1188:
543:
1849:
25:
3037:
3022:
1909:
3094:
3007:
2992:
2977:
2962:
3099:
1893:
1003:
292:
3074:
3054:
3452:
3437:
3079:
3059:
1767:
3514:
3422:
3407:
3392:
3377:
1877:
3509:
1760:
1921:
3494:
3474:
3489:
3469:
2892:
2877:
2832:
2817:
1635:
2862:
2847:
1268:
1254:
1296:
1795:
1289:
1275:
1261:
1247:
306:
1052:
1861:
1282:
93:
1061:
1628:
1621:
1600:
1180:
1131:
1124:
3307:
3292:
3247:
3232:
691:
1117:
1110:
3277:
3262:
1781:
2943:
1774:
1753:
2926:
2909:
1607:
275:
1166:
1159:
1593:
1152:
1145:
3358:
1788:
369:
3341:
3324:
1614:
3215:
3200:
3151:
3136:
2800:
2785:
2736:
2721:
1586:
3183:
3168:
2768:
2753:
998:{\displaystyle {\begin{aligned}S&=12{\sqrt {5}}\,a^{2}&&\approx 26.8328a^{2}\\V&=4{\sqrt {5+2{\sqrt {5}}}}\,a^{3}&&\approx 12.3107a^{3}\\r_{\mathrm {i} }&={\frac {\varphi ^{2}}{\sqrt {1+\varphi ^{2}}}}\,a={\sqrt {1+{\frac {2}{\sqrt {5}}}}}\,a&&\approx 1.37638a\\r_{\mathrm {m} }&=\left(1+{\frac {1}{\sqrt {5}}}\right)\,a&&\approx 1.44721a\end{aligned}}}
168:
386:
1076:
The rhombic triacontahedron has four symmetry positions, two centered on vertices, one mid-face, and one mid-edge. Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the non-periodic tessellation often referred to as
1876:
696:
1860:
1023:
is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron.
1932:
Woodworker Jane
Kostick builds boxes in the shape of a rhombic triacontahedron. The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.
132:
1529:
1519:
1509:
1500:
1490:
1480:
1471:
1451:
1442:
1413:
1403:
1374:
1345:
1335:
1306:
1461:
1432:
1422:
1393:
1384:
1364:
1355:
1326:
1316:
142:
122:
1524:
1514:
1495:
1485:
1466:
1456:
1437:
1427:
1408:
1398:
1379:
1369:
1350:
1340:
1321:
1311:
612:
and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin. The length of its edges is
137:
127:
1884:
1916:
Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light (IQ for "interlocking quadrilaterals").
3567:
1686:
1868:
2276:
2211:
1837:
512:
The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four
Platonic solids. It contains ten
3560:
2697:
1928:
of a rhombic triacontahedral box made of six panels around a cubic hole – zoom into the model to see the hole from the inside
231:
2242:
1892:
1848:
2174:
2149:
2177:
3553:
1201:
The rhombic triacontahedron has 227 fully supported stellations. Another stellation of the
Rhombic triacontahedron is the
393:
to a rhombic triacontahedron by dividing the square faces into 4 squares and splitting middle edges into new rhombic faces.
2304:
2271:
1187:
2114:
1968:
1231:
211:
68:
46:
39:
3974:
291:
97:
2602:
2179:(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p. 285, Rhombic triacontahedron )
2690:
685:
to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:
3766:
3707:
3456:
3083:
1567:
3796:
3756:
3522:
3502:
3482:
3460:
3445:
3430:
3415:
3400:
3385:
3368:
3351:
3334:
3315:
3300:
3285:
3270:
3255:
3240:
3223:
3208:
3191:
3176:
3159:
3144:
3107:
3103:
3087:
3067:
3045:
3030:
3015:
3000:
2996:
2985:
2970:
2953:
2936:
2919:
2900:
2885:
2870:
2855:
2840:
2825:
2808:
2793:
2776:
2761:
2744:
2729:
2492:
2133:
2102:
1677:
150:
114:
3791:
3786:
3041:
3026:
1672:
1667:
2486:
475:
33:
542:
3969:
3897:
3892:
3677:
3426:
2683:
2614:
2542:
2297:
2019:
1196:
2163:(The thirteen semiregular convex polyhedra and their duals, p. 22, Rhombic triacontahedron)
3761:
3702:
3692:
3637:
3441:
3296:
3011:
2548:
2263:
1990:
1562:
1542:
1202:
582:, whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points
50:
2258:
3984:
3781:
3697:
3652:
3600:
3311:
3251:
3236:
2896:
2267:
1693:
symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.
