902:
1928:
1985:
2855:
2844:
305:
2889:
2833:
2822:
1554:
2884:
2871:
893:
296:
2087:
2063:
2866:
1456:
2098:
2052:
2787:
2776:
2743:
1089:
1075:
2157:
2146:
2133:
2122:
172:
2765:
2754:
1117:
1110:
1096:
1082:
1068:
1103:
1606:
1907:
1449:
1442:
1421:
2215:
2076:
188:
1634:
1627:
1620:
1613:
1802:
2811:
1428:
198:
287:
2111:
2798:
1641:
1809:
1414:
2043:
29:
1914:
1648:
1921:
1844:
1837:
1830:
1823:
1816:
1435:
1407:
2681:
686:
2186:
from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wireframe figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron.
512:
435:
803:, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the
771:
330:
2181:
as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed
681:{\displaystyle {\begin{aligned}A&=\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&\approx 29.306a^{2}\\V&={\frac {45+17{\sqrt {5}}}{6}}a^{3}&\approx 13.836a^{3}.\end{aligned}}}
517:
262:
by attaching them to their bases. These rotundas cover their decagonal base so that the resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to one of the
870:
120:
697:
2928:
993:
965:
455:
509:
can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:
1724:
937:
507:
487:
2719:
1003:. (The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.) These
2921:
242:, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such, it is one of the
1988:
A topological icosidodecahedron in truncated cube, inserting 6 vertices in center of octagons, and dissecting them into 2 pentagons and 2 triangles.
785:
2507:
2477:
2438:
2378:
2363:
Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In
Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.).
2914:
2161:
2032:
2634:
2535:
2290:
1746:
1717:
1052:
84:
2461:
430:{\displaystyle (0,0,\pm \varphi ),\qquad \left(\pm {\frac {1}{2}},\pm {\frac {\varphi }{2}},\pm {\frac {\varphi ^{2}}{2}}\right),}
2712:
1350:
1340:
1330:
1321:
1311:
1301:
1292:
1272:
1263:
1234:
1224:
1195:
1166:
1156:
1127:
3335:
1282:
1253:
1243:
1214:
1205:
1185:
1176:
1147:
1137:
2522:
1345:
1335:
1316:
1306:
1287:
1277:
1258:
1248:
1229:
1219:
1200:
1190:
1171:
1161:
1142:
1132:
2370:
1008:
2227:
2091:
2067:
2020:
1710:
1536:
901:
3330:
3127:
3068:
2859:
2705:
2497:
2305:
2102:
2056:
2024:
2016:
1388:
810:
274:, resulting in the pentagonal face connecting to the triangular one. The icosidodecahedron has an alternative name,
3157:
3117:
2428:
2424:
2269:
2150:
2137:
2126:
2028:
1498:
270:. The difference is that the icosidodecahedron is constructed by twisting its rotundas by 36°, a process known as
3152:
3147:
1493:
1488:
267:
94:
2526:
2195:
1020:
1996:
can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has
3258:
3253:
3132:
3038:
2837:
2567:
2203:
2080:
2004:
1948:
1878:
1473:
877:
247:
146:
47:
1927:
1883:
3122:
3063:
3053:
2998:
2848:
2780:
2300:
1868:
1863:
1383:
1363:
3142:
3058:
3013:
2791:
2747:
2493:
2231:
1997:
1956:
1873:
1478:
1373:
320:
2854:
2843:
1984:
489:
can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron
3102:
3028:
2976:
2769:
1853:
1680:
1655:
1563:
1558:
1523:
1514:), progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With
1507:
1056:
777:
89:
2547:
1506:
The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with
995:
if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional
3268:
3137:
3112:
3097:
3033:
2981:
2888:
2832:
2826:
2821:
2223:
1685:
1675:
1553:
1468:
304:
2883:
766:{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62^{\circ },}
3283:
3248:
3107:
3002:
2951:
2115:
2008:
1943:
1527:
259:
43:
2870:
970:
2865:
2086:
2062:
1455:
3263:
3073:
2992:
2893:
2786:
2775:
2742:
2728:
2651:
2630:
2604:
2531:
2503:
2473:
2434:
2395:
2374:
2364:
2235:
2175:
2097:
2051:
1515:
1393:
950:
800:
796:, with the vertices of the icosidodecahedron located at the midpoints of the edges of either.
440:
295:
243:
239:
74:
39:
2455:
936:
used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his
3202:
2764:
2753:
2594:
2465:
2337:
2156:
2145:
2132:
2121:
2012:
1116:
1109:
324:
179:
2349:
1102:
1088:
1074:
892:
313:
The difference between icosidodecahedron and pentagonal orthobirotunda, and its dissection.
