1144:truncated octahedron
1128:icosahedral symmetry
1026:Tetrahedral symmetry
1012:truncated octahedron
905:, , (332), order 12
889:, , (3*2), order 24
798:
690:icosahedral symmetry
499:(Convex) icosahedron
434:Stellated icosahedra
340:regular dodecahedron
287:icosahedral symmetry
283:equilateral triangle
229:—whose faces are 20
2309:triakis icosahedron
2284:tetrakis hexahedron
2269:triakis tetrahedron
2205:rhombicuboctahedron
1328:(plus 2 ends = 20).
1321:(plus 1 base = 20).
1291:rhombic icosahedron
1285:Rhombic icosahedron
1279:Rhombic icosahedron
1273:Rhombic icosahedron
1097:regular icosahedron
1030:isosceles triangles
985:regular icosahedron
321:regular icosahedron
311:Three interlocking
303:Regular icosahedron
259:regular icosahedron
223:regular icosahedron
27:regular icosahedron
2502:Geodesic polyhedra
2279:triakis octahedron
2164:Archimedean solids
1939:regular polyhedron
1933:uniform polyhedron
1895:Hectotriadiohedron
1276:
1243:
1219:scissors congruent
1195:
1162:vertices deleted.
1148:
1005:pseudo-icosahedron
830:
824:
479:Uniform duals
369:
317:
237:Regular icosahedra
42:
30:
2507:Individual graphs
2489:
2488:
2408:
2407:
2245:snub dodecahedron
2220:icosidodecahedron
2075:
2074:
1976:Archimedean solid
1963:convex polyhedron
1871:Icosidodecahedron
1543:
1542:
1239:pseudoicosahedron
981:
980:
977:
976:
696:
695:
534:Great icosahedron
484:Regular compounds
373:great icosahedron
352:Great icosahedron
346:Great icosahedron
313:golden rectangles
291:great icosahedron
273:Great icosahedron
2514:
2367:
2363:Dihedral uniform
2338:Dihedral regular
2261:
2177:
2126:
2102:
2095:
2088:
2079:
1911:elemental things
1889:Enneacontahedron
1859:Icositetrahedron
1709:
1702:
1695:
1686:
1679:
1673:
1667:
1666:
1656:
1650:
1649:
1641:
1632:
1631:
1629:
1605:
1599:
1598:
1578:
1495:
1488:
1481:
1474:
1467:
1460:
1447:
1436:
1425:
1414:
1403:
1392:
1365:
1362:are icosahedra:
1264:Other icosahedra
1167:snub tetrahedron
1125:
1124:
1123:
1119:
1118:
1114:
1113:
1109:
1108:
1104:
1103:
1094:
1093:
1092:
1088:
1087:
1083:
1082:
1078:
1077:
1073:
1072:
1066:
1065:
1064:
1060:
1059:
1055:
1054:
1050:
1049:
1045:
1044:
1037:Coxeter diagrams
1001:snub tetrahedron
960:
953:
946:
936:
839:
837:
836:
831:
829:
828:
782:
776:
775:
774:
770:
769:
765:
764:
760:
759:
755:
754:
748:
742:
741:
740:
736:
735:
731:
730:
726:
725:
721:
720:
712:Coxeter diagrams
703:
675:
668:
661:
654:
647:
640:
633:
626:
619:
610:
603:
596:
589:
582:
575:
568:
561:
554:
544:Final stellation
460:
429:
427:
426:
423:
420:
398:
396:
395:
392:
389:
269:
254:
205:
204:
201:
200:
197:
194:
163:
162:
159:
158:
155:
152:
149:
146:
143:
140:
137:
134:
131:
128:
125:
116:
115:
112:
111:
108:
103:
102:
99:
94:
93:
90:
87:
84:
81:
78:
75:
72:
69:
66:
63:
60:
2524:
2523:
2517:
2516:
2515:
2513:
2512:
2511:
2492:
2491:
2490:
2485:
2474:
2413:Dihedral others
2404:
2383:
2358:
2333:
2262:
2259:
2258:
2249:
2178:
2167:
2166:
2157:
2120:
2118:Platonic solids
2112:
2106:
2076:
2071:
2022:
2011:Star polyhedron