1657:
1552:
2194:
3741:
3667:
3615:
3396:
3281:
3204:
3063:
2966:
2931:
2804:
2159:
1827:
1812:
1712:
1235:
579:
547:
536:
525:
506:
216:
397:
The ratio of the long diagonal to the short diagonal of each face is exactly equal to the
8:
3907:
3776:
3751:
3736:
3672:
3620:
3478:
3411:
3219:
3155:
3140:
3036:
3021:
2981:
2881:
2866:
2836:
2821:
2789:
2740:
2725:
2006:
1882:
A rhombic triacontahedron with an inscribed dodecahedron (blue) and icosahedron (purple).
1832:
1647:
575:
521:
1908:
3979:
3922:
3887:
3746:
3641:
3590:
3187:
3093:
3006:
2991:
2976:
2961:
2851:
2757:
2659:
2536:
2530:
2290:
2056:
1802:
1033:
160:
3098:
2239:
2217:
3902:
3712:
3687:
3631:
3518:
3363:
3120:
3073:
3053:
2649:
2573:
2525:
2498:
2468:
2190:
2170:
2145:
2110:
2060:
2048:
1925:
1707:
1572:
1547:
502:
459:
445:
354:
342:
327:
279:
3451:
3436:
3078:
3058:
1766:
3841:
2654:
2634:
2456:
2251:
2137:
2125:
2040:
1963:
1822:
678:
296:
258:
3513:
3421:
3406:
3391:
3376:
3508:
2608:
2520:
2515:
2480:
2435:
2425:
2415:
2410:
2246:
2198:
2155:
1986:
1759:
490:
478:
of the solid that leaves it occupying the same region of space while moving face
450:
350:
338:
334:
284:
262:
255:
3493:
3473:
1866:
A rhombic triacontahedron with an inscribed tetrahedron (red) and cube (yellow).
3662:
3585:
3488:
3468:
3266:
2891:
2876:
2831:
2816:
2664:
2567:
2430:
2420:
2092:
2074:
Messer, P. W. (1995). "Stellations of the rhombic triacontahedron and Beyond".
1958:
1935:
1634:
1078:
494:
455:
437:
241:
2861:
2846:
1920:
3963:
3867:
3723:
3657:
3381:
3346:
2706:
2585:
2579:
2474:
2405:
2395:
2141:
2052:
1993:
1794:
1690:
1295:
1288:
498:
406:
346:
107:
1942:
1281:
1267:
1253:
1051:
2400:
1662:
1627:
1620:
1537:
1274:
1260:
1246:
1060:
1013:
563:
398:
376:
3306:
3291:
3246:
3231:
1912:
An example of the use of a rhombic triacontahedron in the design of a lamp
1179:
1130:
1123:
305:
3276:
3261:
2440:
2374:
2364:
2354:
2349:
1946:
1642:
1599:
1557:
1116:
1109:
513:
489:
The rhombic triacontahedron is somewhat special in being one of the nine
92:
2942:
2107:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1752:
3932:
3820:
3610:
3577:
3329:
2625:
2379:
2369:
2359:
2344:
2334:
2313:
2044:
1780:
1606:
529:
358:
323:
266:
2925:
2908:
1773:
3927:
3917:
3862:
3846:
3682:
3498:
2639:
2203:
1938:'s "Ball of Whacks" comes in the shape of a rhombic triacontahedron.