2345:
2257:
Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31
2251:
1605:
1448:
1441:
1095:
1081:
1067:
873:
171:
141:
64:
54:
2674:
2214:
2075:
2023:(having the pentagonal faces in common). The vertex arrangement is also shared with the
3023:
2946:
2758:
2655:
1993:
1906:
1420:
691:
492:
472:
263:
130:
126:
2691:
2398:
2366:
Integrability, Quantization, and
Geometry: II. Quantum Theories and Algebraic Geometry
1801:
1619:
1612:
3324:
3228:
3084:
3018:
2802:
2659:
2599:
2582:
2341:
2285:
2239:
1665:
881:
804:
163:
2810:
1633:
1626:
1427:
187:
1483:
1358:
1034:
1023:
944:
929:
915:
789:
458:
2797:
2110:
1413:
2961:
2295:
2273:
2250:
The icosidodecahedron may appears in structural, as in the geodesic dome of the
2183:
2042:
1463:
1378:
1038:
1027:
793:
286:
197:
28:
1808:
1640:
3293:
3181:
2971:
2938:
2686:
2469:
1843:
1836:
1829:
1822:
1815:
1660:
1647:
781:
227:
219:
156:
3288:
3278:
3223:
3207:
3043:
2875:
2664:
2403:
2265:
1913:
1530:
of symmetry, with generator points at the right angle corner of the domain.
2608:
1920:
1434:
323:
for the vertices of an icosidodecahedron with unit edges are given by the
3174:
2199:
2191:
2178:
996:
271:
235:
207:
2430:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1406:
3298:
3273:
2682:
Editable printable net of an icosidodecahedron with interactive 3D view
2242:, meaning that each of its vertex is connected by four other vertices.
1000:
925:
911:
943:
The long radius (center to vertex) of the icosidodecahedron is in the
2697:
1030:, existing as the full-edge truncation between these regular solids.
2906:
2174:
In four-dimensional geometry, the icosidodecahedron appears in the
1011:
which meet at the center, each contributing two radii and an edge.
999:, the three-dimensional icosidodecahedron, and the two-dimensional
2538:(Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
2258:
195:
2966:
258:
One way to construct the icosidodecahedron is to start with two
2910:
2701:
2583:"New Insights into the Structural Mechanisms of the COPII Coat"
2198:
to 3-space about any vertex and all points are normalised, the
773:
determined by calculating the angle of a pentagonal rotunda.
2276:
icosidodecahedra joined at the midpoints of their segments.
2264:
The icosidodecahedron may also found in popular culture. In
2460:. Carbon Materials: Chemistry and Physics. Vol. 10.
2328:
Berman, Martin (1971). "Regular-faced convex polyhedra".
694:
of an icosidodecahedron between pentagon-to-triangle is
2581:
Russell, Christopher; Stagg, Scott (11 February 2010).
2202:
upon which edges fall comprise the icosidodecahedron's
2499:
Synergetics: Explorations in the
Geometry of Thinking
1007:
polytopes can be constructed, with their radii, from
973:
953:
813:
700:
515:
495:
475:
443:
333:
97:
2037:
3241:
3216:
3191:
3166:
3082:
2990:
2945:
1033:The icosidodecahedron contains 12 pentagons of the
178:
162:
152:
140:
125:
83:
73:
63:
53:
35:
21:
2226:of an icosidodecahedron can be represented as the
987:
959:
864:
765:
680:
501:
481:
449:
429:
114:
2272:has the goal of creating a shape with two nested
2019:(having the triangular faces in common), and the
967:if its edge length is 1, and its edge length is
865:{\displaystyle (3\cdot 5)^{2}=3^{2}\cdot 5^{2}}
1703:2 symmetry mutations of quasiregular tilings:
1541:32 orbifold symmetries of quasiregular tilings
2922:
2713:
1718:
938:31 great circles of the spherical icosahedron
238:faces. An icosidodecahedron has 30 identical
8:
2625:. United Kingdom: Cambridge. pp. 79–86
115:{\displaystyle \mathrm {I} _{\mathrm {h} }}
3195:
2929:
2915:
2907:
2720:
2706:
2698:
2419:
2417:
2415:
1725:
1711:
1695:
1532:
1043:
186:
170:
27:
2598:
977:
972:
952:
928:. Projected into a sphere, they define 6
856:
843:
830:
812:
754:
728:
715:
699:
665:
647:
630:
618:
598:
580:
562:
551:
538:
516:
514:
494:
474:
469:The surface area of an icosidodecahedron
442:
408:
402:
386:
370:
332:
105:
104:
99:
96:
2675:"3D convex uniform polyhedra o3x5o - id"
2530:, Third edition, (1973), Dover edition,
2323:
2321:
2213:
1983:
1047:Family of uniform icosahedral polyhedra
2433:. Dover Publications, Inc. p. 86.
2317:
1708:
947:to its edge length; thus its radius is
933:
18:
7:
2502:. MacMillan. p. 183–185.