1993:
1958:
1906:
1883:Hexecontahedron
1865:Triacontahedron
1847:
1838:Enneadecahedron
1828:Heptadecahedron
1818:Pentadecahedron
1813:Tetradecahedron
1786:
1725:
1718:
1713:
1683:
1682:
1674:
1670:
1658:
1657:
1653:
1643:
1642:
1635:
1607:
1606:
1602:
1596:
1580:
1579:
1575:
1570:
1548:
1538:
1536:
1534:
1529:
1527:
1522:
1520:
1515:
1510:
1505:
1503:
1501:
1448:
1437:
1426:
1415:
1404:
1393:
1356:
1311:
1303:face-transitive
1287:
1281:
1266:
1249:
1227:
1212:dihedral angles
1201:
1187:
1173:, 1, 0), where
1136:
1121:
1116:
1111:
1106:
1101:
1099:
1090:
1085:
1080:
1075:
1070:
1068:
1062:
1057:
1052:
1047:
1042:
1040:
993:snub octahedron
937:
911:Dual polyhedron
903:
887:
852:
850:
823:
822:
816:
815:
805:
796:
795:
793:
788:Schläfli symbol
772:
767:
762:
757:
752:
750:
749:
743:(pyritohedral)
738:
733:
728:
723:
718:
716:
701:
436:
424:
421:
418:
417:
415:
408:dual polyhedron
393:
390:
387:
386:
384:
381:Schläfli symbol
354:
348:
336:dual polyhedron
329:Schläfli symbol
325:Platonic solids
305:
299:
279:
278:
277:
276:
275:
270:
262:
261:
255:
246:
245:
239:
227:Platonic solids
191:
187:
122:
118:
105:
96:
57:
53:
19:
12:
11:
5:
2522:
2521:
2518:
2510:
2509:
2504:
2494:
2493:
2487:
2486:
2479:
2476:
2475:
2473:
2472:
2467:
2462:
2457:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2416:
2414:
2410:
2409:
2406:
2405:
2403:
2402:
2397:
2391:
2389:
2385:
2384:
2382:
2381:
2376:
2370:
2364:
2360:
2359:
2357:
2356:
2349:
2341:
2339:
2335:
2334:
2332:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2265:
2263:
2256:Catalan solids
2254:
2251:
2250:
2248:
2247:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2207:
2202:
2197:
2195:truncated cube
2192:
2187:
2181:
2179:
2162:
2159:
2158:
2156:
2155:
2150:
2145:
2140:
2135:
2129:
2127:
2114:
2113:
2107:
2105:
2104:
2097:
2090:
2082:
2073:
2072:
2070:
2069:
2067:parallelepiped
2064:
2059:
2054:
2049:
2044:
2039:
2033:
2031:
2024:
2023:
2021:
2020:
2014:
2008:
2001:
1999:
1995:
1994:
1992:
1991:
1985:
1979:
1973:
1970:Platonic solid
1966:
1964:
1960:
1959:
1957:
1956:
1955:
1954:
1948:
1942:
1930:
1925:
1920:
1914:
1912:
1908:
1907:
1905:
1904:
1898:
1892:
1886:
1880:
1874:
1868:
1862:
1855:
1853:
1849:
1848:
1846:
1845:
1840:
1835:
1833:Octadecahedron
1830:
1825:
1823:Hexadecahedron
1820:
1815:
1810:
1805:
1800:
1794:
1792:
1788:
1787:
1785:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1744:
1739:
1733:
1731:
1727:
1726:
1723:
1720:
1719:
1714:
1712:
1711:
1704:
1697:
1689:
1681:
1680:
1668:
1651:
1633:
1600:
1594:
1572:
1571:
1569:
1566:
1565:
1564:
1559:
1554:
1547:
1544:
1541:
1540:
1531:
1524:
1517:
1512:
1507:
1497:
1496:
1489:
1482:
1475:
1468:
1461:
1453:
1452:
1441:
1430:
1419:
1408:
1397:
1385:
1384:
1381:
1378:
1375:
1372:
1369:
1360:Johnson solids
1355:
1354:Johnson solids
1352:
1351:
1350:
1343:
1336:
1329:
1322:
1310:
1307:
1283:Main article:
1280:
1277:
1265:
1262:
1245:Main article:
1226:
1223:
1197:Main article:
1186:
1183:
1135:
1132:
1130:of order 120.
