2132:
2139:
2125:
2118:
2111:
2153:
2067:
2074:
2060:
2146:
2095:
508:
2104:
2081:
2039:
2160:
2088:
2053:
2046:
1013:
266:
1421:
1442:
251:
553:
499:
1435:
1006:
530:
286:
1449:
29:
486:
1428:
1117:
641:
709:
772:
536:
Net (surface geometry); twelve isosceles pentagrammic pyramids, arranged like the faces of a dodecahedron. Each pyramid folds up like a fan: the dotted lines fold the opposite direction from the solid lines.
1161:
is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two
451:
573:
1642:
647:
1143:
A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the
715:
557:
This polyhedron represents a spherical tiling with a density of 7. (One spherical triangle face is shown above, outlined in blue, filled in yellow)
1635:
1742:
1184:
2194:
1523:
1147:
as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the
2030:
517:
223:
52:
1628:
1860:
1845:
1830:
1747:
1555:
1501:
184:
1875:
1850:
1835:
1378:
1349:
1339:
1310:
1291:
1271:
1252:
1092:
1072:
1063:
1043:
1023:
962:
942:
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904:
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872:
837:
817:
797:
338:
146:
1870:
1865:
1408:
1388:
1369:
1330:
1300:
1261:
1232:
1222:
1102:
982:
934:
368:
348:
176:
156:
1398:
1359:
1320:
1281:
1242:
1082:
1053:
1033:
972:
952:
914:
827:
807:
379:
The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the
358:
166:
2204:
1825:
1541:
1511:
1403:
1364:
1325:
1286:
1247:
1087:
1077:
1058:
1048:
1038:
1028:
977:
957:
947:
919:
909:
832:
822:
812:
802:
363:
216:
171:
1975:
1659:
1393:
1383:
1354:
1344:
1315:
1305:
1276:
1266:
1237:
1227:
1097:
967:
929:
887:
877:
353:
343:
161:
151:
39:
1855:
303:
1792:
1782:
1722:
1712:
1682:
1672:
1175:
1148:
1137:
270:
2131:
2010:
1995:
1797:
1787:
1737:
1196:
2138:
2124:
2117:
2110:
2199:
2015:
2005:
2000:
1990:
1920:
1910:
1807:
1802:
1493:
491:
1952:
1925:
1915:
414:
310:
204:
2066:
2152:
2073:
2059:
1935:
1930:
1717:
1546:
1116:
1965:
1732:
1727:
1189:
1144:
2094:
2025:
1840:
1212:
331:
2145:
2080:
1155:
897:
507:
372:. It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a
2103:
2176:
858:
790:
239:
189:
59:
2038:
2209:
2172:
2159:
2087:
2052:
2045:
1985:
1970:
1767:
636:{\displaystyle {\text{Circumradius}}={{\tfrac {E}{4}}{\Bigl (}{\sqrt {50+22{\sqrt {5}}}}{\Bigr )}}}
1012:
1777:
1772:
1707:
1677:
1163:
1125:
472:
467:
236:
209:
109:
314:
89:
1472:
1420:
265:
1895:
1596:
1573:
1551:
1519:
1497:
1441:
1905:
1900:
1599:
1485:
1133:
850:
544:
498:
477:
1533:
250:
1651:
1529:
786:
552:
275:
138:
1434:
1960:
1759:
863:
521:
400:
119:
1581:
1577:
1005:
529:
2188:
258:
396:
1448:
1129:
854:
285:
242:
28:
1615:
485:
2168:
513:
46:
1604:
1586:
373:
307:
1620:
295:
1427:
704:{\displaystyle {\text{Surface Area}}=3{\sqrt {3}}(5+4{\sqrt {5}})E^{2}}
388:
2167:
The stellation process on the icosahedron creates a number of related
403:
can be seen as its four-dimensional analogue using the same process.