2011:. Of these, two also share the same
924:The icosidodecahedron has 6 central
2562:Read, R. C.; Wilson, R. J. (1998),
2162:Compound of five tetrahemihexahedra
2549:Two Dimensional symmetry Mutations
106:
100:
14:
2330:Journal of the Franklin Institute
2291:Great truncated icosidodecahedron
2218:The graph of an icosidodecahedron
2887:
2882:
2869:
2864:
2853:
2842:
2831:
2820:
2809:
2796:
2785:
2774:
2763:
2752:
2741:
2600:10.1111/j.1600-0854.2009.01026.x
2155:
2144:
2131:
2120:
2109:
2096:
2085:
2074:
2061:
2050:
2041:
1926:
1919:
1912:
1905:
1842:
1835:
1828:
1821:
1814:
1807:
1800:
1646:
1639:
1632:
1625:
1618:
1611:
1604:
1552:
1454:
1447:
1440:
1433:
1426:
1419:
1412:
1405:
1348:
1343:
1338:
1333:
1328:
1319:
1314:
1309:
1304:
1299:
1290:
1285:
1280:
1275:
1270:
1261:
1256:
1251:
1246:
1241:
1232:
1227:
1222:
1217:
1212:
1203:
1198:
1193:
1188:
1183:
1174:
1169:
1164:
1159:
1154:
1145:
1140:
1135:
1130:
1125:
1115:
1108:
1101:
1094:
1087:
1080:
1073:
1066:
900:
891:
303:
294:
285:
202:3D model of an icosidodecahedron
2457:Multi-shell Polyhedral Clusters
361:
16:Archimedean solid with 32 faces
827:
814:
355:
334:
1:
2694:The Encyclopedia of Polyhedra
2371:American Mathematical Society
3309:Degenerate polyhedra are in
2342:10.1016/0016-0032(71)90071-8
2154:
2143:
2130:
2119:
2108:
2095:
2092:Great dodecahemidodecahedron
2084:
2073:
2068:Small dodecahemidodecahedron
2060:
2049:
2040:
2021:small dodecahemidodecahedron
1522:32 all of these tilings are
799:The icosidodecahedron is an
3128:pentagonal icositetrahedron
3069:truncated icosidodecahedron
2860:Truncated icosidodecahedron
2306:Truncated icosidodecahedron
2268:, the Vulcan game of logic
2196:stereographically projected
2103:Great icosihemidodecahedron
2057:Small icosihemidodecahedron
2017:small icosihemidodecahedron
1400:Duals to uniform polyhedra
1019:The icosidodecahedron is a
807:of an icosidodecahedron is
3352:
3158:pentagonal hexecontahedron
3118:deltoidal icositetrahedron
2151:Compound of five octahedra
2138:Great dodecahemicosahedron
2127:Small dodecahemicosahedron
988:{\displaystyle 1/\varphi }
3307:
3198:
3153:disdyakis triacontahedron
3148:deltoidal hexecontahedron
2735:
2692:Virtual Reality Polyhedra
2470:10.1007/978-3-319-64123-2
2261:coat-protein formations.
2234:and 60 edges, one of the
1750:
1745:
1735:
1698:
1570:
1562:
1551:
1535:
1399:
1051:
1046:
776:An icosidodecahedron has
268:pentagonal orthobirotunda
246:and more particularly, a
185:
169:
26:
1037:and 20 triangles of the
960:{\displaystyle \varphi }
918:in the spherical tiling.
450:{\displaystyle \varphi }
276:pentagonal gyrobirotunda
216:pentagonal gyrobirotunda
3259:gyroelongated bipyramid
3133:rhombic triacontahedron
3039:truncated cuboctahedron
2838:Truncated cuboctahedron
2568:Oxford University Press
2204:barycentric subdivision
2081:Great icosidodecahedron
2033:five tetrahemihexahedra
878:rhombic triacontahedron
248:quasiregular polyhedron
147:Rhombic triacontahedron
48:Quasiregular polyhedron
3336:Quasiregular polyhedra
3254:truncated trapezohedra
3123:disdyakis dodecahedron
3089:(duals of Archimedean)
3064:rhombicosidodecahedron
3054:truncated dodecahedron
2849:Rhombicosidodecahedron
2781:Truncated dodecahedron
2454:Diudea, M. V. (2018).
2301:Rhombicosidodecahedron
2219:
2005:uniform star polyhedra
1989:
989:
961:
866:
767:
682:
503:
483:
451:
431:
203:
116:
3143:pentakis dodecahedron
3059:truncated icosahedron
3014:truncated tetrahedron
2792:Truncated icosahedron
2748:Truncated tetrahedron
2687:The Uniform Polyhedra
2621:Cromwell, P. (1997).
2217:
1998:pyritohedral symmetry
1987:
1508:vertex configurations
1026:and also a rectified
990:
962:
867:
768:
683:
504:
484:
452:
432:
321:Cartesian coordinates
201:
117:
3103:rhombic dodecahedron
3029:truncated octahedron
2770:Truncated octahedron
1524:wythoff construction
971:
951:
910:The 60 edges form 6
811:
778:icosahedral symmetry
698:
513:
493:
473:
441:
331:
95:
90:Icosahedral symmetry
3138:triakis icosahedron
3113:tetrakis hexahedron
3098:triakis tetrahedron
3034:rhombicuboctahedron
2827:Rhombicuboctahedron
2673:Klitzing, Richard.