979:
978:
975:
974:
962:
961:
954:
942:
941:
929:
928:
923:
919:
918:
913:
907:
906:
901:
897:
895:Rotation group
891:
890:
885:
881:
879:Symmetry group
875:
874:
871:
865:
864:
861:
855:
854:
847:
841:
840:
827:
821:
818:
817:
814:
811:
810:
808:
803:
790:
784:
783:
777:(tetrahedral)
714:
708:
707:
700:
697:
694:
693:
677:
676:
669:
662:
655:
648:
641:
634:
627:
620:
612:
611:
604:
597:
590:
583:
576:
569:
562:
555:
547:
546:
541:
536:
531:
526:
521:
516:
511:
506:
501:
495:
494:
491:
486:
481:
476:
470:
469:
445:In their book
435:
432:
350:Main article:
347:
344:
301:Main article:
298:
295:
271:
264:
263:
256:
249:
248:
247:
243:
242:
241:
240:
238:
235:
17:
13:
10:
9:
6:
4:
3:
2:
2520:
2519:
2508:
2505:
2503:
2500:
2499:
2497:
2483:
2477:
2471:
2468:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2417:
2415:
2411:
2401:
2398:
2396:
2393:
2392:
2390:
2386:
2380:
2377:
2375:
2372:
2371:
2368:
2365:
2361:
2355:
2354:
2350:
2348:
2347:
2343:
2342:
2340:
2336:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2266:
2264:
2257:
2252:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2191:
2190:cuboctahedron
2188:
2186:
2183:
2182:
2180:
2175:
2171:
2165:
2160:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2130:
2128:
2124:
2119:
2115:
2111:
2103:
2098:
2096:
2091:
2089:
2084:
2083:
2080:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2034:
2032:
2029:
2025:
2018:
2015:
2012:
2009:
2006:
2003:
2002:
2000:
1996:
1989:
1988:Johnson solid
1986:
1983:
1982:Catalan solid
1980:
1977:
1974:
1971:
1968:
1967:
1965:
1961:
1952:
1949:
1946:
1943:
1940:
1937:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1919:
1916:
1915:
1913:
1909:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1877:Hexoctahedron
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1856:
1854:
1850:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1808:Tridecahedron
1806:
1804:
1801:
1799:
1798:Hendecahedron
1796:
1795:
1793:
1789:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1734:
1732:
1728:
1721:
1717:
1710:
1705:
1703:
1698:
1696:
1691:
1690:
1687:
1678:on Mathworld.