1115:
767:{\displaystyle {\text{Volume}}={\tfrac {25+9{\sqrt {5}}}{4}}E^{3}}
283:
395:-polytope (equilateral triangles for the great icosahedron, and
1624:
399:
for the pentagram) until the figure regains regular faces. The
896:. It can also be constructed with 2 colors of triangles and
1166:
inscribed within and sharing the edges of the icosahedron.
516:
of the icosahedron, counted by
Wenninger as model and the
503:
It has a density of 7, as shown in this cross-section.
490:
A transparent model of the great icosahedron (See also
1120:
Animated truncation sequence from {5/2, 3} to {3, 5/2}
728:
587:
718:
650:
576:
417:
1946:
1514:; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999).
1168:
542:
460:
18:
1884:
1816:
1756:
1696:
1658:
766:
703:
635:
445:
627:
600:
1636:
8:
568:For a great icosahedron with edge length E,
446:{\displaystyle {\frac {7+3{\sqrt {5}}}{2}}}
1643:
1629:
1621:
411:The edge length of a great icosahedron is
758:
740:
727:
719:
717:
695:
681:
662:
651:
649:
626:
625:
616:
605:
599:
598:
586:
585:
577:
575:
518:16th of 17 stellations of the icosahedron
430:
418:
416:
1600:"Fifteen stellations of the icosahedron"
992:
789:, with different colored faces and only
453:times that of the original icosahedron.
1463:
16:Kepler-Poinsot polyhedron with 20 faces
1743:nonconvex great rhombicosidodecahedron
1550:, (3rd edition, 1973), Dover edition,
1538:(1st Edn University of Toronto (1938))
1185:Truncated great stellated dodecahedron
1473:"uniform polyhedra Great icosahedron"
7:
841:. This construction can be called a
14:
1861:great stellapentakis dodecahedron
1846:medial pentagonal hexecontahedron
1831:small stellapentakis dodecahedron
1748:great truncated icosidodecahedron
2158:
2151:
2144:
2137:
2130:
2123:
2116:
2109:
2102:
2093:
2086:
2079:
2072:
2065:
2058:
2051:
2044:
2037:
1876:great pentagonal hexecontahedron
1851:medial disdyakis triacontahedron
1836:medial deltoidal hexecontahedron
1447:
1440:
1433:
1426:
1419:
1406:
1401:
1396:
1391:
1386:
1381:
1376:
1367:
1362:
1357:
1352:
1347:
1342:
1337:
1328:
1323:
1318:
1313:
1308:
1303:
1298:
1289:
1284:
1279:
1274:
1269:
1264:
1259:
1250:
1245:
1240:
1235:
1230:
1225:
1220:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1061:
1056:
1051:
1046:
1041:
1036:
1031:
1026:
1021:
1011:
1004:
980:
975:
970:
965:
960:
955:
950:
945:
940:
932:
927:
922:
917:
912:
907:
902:
890:
885:
880:
875:
870:
835:
830:
825:
820:
815:
810:
805:
800:
795:
785:can be constructed as a uniform
551:
528:
506:
497:
484:
366:
361:
356:
351:
346:
341:
336:
264:
249:
174:
169:
164:
159:
154:
149:
144:
27:
1871:great disdyakis triacontahedron
1866:great deltoidal hexecontahedron
1512:Coxeter, Harold Scott MacDonald
857:, as a partial faceting of the
290:3D model of a great icosahedron
1953:stellations of the icosahedron
1826:medial rhombic triacontahedron
1560:Stellating the Platonic solids
688:
669:
1:
1856:great rhombic triacontahedron
520:and 7th of 59 stellations by
1793:great dodecahemidodecahedron
1783:small dodecahemidodecahedron
1723:truncated dodecadodecahedron
1713:truncated great dodecahedron
1683:great stellated dodecahedron
1673:small stellated dodecahedron
1210:
1159:great stellated dodecahedron
1149:great stellated dodecahedron
1138:small stellated dodecahedron
271:Great stellated dodecahedron
2011:Compound of five tetrahedra
1996:Medial triambic icosahedron
1798:great icosihemidodecahedron
1788:small icosihemidodecahedron
1738:truncated great icosahedron
1616:Uniform polyhedra and duals
2226:
2166:
2016:Compound of ten tetrahedra
2006:Compound of five octahedra
2001:Great triambic icosahedron
1991:Small triambic icosahedron
1949:
1921:great dodecahemidodecacron
1911:small dodecahemidodecacron
1808:small dodecahemicosahedron
1803:great dodecahemicosahedron
1494:Cambridge University Press
1132:. It also shares the same
847:retrosnub tetratetrahedron
482:
2115:
2050:
1979:
1969:
1964:
1926:great icosihemidodecacron
1916:small icosihemidodecacron
1518:(3rd ed.). Tarquin.