2399:"Icosahedral group"
3331:Archimedean solids
3108:triakis octahedron
2993:Archimedean solids
2729:Archimedean solids
2652:Weisstein, Eric W.
2627:Archimedean solids
2564:An Atlas of Graphs
2396:Weisstein, Eric W.
2266:Star Trek universe
2236:Archimedean graphs
2220:
2116:Dodecadodecahedron
2047:Icosidodecahedron
2009:vertex arrangement
1990:
1528:fundamental domain
985:
957:
862:
763:
678:
676:
499:
479:
447:
427:
260:pentagonal rotunda
244:Archimedean solids
230:faces and twelve (
204:
112:
44:Uniform polyhedron
3318:
3317:
3237:
3236:
3074:snub dodecahedron
3049:icosidodecahedron
2904:
2903:
2899:
2898:
2894:Snub dodecahedron
2816:Icosidodecahedron
2660:Archimedean solid
2656:Icosidodecahedron
2527:Regular Polytopes
2509:978-0-02-065320-2
2479:978-3-319-64123-2
2440:978-0-486-23729-9
2380:978-1-4704-5592-7
2170:Related polychora
2167:
2166:
1980:Related polyhedra
1977:
1976:
1694:
1693:
1516:orbifold notation
1504:
1503:
1015:Related polytopes
914:corresponding to
801:Archimedean solid
740:
739:
733:
641:
635:
569:
567:
543:
502:{\displaystyle V}
482:{\displaystyle A}
417:
394:
378:
325:even permutations
212:icosidodecahedron
194:
193:
40:Archimedean solid
22:Icosidodecahedron
3343:
3196:
3192:Dihedral uniform
3167:Dihedral regular
3090:
3006:
2955:
2931:
2924:
2917:
2908:
2891:
2886:
2873:
2868:
2857:
2846:
2835:
2824:
2813:
2800:
2789:
2778:
2767:
2756:
2745:
2738:
2737:
2722:
2715:
2708:
2699:
2678:
2669:
2640:
2613:
2612:
2602:
2578:
2572:
2571:
2559:
2553:
2545:
2539:
2520:
2514:
2513:
2490:
2484:
2483:
2451:
2445:
2444:
2425:Williams, Robert
2421:
2410:
2409:
2408:
2391:
2385:
2384:
2360:
2354:
2353:
2325:
2159:
2148:
2135:
2124:
2113:
2100:
2089:
2078:
2065:
2054:
2045:
2038:
2013:edge arrangement
1930:
1923:
1916:
1909:
1846:
1839:
1832:
1825:
1818:
1811:
1804:
1727:
1720:
1713:
1696:
1650:
1643:
1636:
1629:
1622:
1615:
1608:
1556:
1533:
1458:
1451:
1444:
1437:
1430:
1423:
1416:
1409:
1353:
1352:
1351:
1347:
1346:
1342:
1341:
1337:
1336:
1332:
1331:
1324:
1323:
1322:
1318:
1317:
1313:
1312:
1308:
1307:
1303:
1302:
1295:
1294:
1293:
1289:
1288:
1284:
1283:
1279:
1278:
1274:
1273:
1266:
1265:
1264:
1260:
1259:
1255:
1254:
1250:
1249:
1245:
1244:
1237:
1236:
1235:
1231:
1230:
1226:
1225:
1221:
1220:
1216:
1215:
1208:
1207:
1206:
1202:
1201:
1197:
1196:
1192:
1191:
1187:
1186:
1179:
1178:
1177:
1173:
1172:
1168:
1167:
1163:
1162:
1158:
1157:
1150:
1149:
1148:
1144:
1143:
1139:
1138:
1134:
1133:
1129:
1128:
1119:
1112:
1105:
1098:
1091:
1084:
1077:
1070:
1044:
1009:golden triangles
994:
992:
991:
986:
981:
966:
964:
963:
958:
904:
895:
871:
869:
868:
863:
861:
860:
848:
847:
835:
834:
780:, and its first
772:
770:
769:
764:
759:
758:
746:
742:
741:
735:
734:
729:
717:
716:
687:
685:
684:
679:
677:
670:
669:
652:
651:
642:
637:
636:
631:
619:
603:
602:
585:
584:
575:
571:
570:
568:
563:
552:
544:
539:
508:
506:
505:
500:
488:
486:
485:
480:
456:
454:
453:
448:
436:
434:
433:
428:
423:
419:
418:
413:
412:
403:
395:
387:
379:
371:
307:
298:
289:
200:
190:
174:
121:
119:
118:
113:
111:
110:
109:
103:
31:
19:
3351:
3350:
3346:
3345:
3344:
3342:
3341:
3340:
3321:
3320:
3319:
3314:
3303:
3242:Dihedral others
3233:
3212:
3187:
3162:
3091:
3088:
3087:
3078:
3007:
2996:
2995:
2986:
2949:
2947:Platonic solids
2941:
2935:
2905:
2900:
2892:
2874:
2858:
2847:
2836:
2825:
2814:
2801:
2790:
2779:
2768:
2757:
2746:
2731:
2726:
2672:
2650:
2647:
2637:
2620:
2617:
2616:
2580:
2579:
2575:
2561:
2560:
2556:
2551:by Daniel Huson
2546:
2542:
2521:
2517:
2510:
2492:
2491:
2487:
2480:
2453:
2452:
2448:
2441:
2423:
2422:
2413:
2394:
2393:
2392:
2388:
2381:
2373:. p. 477.