1592:
1205:. The total number of stellations of the rhombic triacontahedron is
1165:
1158:
3114:
2223:
1151:
1144:
274:
3813:
3545:
3357:
2339:
1020:
574:
and cyclic permutations of these coordinates are the vertices of a
471:
3937:
3912:
3340:
3323:
2675:
2644:
1787:
682:
331:
3214:
3199:
3150:
3135:
2799:
2784:
2735:
2720:
2189:
368:
3182:
3167:
2767:
2752:
1613:
674:
412:
1585:
167:
2228:
435:, or approximately 63.43°. A rhombus so obtained is called a
2282:
2169:
2008, John H. Conway, Heidi
Burgiel, Chaim Goodman-Strauss,
3605:
3172:
2914:
2772:
1807:
517:
390:
385:
2233:
1191:
An example of stellations of the rhombic triacontahedron.
2229:
462:
on the set of faces. This means that for any two faces,
2252:
a wooden construction of a rhombic triacontahedron box
1032:
The rhombic triacontahedron can be dissected into 20
694:
669:If the edge length of a rhombic triacontahedron is
539:by dividing the hexagonal faces into three rhombi:
997:
373:A face of the rhombic triacontahedron. The lengths
1699:Symmetry mutations of dual quasiregular tilings:
3961:
2031:Pawley, G. S. (1975). "The 227 triacontahedra".
3561:
2691:
2298:
2007:"How to make golden rhombohedra out of paper"
1945:" thirty-sided die, sometimes useful in some
389:This animation shows a transformation from a
2039:(2–4). Kluwer Academic Publishers: 221–232.
1941:The rhombic triacontahedron is used as the "
493:convex polyhedra, the others being the five
1685:This polyhedron is a part of a sequence of
3568:
3554:
2698:
2684:
2305:
2291:
2277:A viper drawn on a rhombic triacontahedron
2124:
2020:Dissection of the rhombic triacontahedron
1071:
973:
904:
875:
788:
719:
546:A topological rhombic triacontahedron in
69:Learn how and when to remove this message
2101:
1918:
1907:
1226:Family of uniform icosahedral polyhedra
1186:
1178:
637:. Its faces have diagonals with lengths
553:
541:
528:. The centers of the faces contain five
384:
303:
32:This article includes a list of general
1898:Fully truncated rhombic triacontahedron
3962:
2224:Stellations of Rhombic Triacontahedron
2073:
2030:
322:as it is the most common thirty-faced
3549:
3119:
3118:
2679:
2286:
310:3D model of a rhombic triacontahedron
3575:
2236:—Danish designer Holger Strøm's lamp
1218:
1036:: 10 acute ones and 10 obtuse ones.
18:
2093:triacontahedron box - KO Sticks LLC
13:
2705:
2322:Listed by number of faces and type
2272:The Wolfram Demonstrations Project
2254:– by woodworker Jane Kostick
932:
830:
166:
38:it lacks sufficient corresponding
14:
3996:
2183:
1969:Truncated rhombic triacontahedron
1854:Spherical rhombic triacontahedron
448:, the rhombic triacontahedron is
3512:
3507:
3492:
3487:
3472:
3467:
3450:
3435:
3420:
3405:
3390:
3375:
3356:
3339:
3322:
3305:
3290:
3275:
3260:
3245:
3230:
3213:
3198:
3181:
3166:
3149:
3134:
3097:
3092:
3077:
3072:
3057:
3052:
3035:
3020:
3005:
2990:
2975:
2960:
2941:
2924:
2907:
2890:
2875:
2860:
2845:
2830:
2815:
2798:
2783:
2766:
2751:
2734:
2719:
1891:
1875:
1859:
1847:
1793:
1786:
1779:
1772:
1765:
1758:
1751:
1633:
1626:
1619:
1612:
1605:
1598:
1591:
1584:
1527:
1522:
1517:
1512:
1507:
1498:
1493:
1488:
1483:
1478:
1469:
1464:
1459:
1454:
1449:
1440:
1435:
1430:
1425:
1420:
1411:
1406:
1401:
1396:
1391:
1382:
1377:
1372:
1367:
1362:
1353:
1348:
1343:
1338:
1333:
1324:
1319:
1314:
1309:
1304:
1294:
1287:
1280:
1273:
1266:
1259:
1252:
1245:
1164:
1157:
1150:
1143:
1129:
1122:
1115:
1108:
1059:
1050:
367:
290:
273:
140:
135:
130:
125:
120:
91:
23:
2220:– The Encyclopedia of Polyhedra
1919:
1885:(Click here for rotating model)
1869:(Click here for rotating model)
304:
98:(Click here for rotating model)
2214:– Interactive Polyhedron Model
2086:
2067:
2024:
2013:
1999:
1980:
1174:
318:, sometimes simply called the
1:
2533:(two infinite groups and 75)
2312:
1974:
1027:
664:
3948:Degenerate polyhedra are in
2551:(two infinite groups and 50)
2264:30+12 Rhombic Triacontahedra
1088:
1048:
673:, surface area, volume, the
586:together with the 12 points
375:of the diagonals are in the
82:
7:
3767:pentagonal icositetrahedron
3708:truncated icosidodecahedron
3457:Truncated icosidodecahedron
3084:Pentagonal icositetrahedron
2109:. Dover Publications, Inc.
1996:. Retrieved 7 January 2013.
1952:
1579:Duals to uniform polyhedra
16:Catalan solid with 30 faces
10:
4001:
3797:pentagonal hexecontahedron
3757:deltoidal icositetrahedron
3397:Truncated tetratetrahedron
3104:Pentagonal hexecontahedron
2997:Deltoidal icositetrahedron
2259:120 Rhombic Triacontahedra
2134:Cambridge University Press
1698:
1194:
366:
3946:
3880:
3855:
3837:
3830:
3805:
3792:disdyakis triacontahedron
3787:deltoidal hexecontahedron
3721:
3629:
3584:
3506:
3486:
3466:
3355:
3338:
3321:
3195:
3163:
3128:
3113:
3091:
3071:
3051:
3042:Disdyakis triacontahedron
3027:Deltoidal hexecontahedron
2940:
2923:
2906:
2780:
2748:
2713:
2624:
2603:Kepler–Poinsot polyhedron
2595:
2560:
2508:
2449:
2388:
2327:
2320:
2268:12 Rhombic Triacontahedra
2218:Virtual Reality Polyhedra
1719:
1711:
1706:
1578:
1230:
1225:
566:. The 12 points given by
90:
85:
2212:Rhombic Triacontrahedron
2167:The Symmetries of Things
2142:10.1017/CBO9780511569371
86:Rhombic triacontahedron
3898:gyroelongated bipyramid
3772:rhombic triacontahedron
3678:truncated cuboctahedron
3427:Truncated cuboctahedron
3064:Pentagonal dodecahedron
2949:Rhombic triacontahedron
2615:Uniform star polyhedron
2543:quasiregular polyhedron
2195:Rhombic triacontahedron
1949:games or other places.
1903:
1197:Rhombic hexecontahedron
1183:Rhombic hexecontahedron
1085:Orthogonal projections
316:rhombic triacontahedron
53:more precise citations.