1677:
1672:
1669:
1664:
1663:
1655:
1652:
1647:
1646:"Fool's Gold"
1640:
1638:
1634:
1628:
1623:
1619:
1615:
1611:
1604:
1601:
1597:
1595:3-12-539683-2
1591:
1587:
1583:
1582:Jones, Daniel
1577:
1574:
1567:
1563:
1560:
1558:
1555:
1553:
1550:
1549:
1545:
1532:
1530:10 pentagons
1525:
1523:10 pentagons
1518:
1513:
1508:
1499:
1494:
1490:
1487:
1483:
1480:
1476:
1473:
1469:
1466:
1462:
1459:
1455:
1454:
1451:
1446:
1442:
1440:
1435:
1431:
1429:
1424:
1420:
1418:
1413:
1409:
1407:
1402:
1398:
1396:
1391:
1387:
1382:
1379:
1376:
1373:
1370:
1367:
1366:
1363:
1361:
1353:
1348:
1347:trapezohedron
1344:
1341:
1337:
1334:
1330:
1327:
1323:
1320:
1316:
1315:
1314:
1308:
1306:
1304:
1300:
1296:
1292:
1286:
1278:
1274:
1270:
1263:
1261:
1259:
1254:
1253:cuboctahedron
1248:
1240:
1236:
1231:
1225:Cuboctahedron
1224:
1222:
1220:
1215:
1213:
1210:
1206:
1200:
1191:
1182:
1180:
1176:
1172:
1168:
1163:
1161:
1157:
1153:
1145:
1140:
1133:
1131:
1129:
1098:
1038:
1033:
1031:
1027:
1023:
1019:
1017:
1013:
1010:
1006:
1002:
998:
994:
990:
986:
972:
968:
967:cuboctahedron
963:
959:
955:
952:
948:
947:
943:
940:
935:
930:
927:
924:
920:
917:
914:
912:
908:
904:
898:
896:
892:
888:
882:
880:
876:
872:
870:
866:
862:
860:
856:
853:12 isosceles
851:8 equilateral
849:20 triangles:
848:
846:
842:
825:
819:
812:
806:
801:
791:
789:
785:
781:
747:
715:
713:
709:
704:
698:
691:
687:
683:
678:
674:
670:
667:
663:
660:
656:
653:
649:
646:
642:
639:
635:
632:
625:
621:
618:
614:
613:
609:
605:
602:
598:
595:
591:
588:
584:
581:
577:
574:
570:
567:
560:
556:
553:
549:
548:
545:
542:
540:
537:
535:
532:
530:
527:
525:
522:
520:
517:
515:
512:
510:
507:
505:
502:
500:
497:
496:
490:
487:
485:
480:
475:
472:
471:
468:
467:
461:
458:
455:
452:
450:
449:
443:
440:
433:
431:
413:
409:
404:
402:
382:
378:
374:
367:
363:
358:
353:
345:
343:
341:
337:
332:
330:
326:
322:
314:
309:
304:
296:
294:
292:
288:
284:
274:
268:
260:
253:
236:
234:
232:
228:
224:
220:
216:
212:
207:
203:
185:
181:
178:
174:
171:
170:Ancient Greek
167:
161:
114:
51:
47:
39:
34:
28:
23:
16:
2481:
2400:trapezohedra
2351:
2344:
2148:dodecahedron
1901:Apeirohedron
1852:>20 faces
1842:
1803:Dodecahedron
1671:
1661:
1654:
1617:
1613:
1603:
1585:
1576:
1533:13 triangles
1526:10 triangles
1519:10 triangles
1500:16 triangles
1357:
1312:
1288:
1258:double cover
1250:
1216:
1204:
1202:
1179:golden ratio
1174:
1170:
1166:
1164:
1149:
1034:
1020:
1004:
1000:
996:
992:
989:pyritohedral
984:
982:
916:Pyritohedron
489:Regular star
463:
456:
453:
446:
444:
437:
405:
370:
364:monument in
360:A detail of
333:
320:
318:
290:
280:
225:—one of the
208:
183:
180:
176:
173:
49:
43:
15:
2170:semiregular
2153:icosahedron
2133:tetrahedron
1843:Icosahedron
1791:11–20 faces
1777:Enneahedron
1767:Heptahedron
1757:Pentahedron
1752:Tetrahedron
1676:Icosahedron
1537:3 pentagons
1516:12 squares
1514:8 triangles
1511:12 squares
1509:8 triangles
1126:, (*532),
1016:equilateral
794:sr{3,3} or
50:icosahedron
40:icosahedron
2496:Categories
2465:prismatoid
2395:bipyramids
2379:antiprisms
2353:hosohedron
2143:octahedron
2028:prismatoid
2013:(infinite)
1782:Decahedron
1772:Octahedron
1762:Hexahedron
1737:Monohedron
1730:1–10 faces
1620:(2): 139.