2362:
2361:
2357:
2327:
2326:
2319:
2314:
2282:
2252:Hoberman sphere
2248:
2212:
2172:
2160:
2149:
2136:
2125:
2114:
2101:
2090:
2079:
2066:
2055:
2046:
2007:share the same
1982:
1901:
1792:
1788:
1783:
1779:
1775:
1771:
1767:
1763:
1743:
1737:
1731:
1600:
1557:
1349:
1344:
1339:
1334:
1329:
1327:
1320:
1315:
1310:
1305:
1300:
1298:
1291:
1286:
1281:
1276:
1271:
1269:
1262:
1257:
1252:
1247:
1242:
1240:
1233:
1228:
1223:
1218:
1213:
1211:
1204:
1199:
1194:
1189:
1184:
1182:
1175:
1170:
1165:
1160:
1155:
1153:
1146:
1141:
1136:
1131:
1126:
1124:
1017:
1005:radially golden
969:
968:
949:
948:
922:
921:
920:
919:
907:
906:
905:
897:
896:
874:dual polyhedron
852:
839:
826:
809:
808:
750:
718:
711:
707:
696:
695:
675:
674:
661:
653:
643:
620:
611:
605:
604:
594:
586:
576:
534:
530:
523:
511:
510:
491:
490:
471:
470:
467:
439:
438:
404:
366:
362:
329:
328:
317:
316:
315:
314:
310:
309:
308:
300:
299:
291:
290:
256:
196:
142:Dual polyhedron
98:
93:
92:
46:
42:
17:
12:
11:
5:
3349:
3347:
3339:
3338:
3333:
3323:
3322:
3316:
3315:
3308:
3305:
3304:
3302:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3256:
3251:
3245:
3243:
3239:
3238:
3235:
3234:
3232:
3231:
3226:
3220:
3218:
3214:
3213:
3211:
3210:
3205:
3199:
3193:
3189:
3188:
3186:
3185:
3178:
3170:
3168:
3164:
3163:
3161:
3160:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3094:
3092:
3085:Catalan solids
3083:
3080:
3079:
3077:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3026:
3024:truncated cube
3021:
3016:
3010:
3008:
2991:
2988:
2987:
2985:
2984:
2979:
2974:
2969:
2964:
2958:
2956:
2943:
2942:
2936:
2934:
2933:
2926:
2919:
2911:
2902:
2901:
2897:
2896:
2879:
2878:
2862:
2851:
2840:
2829:
2818:
2806:
2805:
2794:
2783:
2772:
2761:
2759:Truncated cube
2750:
2736:
2733:
2732:
2727:
2725:
2724:
2717:
2710:
2702:
2696:
2695:
2689:
2684:
2679:
2670:
2646:
2645:External links
2643:
2642:
2641:
2635:
2615:
2614:
2593:(3): 303–310.
2573:
2554:
2540:
2515:
2508:
2485:
2478:
2464:. p. 39.
2446:
2439:
2411:
2386:
2379:
2355:
2336:(5): 329–352.