3975:Quasiregular polyhedra
3893:truncated trapezohedra
3762:disdyakis dodecahedron
3728:(duals of Archimedean)
3703:rhombicosidodecahedron
3693:truncated dodecahedron
3442:Rhombicosidodecahedron
3382:Rhombitetratetrahedron
3297:Truncated dodecahedron
3012:Disdyakis dodecahedron
2967:Deltoidal dodecahedron
2549:semiregular polyhedron
1929:
1913:
1203:compound of five cubes
1192:
1184:
1072:Orthogonal projections
999:
550:
535:It can be made from a
394:
345:of two types. It is a
311:
171:
3782:pentakis dodecahedron
3698:truncated icosahedron
3653:truncated tetrahedron
3312:Truncated icosahedron
3252:Truncated tetrahedron
3237:Truncated tetrahedron
2897:Pentakis dodecahedron
2596:non-convex polyhedron
1924:
1911:
1195:Further information:
1190:
1182:
1000:
554:Cartesian coordinates
545:
444:Being the dual of an
409:on each face measure
388:
309:
170:
3742:rhombic dodecahedron
3668:truncated octahedron
3282:Truncated octahedron
2982:Disdyakis hexahedron
2932:Rhombic dodecahedron
2245:17 July 2007 at the
692:
580:regular dodecahedron
548:truncated octahedron
537:truncated octahedron
507:rhombic dodecahedron
3777:triakis icosahedron
3752:tetrakis hexahedron
3737:triakis tetrahedron
3673:rhombicuboctahedron
3412:Rhombicuboctahedron
2882:Triakis icosahedron
2867:Tetrakis hexahedron
2837:Triakis tetrahedron
2822:Triakis tetrahedron
2076:Structural Topology
2033:Geometriae Dedicata
1086:
576:regular icosahedron
3747:triakis octahedron
3632:Archimedean solids
2915:Rhombic hexahedron
2852:Triakis octahedron
2537:regular polyhedron
2531:uniform polyhedron
2493:Hectotriadiohedron
2191:Weisstein, Eric W.
2045:10.1007/BF00148756
1930:
1914:
1689:and tilings with
1193:
1185:
1084:
1034:golden rhombohedra
995:
993:
551:
458:of the solid acts
395:
312:
172:
3957:
3956:
3876:
3875:
3713:snub dodecahedron
3688:icosidodecahedron
3543:
3542:
3538:
3537:
3533:
3532:
3528:
3527:
3519:Snub dodecahedron
3364:Icosidodecahedron
2673:
2672:
2574:Archimedean solid
2561:convex polyhedron
2469:Icosidodecahedron
2270:by Sándor Kabai,
2175:978-1-56881-220-5
2151:978-0-521-54325-5
2126:Wenninger, Magnus
1843:
1842:
1687:rhombic polyhedra
1683:
1682:
1219:Related polyhedra
1172:
1171:
1069:
1068:
966:
965:
902:
900:
899:
873:
872:
786:
784:
717:
503:icosidodecahedron
446:Archimedean solid
383:
382:
355:icosidodecahedron
328:convex polyhedron
302:
301:
280:Icosidodecahedron
79:
78:
71:
3992:
3835:
3834:
3831:Dihedral uniform
3806:Dihedral regular
3729:
3645:
3594:
3570:
3563:
3556:
3547:
3546:
3516:
3511:
3496:
3491:
3479:Snub tetrahedron
3476:
3471:
3454:
3439:
3424:
3409:
3394:
3379:
3360:
3343:
3330:Tetratetrahedron
3326:
3309:
3294:
3279:
3264:
3249:
3234:
3217:
3202:
3185:
3170:
3153:
3138:
3131:
3130:
3116:
3115:
3101:
3096:
3081:
3076:
3061:
3056:
3039:
3024:
3009:
2994:
2979:
2964:
2945:
2928:
2911:
2894:
2879:
2864:
2849:
2834:
2819:
2802:
2787:
2770:
2755:
2738:
2723:
2716:
2715:
2700:
2693:
2686:
2677:
2676:
2509:elemental things
2487:Enneacontahedron
2457:Icositetrahedron
2307:
2300:
2293:
2284:
2283:
2208:
2162:
2120:
2103:Williams, Robert
2095:
2090:
2084:
2083:
2071:
2065:
2064:
2028:
2022:
2017:
2011:
2010:
2003:
1997:
1984:
1964:Rhombille tiling
1923:
1895:
1879:
1863:
1851:
1797:
1790:
1783:
1776:
1769:
1762:
1755:
1696:
1695:
1637:
1630:
1623:
1616:
1609:
1602:
1595:
1588:
1532:
1531:
1530:
1526:
1525:
1521:
1520:
1516:
1515:
1511:
1510:
1503:
1502:
1501:
1497:
1496:
1492:
1491:
1487:
1486:
1482:
1481:
1474:
1473:
1472:
1468:
1467:
1463:
1462:
1458:
1457:
1453:
1452:
1445:
1444:
1443:
1439:
1438:
1434:
1433:
1429:
1428:
1424:
1423:
1416:
1415:
1414:
1410:
1409:
1405:
1404:
1400:
1399:
1395:
1394:
1387:
1386:
1385:
1381:
1380:
1376:
1375:
1371:
1370:
1366:
1365:
1358:
1357:
1356:
1352:
1351:
1347:
1346:
1342:
1341:
1337:
1336:
1329:
1328:
1327:
1323:
1322:
1318:
1317:
1313:
1312:
1308:
1307:
1298:
1291:
1284:
1277:
1270:
1263:
1256:
1249:
1223:
1222:
1214:
1213:
1210:
1168:
1161:
1154:
1147:
1133:
1126:
1119:
1112:
1087:
1083:
1063:
1054:
1039:
1038:
1011:
1004:
1002:
1001:
996:
994:
978:
972:
968:
967:
961:
957:
937:
936:
935:
909:
903:
901:
895:
891:
883:
874:
871:
870:
855:
854:
853:
844:
835:
834:
833:
819:
818:
800:
798:
797:
787:
785:
780:
769:
750:
749:
731:
729:
728:
718:
713:
679:inscribed sphere
672:
660:
659:
657:
656:
651:
648:
640:
636:
635:
634:
631:
628:
622:
621:
611:
609:
607:
606:
601:
598:
585:
573:
561:
485:
481:
469:
465:
434:
432:
430:
429:
424:
421:
404:
371:
364:
363:
308:
294:
277:
203:Vertices by type
145:
144:
143:
139:
138:
134:
133:
129:
128:
124:
123:
95:
83:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
4000:
3999:
3995:
3994:
3993:
3991:
3990:
3989:
3960:
3959:
3958:
3953:
3942:
3881:Dihedral others
3872:
3851:
3826:
3801:
3730:
3727:
3726:
3717:
3646:
3635:
3634:
3625:
3588:
3586:Platonic solids
3580:
3574:
3544:
3539:
3534:
3529:
3521:
3517:
3501:
3497:
3481:
3477:
3459:
3455:
3444:
3440:
3429:
3425:
3414:
3410:
3399:
3395:
3384:
3380:
3367:
3366:
3361:
3350:
3349:
3344:
3333:
3332:
3327:
3314:
3310:
3299:
3295:
3284:
3280:
3269:
3265:
3254:
3250:
3239:
3235:
3222:
3218:
3207:
3203:
3190:
3186:
3175:
3171:
3158:
3154:
3143:
3139:
3124:
3106:
3102:
3086:
3082:
3066:
3062:
3044:
3040:
3029:
3025:
3014:
3010:
2999:
2995:
2984:
2980:
2969:
2965:
2952:
2951:
2946:
2935:
2934:
2929:
2918:
2917:
2912:
2899:
2895:
2884:
2880:
2869:
2865:
2854:
2850:
2839:
2835:
2824:
2820:
2807:
2803:
2792:
2788:
2775:
2771:
2760:
2756:
2743:
2739:
2728:
2724:
2709:
2704:
2674:
2669:
2620:
2609:Star polyhedron
2591:
2556:
2504:
2481:Hexecontahedron
2463:Triacontahedron
2445:
2436:Enneadecahedron
2426:Heptadecahedron
2416:Pentadecahedron
2411:Tetradecahedron