1568:References
1539:1 hexagon
1506:1 hexagon
1295:zonohedron
1235:octahedron
1160:alternated
1009:alternated
922:Properties
439:Stellation
166:polyhedron
38:tensegrity
2460:birotunda
2450:bifrustum
2215:snub cube
2110:polyhedra
2042:antiprism
1747:Trihedron
1716:Polyhedra
1535:3 squares
1502:3 squares
1345:10-sided
1340:bipyramid
1338:10-sided
1333:antiprism
1324:18-sided
1317:19-sided
686:compounds
682:polyhedra
401:pentagram
366:Amsterdam
219:stellated
2440:bicupola
2420:pyramids
2346:dihedron
1742:Dihedron
1557:600-cell
1546:See also
1358:Several
1331:9-sided
869:Vertices
464:Notable
177:(eíkosi)
46:geometry
2482:italics
2470:scutoid
2455:rotunda
2445:frustum
2174:uniform
2123:regular
2108:Convex
2062:pyramid
2047:frustum
1562:Icosoku
1319:pyramid
1177:is the
493:Others
474:Regular
428:
416:
410:is the
397:
385:
383:is {3,
362:Spinoza
338:is the
257:Convex
211:similar
184:(hédra)
164:) is a
25:Convex
2435:cupola
2388:duals:
2374:prisms
2052:cupola
1928:vertex
1592:
1528:
1521:
1504:
1217:It is
1003:, and
926:convex
792:s{3,4}
379:. Its
217:, non-
215:convex
175:εἴκοσι
2057:wedge
2037:prism
1897:(132)
1326:prism
1293:is a
1209:right
1158:with
859:Edges
845:Faces
688:with
172:
48:, an
2138:cube
2019:(57)
1990:(92)
1984:(13)
1978:(13)
1947:(16)
1923:edge
1918:face
1891:(90)
1885:(60)
1879:(48)
1873:(32)
1867:(30)
1861:(24)
1590:ISBN
1383:J92
1380:J60
1377:J59
1374:J36
1371:J35
1368:J22
1289:The
1150:The
1067:and
684:and
406:Its
371:The
334:Its
182:ἕδρα
2172:or
2007:(4)
1972:(5)
1941:(9)
1903:(∞)
1622:doi
939:Net
873:12
117:or
104:-,-
44:In
2498::
2030:s
1636:^
1618:21
1616:.
1612:.
1305:.
1237:,
1214:.
1181:.
1039::
1032:.
999:,
995:,
983:A
973:.
692:.
293:.
233:.
221:)
190:/-
157:ən
148:iː
124:aɪ
113:-/
110:oʊ
95:,-
92:ən
83:iː
62:aɪ
36:A
2484:.
2176:)
2168:(
2125:)
2121:(
2101:e
2094:t
2087:v
1708:e
1701:t
1694:v
1648:.
1630:.
1624::
1175:ϕ
1171:ϕ
902:d
900:T
886:h
884:T
826:}
820:3
813:3
807:{
802:s
425:2
422:/
419:5
414:{
394:2
391:/
388:5
202:/
199:ə
196:r
193:d
160:/
154:r
151:d
145:h
142:ˈ
139:ə
136:s
133:ɒ
130:k
127:ˌ
121:/
107:k
101:ə
98:k
89:r
86:d
80:h
77:ˈ
74:ə
71:s
68:ɒ
65:k
59:ˌ
56:/
52:(
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