2316:
2315:
2313:
2310:
2309:
2308:
2303:
2298:
2293:
2288:
2281:
2278:
2247:
2244:
2211:
2208:
2171:
2168:
2165:
2164:
2153:
2141:
2140:
2129:
2118:
2106:
2105:
2094:
2083:
2071:
2070:
2059:
2048:
2029:five octahedra
1994:truncated cube
1981:
1978:
1975:
1974:
1971:
1968:
1965:
1962:
1959:
1954:
1951:
1946:
1940:
1939:
1937:
1935:
1933:
1931:
1924:
1917:
1910:
1903:
1897:
1896:
1889:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1850:
1849:
1847:
1840:
1833:
1826:
1819:
1812:
1805:
1798:
1794:
1793:
1789:
1785:
1780:
1776:
1772:
1768:
1764:
1759:
1758:
1755:
1752:
1749:
1744:
1733:
1732:
1730:
1729:
1722:
1715:
1707:
1692:
1691:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1652:
1651:
1644:
1637:
1630:
1623:
1616:
1609:
1602:
1596:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1573:
1572:
1569:
1566:
1561:
1549:
1548:
1502:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1460:
1459:
1452:
1445:
1438:
1431:
1424:
1417:
1410:
1402:
1401:
1397:
1396:
1391:
1386:
1381:
1376:
1371:
1366:
1361:
1355:
1354:
1325:
1296:
1267:
1238:
1209:
1180:
1151:
1121:
1120:
1113:
1106:
1099:
1092:
1085:
1078:
1071:
1063:
1062:
1059:
1049:
1048:
1016:
1013:
984:
980:
976:
956:
909:
908:
899:
898:
890:
889:
888:
887:
886:
859:
855:
851:
846:
842:
838:
833:
829:
825:
822:
819:
816:
762:
757:
753:
749:
745:
738:
732:
727:
724:
721:
714:
710:
706:
703:
692:dihedral angle
673:
668:
664:
660:
657:
654:
650:
646:
640:
634:
629:
626:
623:
617:
614:
612:
610:
607:
606:
601:
597:
593:
590:
587:
583:
579:
574:
566:
561:
558:
555:
550:
547:
542:
537:
533:
529:
526:
524:
522:
519:
518:
498:
478:
466:
463:
446:
426:
422:
416:
411:
407:
401:
398:
393:
390:
385:
382:
377:
374:
369:
365:
360:
357:
354:
351:
348:
345:
342:
339:
336:
312:
311:
302:
301:
293:
292:
284:
283:
282:
281:
280:
264:Johnson solids
255:
252:
192:
191:
183:
182:
176:
175:
167:
166:
160:
159:
154:
150:
149:
144:
138:
137:
134:
127:Dihedral angle
123:
122:
108:
102:
87:
85:Symmetry group
81:
80:
77:
71:
70:
67:
61:
60:
57:
51:
50:
37:
33:
32:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
3348:
3337:
3334:
3332:
3329:
3328:
3326:
3312:
3306:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3246:
3244:
3240:
3230:
3227:
3225:
3222:
3221:
3219:
3215:
3209:
3206:
3204:
3201:
3200:
3197:
3194:
3190:
3184:
3183:
3179:
3177:
3176:
3172:
3171:
3169:
3165:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3095:
3093:
3086:
3081:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3019:cuboctahedron
3017:
3015:
3012:
3011:
3009:
3004:
3000:
2994:
2989:
2983:
2980:
2978:
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2959:
2957:
2953:
2948:
2944:
2940:
2932:
2927:
2925:
2920:
2918:
2913:
2912:
2909:
2895:
2890:
2885:
2881:
2880:
2877:
2872:
2867:
2863:
2861:
2856:
2852:
2850:
2845:
2841:
2839:
2834:
2830:
2828:
2823:
2819:
2817:
2812:
2808:
2807:
2804:
2803:Cuboctahedron
2799:
2795:
2793:
2788:
2784:
2782:
2777:
2773:
2771:
2766:
2762:
2760:
2755:
2751:
2749:
2744:
2740:
2739:
2734:
2730:
2723:
2718:
2716:
2711:
2709:
2704:
2703:
2700:
2693:
2690:
2688:
2685:
2683:
2680:
2676:
2671:
2667:
2666:
2661:
2657:
2653:
2649:
2648:
2644:
2638:
2636:0-521-55432-2
2632:
2628:
2624:
2619:
2618:
2610:
2606:
2601:
2596:
2592:
2588:
2584:
2577:
2574:
2570:, p. 269
2569:
2565:
2558:
2555:
2552:
2550:
2544:
2541:
2537:
2536:0-486-61480-8
2533:
2529:
2528:
2524:
2519:
2516:
2511:
2505:
2501:
2500:
2495:
2494:Fuller, R. B.