1516:The fifty-nine icosahedra
40:Kepler–Poinsot polyhedron
35:
26:
21:
2195:Kepler–Poinsot polyhedra
1936:small dodecahemicosacron
1931:great dodecahemicosacron
1718:rhombidodecadodecahedron
1652:Star-polyhedra navigator
304:Kepler–Poinsot polyhedra
1733:great icosidodecahedron
1728:snub dodecadodecahedron
1145:great icosidodecahedron
2026:Excavated dodecahedron
1887:uniform polyhedra with
1841:small rhombidodecacron
1128:as the regular convex
1121:
768:
705:
637:
447:
332:Coxeter-Dynkin diagram
291:
2205:Polyhedral stellation
1119:
898:pyritohedral symmetry
843:retrosnub tetrahedron
769:
706:
638:
448:
289:
2177:icosahedral symmetry
1986:(Convex) icosahedron
1889:infinite stellations
1697:Uniform truncations
988:retrosnub octahedron
859:truncated octahedron
791:tetrahedral symmetry
716:
648:
574:
415:
1817:Duals of nonconvex
1768:tetrahemihexahedron
1471:Klitzing, Richard.
1124:It shares the same
547:
1966:Uniform duals
1885:Duals of nonconvex
1778:octahemioctahedron
1773:cubohemioctahedron
1757:Nonconvex uniform
1708:dodecadodecahedron
1699:of Kepler-Poinsot
1678:great dodecahedron
1666:regular polyhedra)
1597:Weisstein, Eric W.
1582:Uniform polyhedron
1574:Weisstein, Eric W.
1126:vertex arrangement
1122:
986:, and is called a
764:
752:
701:
633:
596:
543:
473:Stellation diagram
464:Transparent model
443:
391:faces of the core
292:
110:Face configuration
22:Great icosahedron
2200:Regular polyhedra
2183:
2182:
2021:Great icosahedron
1971:Regular compounds
1944:
1943:
1896:tetrahemihexacron
1819:uniform polyhedra
1688:great icosahedron
1578:Great icosahedron
1562:, pp. 96–104
1547:Regular Polytopes
1525:978-1-899618-32-3
1490:Polyhedron Models
1486:Wenninger, Magnus
1455:
1454:
1192:icosidodecahedron
1164:great dodecahedra
1112:Related polyhedra
1109:
1108:
849:, similar to the
783:great icosahedron
751:
745:
722:
686:
667:
654:
623:
621:
595:
580:
561:
560:
541:
540:
441:
435:
311:regular polyhedra
300:great icosahedron
282:
281:
2217:
2162:
2155:
2148:
2141:
2134:
2127:
2120:
2113:
2106:
2097:
2090:
2083:
2076:
2069:
2062:
2055:
2048:
2041:
2031:Final stellation
1947:
1906:octahemioctacron
1901:hexahemioctacron
1645:
1638:
1631:
1622:
1610:
1609:
1591:
1537:
1507:
1477:
1476:
1468:
1451:
1444:
1437:
1430:
1423:
1411:
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1409:
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1404:
1400:
1399:
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1380:
1379:
1372:
1371:
1370:
1366:
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1360:
1356:
1355:
1351:
1350:
1346:
1345:
1341:
1340:
1333:
1332:
1331:
1327:
1326:
1322:
1321:
1317:
1316:
1312:
1311:
1307:
1306:
1302:
1301:
1294:
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1288:
1287:
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1278:
1277:
1273:
1272:
1268:
1267:
1263:
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1255:
1254:
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1249:
1248:
1244:
1243:
1239:
1238:
1234:
1233:
1229:
1228:
1224:
1223:
1169:
1134:edge arrangement
1105:
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1103:
1099:
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1083:
1079:
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1015:
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985:
984:
983:
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978:
974:
973:
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968:
964:
963:
959:
958:
954:
953:
949:
948:
944:
943:
937:
936:
935:
931:
930:
926:
925:
921:
920:
916:
915:
911:
910:
906:
905:
895:
894:
893:
889:
888:
884:
883:
879:
878:
874:
873:
853:symmetry of the
851:snub tetrahedron
840:
839:
838:
834:
833:
829:
828:
824:
823:
819:
818:
814:
813:
809:
808:
804:
803:
799:
798:
773:
771:
770:
765:
763:
762:
753:
747:
746:
741:
729:
723:
720:
710:
708:
707:
702:
700:
699:
687:
682:
668:
663:
655:
652:
642:
640:
639:
634:
632:
631:
630:
624:
622:
617:
606:
604:
603:
597:
588:
581:
578:
555:
548:
545:Spherical tiling
534:
532:
510:
501:
488:
461:
452:
450:
449:
444:
442:
437:
436:
431:
419:
394:
386:
371:
370:
369:
365:
364:
360:
359:
355:
354:
350:
349:
345:
344:
340:
339:
329:
327:
326:
322:
288:
268:
253:
179:
178:
177:
173:
172:
168:
167:
163:
162:
158:
157:
153:
152:
148:
147:
132:
131:
127:
103:
102:
98:
31:
19:
2225:
2224:
2220:
2219:
2218:
2216:
2215:
2214:
2185:
2184:
1945:
1940:
1888:
1886:
1880:
1818:
1812:
1758:
1752:
1700:
1698:
1692:
1665:
1661:
1660:Kepler-Poinsot
1654:
1649:
1595:
1594:
1572:
1569:
1526:
1510:
1504:
1484:
1481:
1480:
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1460:
1407:
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1397:
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1387:
1382:
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1358:
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1338:
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1246:
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1226:
1221:
1219:
1214:
1206:
1200:
1198:
1191:
1179:
1177:
1114:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1069:
1062:
1057:
1052:
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1037:
1032:
1027:
1022:
1020:
981:
976:
971:
966:
961:
956:
951:
946:
941:
939:
933:
928:
923:
918:
913:
908:
903:
901:
891:
886:
881:
876:
871:
869:
836:
831:
826:
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811:
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754:
730:
714:
713:
691:
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645:
572:
571:
566:
556:
535:
527:
511:
502:
489:
459:
420:
413:
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409:
392:
380:
367:
362:
357:
352:
347:
342:
337:
335:
324:
320:
319:
317:
315:Schläfli symbol
302:is one of four
284:
276:dual polyhedron
273:
269:
256:
254:
228:
221:
214:
198:
193:
175:
170:
165:
160:
155:
150:
145:
143:
139:Coxeter diagram
129:
125:
124:
100:
96:
95:
90:Schläfli symbol
72:
17:
12:
11:
5:
2223:
2221:
2213:
2212:
2207:
2202:
2197:
2187:
2186:
2181:
2180:
2164:
2163:
2156:
2149:
2142:
2135:
2128:
2121:
2114:
2107:
2099:
2098:
2091:
2084:
2077:
2070:
2063:
2056:
2049:
2042:
2034:
2033:
2028:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1982:
1981:
1978:
1973:
1968:
1963:
1957:
1956:
1942:
1941:
1939:
1938:
1933:
1928:
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1913:
1908:
1903:
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1647:
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1633:
1625:
1619:
1618:
1613:
1612:
1611:
1568:
1567:External links
1565:
1564:
1563:
1542:H.S.M. Coxeter
1539:
1524:
1508:
1502:
1479:
1478:
1462:
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1445:
1438:
1431:
1424:
1417:
1413:
1412:
1373:
1334:
1295:
1256:
1217:
1213:Coxeter-Dynkin
1209:
1208:
1203:
1194:
1187:
1182:
1173:
1113:
1110:
1107:
1106:
1067:
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677:
674:
671:
666:
661:
658:
629:
620:
615:
612:
609:
602:
594:
591:
584:
565:
562:
559:
558:
539:
538:
525:
504:
495:
481:
480:
475:
470:
465:
458:
455:
440:
434:
429:
426:
423:
408:
405:
401:grand 600-cell
280:
279:
262:
246:
245:
234:
230:
229:
226:
219:
212:
207:
201:
200:
196:
191:
187:
185:Symmetry group
181:
180:
141:
135:
134:
122:
120:Wythoff symbol
116:
115:
112:
106:
105:
92:
86:
85:
82:
81:Faces by sides
78:
77:
62:
56:
55:
50:
43:
42:
37:
33:
32:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
2222:
2211:
2208:
2206:
2203:
2201:
2198:
2196:
2193:
2192:
2190:
2178:
2174:
2170:
2165:
2161:
2157:
2154:
2150:
2147:
2143:
2140:
2136:
2133:
2129:
2126:
2122:
2119:
2112:
2108:
2105:
2101:
2100:
2096:
2092:
2089:
2085:
2082:
2078:
2075:
2071:
2068:
2064:
2061:
2057:
2054:
2047:
2043:
2040:
2036:
2035:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1983:
1977:
1974:
1972:
1967:
1962:
1959:
1958:
1955:
1954:
1948:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1917:
1914:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1893:
1891:
1883:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1823:
1821:
1815:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1779:
1776:
1774:
1771:
1769:
1766:
1765:
1763:
1761:
1760:hemipolyhedra
1755:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1705:
1703:
1695:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1670:
1668:
1663:
1657:
1653:
1646:
1641:
1639:
1634:
1632:
1627:
1626:
1623:
1617:
1614:
1607:
1606:
1601:
1598:
1593:
1592:
1589:
1588:
1583:
1579:
1575:
1571:
1570:
1566:
1561:
1557:
1556:0-486-61480-8
1553:
1549:
1548:
1543:
1540:
1535:
1531:
1527:
1521:
1517:
1513:
1509:
1505:
1503:0-521-09859-9
1499:
1495:
1491:
1487:
1483:
1482:
1474:
1467:
1464:
1457:
1450:
1446:
1443:
1439:
1436:
1432:
1429:
1425:
1422:
1418:
1415:
1414:
1374:
1335:
1296:
1257:
1218:
1216:
1211:
1204:
1202:
1195:
1193:
1188:
1186:
1183:
1181:
1174:
1171:
1170:
1167:
1165:
1160:
1157:
1152:
1150:
1146:
1141:
1139:
1135:
1131:
1127:
1118:
1111:
1068:
1019:
1018:
1014:
1010:
1007:
1003:
1002:
999:Pyritohedral
998:
995:
994:
991:
989:
899:
867:
865:
864:omnitruncated
860:
856:
852:
848:
844:
792:
788:
784:
776:
774:
759:
755:
748:
742:
737:
734:
731:
724:
711:
696:
692:
683:
678:
675:
672:
664:
659:
656:
643:
618:
613:
610:
607:
592:
589:
582:
569:
563:
554:
550:
549:
546:
531:
526:
523:
519:
515:
509:
505:
500:
496:
493:
487:
483:
479:
476:
474:
471:
469:
466:
463:
462:
456:
454:
438:
432:
427:
424:
421:
406:
404:
402:
398:
397:line segments
390:
387:-dimensional
384:
377:
375:
333:
316:
312:
309:
305:
301:
297:
287:
277:
272:
267:
263:
260:
259:Vertex figure
252:
248:
247:
244:
241:
238:
235:
232:
231:
225:
218:
211:
208:
206:
203:
202:
194:
188:
186:
183:
182:
142:
140:
137:
136:
123:
121:
118:
117:
113:
111:
108:
107:
93:
91:
88:
87:
83:
80:
79:
76:= 12 (χ = 2)
75:
70:
66:
63:
61:
58:
57:
54:
51:
48:
45:
44:
41:
38:
34:
30:
25:
20:
2020:
1976:Regular star
1950:
1687:
1603:
1585:
1559:
1545:
1515:
1489:
1466:
1207:icosahedron
1180:dodecahedron
1158:
1153:
1142:
1123:
996:Tetrahedral
987:
862:
846:
842:
782:
780:
712:
653:Surface Area
644:
579:Circumradius
570:
567:
410:
407:Construction
382:
378:
374:pentagrammic
299:
293:
73:
68:
64:
1664:(nonconvex
1201:icosahedron
1130:icosahedron
866:tetrahedron
855:icosahedron
243:deltahedron
199:, , (*532)
133:| 2 3
53:icosahedron
2210:Deltahedra
2189:Categories
1558:, 3.6 6.2
1458:References
514:stellation
376:sequence.
233:Properties
205:References
47:Stellation
2173:compounds
2169:polyhedra
1701:polyhedra
1662:polyhedra
1605:MathWorld
1587:MathWorld
1197:Truncated
1178:stellated
1156:truncated
777:As a snub
492:Animation
308:nonconvex
240:nonconvex
1951:Notable
1488:(1974).
1416:Picture
564:Formulas
512:It is a
313:), with
296:geometry
60:Elements
1980:Others
1961:Regular
1534:0676126
1215:diagram
1136:as the
522:Coxeter
468:Density
389:simplex
323:⁄
237:Regular
128:⁄
114:V(5)/2
99:⁄
1584:") at
1554:
1532:
1522:
1500:
721:Volume
457:Images
298:, the
84:20{3}
67:= 20,
2175:with
1205:Great
1199:great
1190:Great
1176:Great
1172:Name
255:(3)/2
2171:and
1580:" ("
1552:ISBN
1520:ISBN
1498:ISBN
1154:The
900:as,
861:(or
787:snub
781:The
533:Ă— 12
330:and
71:= 30
49:core
36:Type
1576:, "
938:or
868:):
845:or
478:Net
385:–1)
334:of
318:{3,
294:In
195:, H
94:{3,
2191::
2179:.
1602:.
1544:,
1530:MR
1528:.
1496:.
1492:.
1151:.
1140:.
990:.
793::
732:25
614:22
608:50
524:.
494:)
328:}
278:)
261:)
227:41
222:,
220:69
215:,
213:53
104:}
1644:e
1637:t
1630:v
1608:.
1590:.
1536:.
1506:.
1475:.
760:3
756:E
749:4
743:5
738:9
735:+
725:=
697:2
693:E
689:)
684:5
679:4
676:+
673:5
670:(
665:3
660:3
657:=
628:)
619:5
611:+
601:(
593:4
590:E
583:=
439:2
433:5
428:3
425:+
422:7
393:n
383:n
381:(
325:2
321:5
306:(
274:(
257:(
224:W
217:C
210:U
197:3
192:h
190:I
130:2
126:5
101:2
97:5
74:V
69:E
65:F
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