2489:
2486:
2481:
2475:
2471:
2467:
2463:
2459:
2458:
2450:
2447:
2442:
2436:
2432:
2431:
2426:
2420:
2418:
2416:
2412:
2406:
2405:
2400:
2397:
2390:
2387:
2382:
2376:
2372:
2368:
2367:
2359:
2356:
2351:
2347:
2343:
2339:
2335:
2331:
2324:
2322:
2318:
2311:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2286:Cuboctahedron
2284:
2283:
2279:
2277:
2275:
2271:
2267:
2262:
2260:
2255:
2253:
2245:
2243:
2241:
2237:
2233:
2229:
2225:
2216:
2209:
2207:
2205:
2201:
2197:
2193:
2188:
2185:
2180:
2177:
2169:
2163:
2158:
2152:
2147:
2142:
2139:
2134:
2128:
2123:
2117:
2112:
2107:
2104:
2099:
2093:
2088:
2082:
2077:
2072:
2069:
2064:
2058:
2053:
2044:
2039:
2036:
2034:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2001:
1999:
1995:
1986:
1979:
1973:V(5.∞)
1972:
1970:V(5.∞)
1969:
1966:
1963:
1960:
1958:
1955:
1952:
1950:
1947:
1945:
1942:
1941:
1938:
1936:
1934:
1932:
1929:
1925:
1922:
1918:
1915:
1911:
1908:
1904:
1899:
1898:
1894:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1851:
1848:
1845:
1841:
1838:
1834:
1831:
1827:
1824:
1820:
1817:
1813:
1810:
1806:
1803:
1799:
1796:
1795:
1790:
1786:
1781:
1777:
1773:
1769:
1765:
1761:
1760:
1756:
1753:
1748:
1741:
1734:
1728:
1723:
1721:
1716:
1714:
1709:
1706:
1702:
1697:
1689:
1687:
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1653:
1649:
1645:
1642:
1638:
1635:
1631:
1628:
1624:
1621:
1617:
1614:
1610:
1607:
1603:
1598:
1597:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1574:
1567:
1565:
1560:
1555:
1550:
1546:
1542:
1540:
1534:
1531:
1529:
1525:
1521:
1518:symmetry of *
1517:
1513:
1509:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1461:
1457:
1453:
1450:
1446:
1443:
1439:
1436:
1432:
1429:
1425:
1422:
1418:
1415:
1411:
1408:
1404:
1403:
1398:
1395:
1392:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1362:
1360:
1357:
1356:
1326:
1297:
1268:
1239:
1210:
1181:
1152:
1123:
1122:
1118:
1114:
1111:
1107:
1104:
1100:
1097:
1093:
1090:
1086:
1083:
1079:
1076:
1072:
1069:
1065:
1064:
1060:
1057:
1054:
1050:
1045:
1042:
1040:
1036:
1031:
1029:
1025:
1022:
1014:
1012:
1010:
1006:
1002:
998:
982:
978:
974:
954:
946:
941:
939:
935:
934:Fuller (1975)
931:
930:great circles
927:
917:
916:great circles
913:
903:
894:
885:
883:
882:Catalan solid
879:
875:
857:
853:
849:
844:
840:
836:
831:
823:
820:
817:
806:
805:vertex figure
802:
797:
795:
792:and its dual
791:
787:
783:
779:
774:
760:
755:
751:
747:
743:
736:
730:
725:
722:
719:
712:
708:
704:
701:
693:
688:
671:
666:
662:
658:
655:
648:
644:
638:
632:
627:
624:
621:
615:
613:
608:
599:
595:
591:
588:
581:
577:
572:
564:
559:
556:
553:
548:
545:
540:
535:
531:
527:
525:
520:
496:
476:
464:
462:
460:
444:
424:
420:
414:
409:
405:
399:
396:
391:
388:
383:
380:
375:
372:
367:
363:
358:
352:
349:
346:
343:
340:
337:
326:
322:
306:
297:
288:
279:
277:
273:
269:
265:
261:
253:
251:
249:
245:
241:
237:
233:
229:
225:
222:with twenty (
221:
217:
213:
209:
199:
189:
184:
181:
177:
173:
168:
165:
164:Vertex figure
161:
158:
155:
151:
148:
145:
143:
139:
135:
132:
128:
124:
91:
88:
86:
82:
78:
76:
72:
68:
66:
62:
58:
56:
52:
49:
45:
41:
38:
34:
30:
25:
20:
3310:
3229:trapezohedra
3180:
3173:
3048:
2977:dodecahedron
2815:
2663:
2626:
2622:
2590:
2586:
2576:
2563:
2557:
2548:
2543:
2525:
2518:
2498:
2488:
2456:
2449:
2429:
2402:
2389:
2365:
2358:
2333:
2329:
2263:
2256:
2249:
2221:
2189:
2173:
2002:
1991:
1892:
1888:(5.∞)
1858:
1754:Paracompact
1739:
1704:
1700:
1690:(3.∞)
1670:
1599:Quasiregular
1559:Construction
1544:
1538:
1519:
1511:
1505:
1368:
1035:dodecahedron
1032:
1024:dodecahedron
1018:
1004:
945:golden ratio
942:
923:
798:
790:dodecahedron
775:
689:
468:
459:golden ratio
457:denotes the
318:
275:
257:
254:Construction
231:
223:
215:
211:
205:
2999:semiregular
2982:icosahedron
2962:tetrahedron
2296:Icosahedron
2274:holographic
2184:hypersphere
1757:Noncompact
1751:Hyperbolic
1594:*∞32
1571:Hyperbolic
1039:icosahedron
1028:icosahedron
794:icosahedron
319:Convenient
3325:Categories
3294:prismatoid
3224:bipyramids
3208:antiprisms
3182:hosohedron
2972:octahedron
2312:References
2246:Appearance
1787:*∞52
1568:Euclidean
1499:V3.3.3.3.5
1484:V3.3.3.3.3
782:stellation
465:Properties
236:pentagonal
228:triangular
220:polyhedron
153:Properties
3289:birotunda
3279:bifrustum
3044:snub cube
2939:polyhedra
2876:Snub cube
2665:MathWorld
2623:Polyhedra
2404:MathWorld
2200:geodesics
2025:compounds
1747:Spherical
1564:Spherical
1526:within a
1058:, (*532)
1021:rectified
983:φ
955:φ
850:⋅
821:⋅
756:∘
748:≈
713:−
705:
656:≈
589:≈
445:φ
406:φ
400:±
389:φ
384:±
368:±
353:φ
350:±
3269:bicupola
3249:pyramids
3175:dihedron
2609:20070605
2496:(1975).
2462:Springer
2427:(1979).
2280:See also
2238:. It is
2232:vertices
2230:with 30
2224:skeleton
2192:600-cell
2179:600-cell
1902:figures
1797:Figures
1736:Symmetry
1601:figures
1591:*832...
1489:V3.4.5.4
1474:V3.5.3.5
1469:V3.10.10
1061:, (532)
1053:Symmetry
997:600-cell
926:decagons
912:decagons
786:compound
272:gyration
240:vertices
208:geometry
75:Vertices
3311:italics
3299:scutoid
3284:rotunda
3274:frustum
3003:uniform
2952:regular
2937:Convex
2587:Traffic
2523:Coxeter
2350:0290245
2270:Kal-Toh
2240:quartic
2176:regular
2031:and of
1967:V(5.8)
1964:V(5.7)
1961:V(5.6)
1953:V(5.4)
1944:Config.
1900:Rhombic
1854:Config.
1494:V4.6.10
1394:sr{5,3}
1389:tr{5,3}
1384:rr{5,3}
1001:decagon
784:is the
136:142.62°
131:degrees
3264:cupola
3217:duals:
3203:prisms
2662:") at
2633:
2607:
2534:
2506:
2476:
2437:
2377:
2348:
2015:: the
2003:Eight
1957:V(5.5)
1949:V(5.3)
1791:
1656:Vertex
1479:V5.6.6
1464:V5.5.5
1374:t{3,5}
1369:r{5,3}
1364:t{5,3}
872:. Its
752:142.62
702:arccos
659:13.836
592:29.306
437:where
266:, the
232:dodeca
157:convex
2259:COPII
2228:graph
2210:Graph
2190:If a
1884:(5.8)
1879:(5.7)
1874:(5.6)
1869:(5.5)
1864:(5.4)
1859:(5.3)
1705:(5.n)
1686:(3.8)
1681:(3.7)
1676:(3.6)
1671:(3.5)
1666:(3.4)
1661:(3.3)
1588:*732
1585:*632
1582:*532
1579:*432
1576:*332
1543:: (3.
1379:{3,5}
1359:{5,3}
788:of a
224:icosi
218:is a
210:, an
65:Edges
55:Faces
2967:cube
2658:" ("
2631:ISBN
2605:PMID
2532:ISBN
2504:ISBN
2474:ISBN
2435:ISBN
2375:ISBN
2222:The
1992:The
1784:...
1782:*852
1778:*752
1774:*652
1770:*552
1766:*452
1762:*352
880:, a
690:The
327:of:
36:Type
3001:or
2654:, "
2595:doi
2466:doi
2338:doi
2334:291
2194:is
2027:of
1895:i)
1891:(5.
1510:(3.
876:is
214:or
206:In
180:Net
3327::
2629:.
2603:.
2591:11
2589:.
2585:.
2566:,
2472:.
2414:^
2401:.
2369:.
2346:MR
2344:.
2332:.
2320:^
2254:.
2206:.
2035:.
2000:.
1738:*5
1699:*5
1547:)
1055::
1041::
940:.
932:.
884:.
737:15
628:17
622:45
560:10
554:25
461:.
278:.
250:.
234:)
226:)
79:30
69:60
59:32
3313:.
3005:)
2997:(
2954:)
2950:(
2930:e
2923:t
2916:v
2721:e
2714:t
2707:v
2677:.
2668:.
2639:.
2611:.
2597::
2512:.
2482:.
2468::
2443:.
2407:.
2383:.
2352:.
2340::
1893:n
1742:2
1740:n
1726:e
1719:t
1712:v
1701:n
1545:n
1539:n
1537:*
1520:n
1512:n
979:/
975:1
858:2
854:5
845:2
841:3
837:=
832:2
828:)
824:5
818:3
815:(
761:,
744:)
731:5
726:2
723:+
720:5
709:(
672:.
667:3
663:a
649:3
645:a
639:6
633:5
625:+
616:=
609:V
600:2
596:a
582:2
578:a
573:)
565:5
557:+
549:3
546:+
541:3
536:5
532:(
528:=
521:A
497:V
477:A
425:,
421:)
415:2
410:2
397:,
392:2
381:,
376:2
373:1
364:(
359:,
356:)
347:,
344:0
341:,
338:0
335:(
133:)
129:(
107:h
101:I
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