2384:
2323:
2316:
2311:
2247:Wayback Machine
2186:
2152:
2117:
2098:
2091:
2087:
2072:
2068:
2029:
2025:
2018:
2014:
2005:
2004:
2000:
1987:Stephen Wolfram
1985:
1981:
1977:
1955:
1906:
1899:
1896:
1887:
1883:
1880:
1871:
1867:
1864:
1855:
1852:
1528:
1523:
1518:
1513:
1508:
1506:
1499:
1494:
1489:
1484:
1479:
1477:
1470:
1465:
1460:
1455:
1450:
1448:
1441:
1436:
1431:
1426:
1421:
1419:
1412:
1407:
1402:
1397:
1392:
1390:
1383:
1378:
1373:
1368:
1363:
1361:
1354:
1349:
1344:
1339:
1334:
1332:
1325:
1320:
1315:
1310:
1305:
1303:
1221:
1211:
1208:
1206:
1199:
1177:
1139:
1091:
1074:
1064:
1055:
1030:
1009:
992:
991:
977:
956:
949:
945:
938:
931:
930:
926:
923:
922:
908:
890:
882:
866:
862:
849:
845:
843:
836:
829:
828:
824:
821:
820:
814:
810:
799:
793:
789:
779:
768:
758:
752:
751:
745:
741:
730:
724:
720:
712:
702:
695:
693:
690:
689:
670:
667:
652:
649:
646:
645:
643:
642:
638:
632:
629:
626:
624:
616:
614:
613:
602:
599:
596:
595:
593:
587:
583:
567:
559:
556:
495:Platonic solids
491:edge-transitive
483:
479:
467:
463:
451:face-transitive
425:
422:
419:
418:
416:
410:
402:
374:
372:
351:dual polyhedron
320:triacontahedron
295:
285:dual polyhedron
282:
278:
256:face-transitive
225:
220:
173:
151:Conway notation
141:
136:
131:
126:
121:
119:
115:Coxeter diagram
96:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
3998:
3988:
3987:
3982:
3977:
3972:
3970:Catalan solids
3955:
3954:
3947:
3944:
3943:
3941:
3940:
3935:
3930:
3925:
3920:
3915:
3910:
3905:
3900:
3895:
3890:
3884:
3882:
3878:
3877:
3874:
3873:
3871:
3870:
3865:
3859:
3857:
3853:
3852:
3850:
3849:
3844:
3838:
3832:
3828:
3827:
3825:
3824:
3817:
3809:
3807:
3803:
3802:
3800:
3799:
3794:
3789:
3784:
3779:
3774:
3769:
3764:
3759:
3754:
3749:
3744:
3739:
3733:
3731:
3724:Catalan solids
3722:
3719:
3718:
3716:
3715:
3710:
3705:
3700:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3663:truncated cube
3660:
3655:
3649:
3647:
3630:
3627:
3626:
3624:
3623:
3618:
3613:
3608:
3603:
3597:
3595:
3582:
3581:
3573:
3572:
3565:
3558:
3550:
3541:
3540:
3536:
3535:
3531:
3530:
3526:
3525:
3505:
3485:
3464:
3463:
3448:
3433:
3418:
3403:
3388:
3372:
3371:
3362:
3354:
3345:
3337:
3328:
3319:
3318:
3303:
3288:
3273:
3267:Truncated cube
3258:
3243:
3227:
3226:
3211:
3196:
3194:
3179:
3164:
3162:
3147:
3129:
3126:
3125:
3111:
3110:
3090:
3070:
3049:
3048:
3033:
3018:
3003:
2988:
2973:
2957:
2956:
2947:
2939:
2930:
2922:
2913:
2904:
2903:
2888:
2873:
2858:
2843:
2828:
2812:
2811:
2796:
2781:
2779:
2764:
2749:
2747:
2732:
2714:
2711:
2710:
2707:Catalan solids
2703:
2702:
2695:
2688:
2680:
2671:
2670:
2668:
2667:
2665:parallelepiped
2662:
2657:
2652:
2647:
2642:
2637:
2631:
2629:
2622:
2621:
2619:
2618:
2612:
2606:
2599:
2597:
2593:
2592:
2590:
2589:
2583:
2577:
2571:
2568:Platonic solid
2564:
2562:
2558:
2557:
2555:
2554:
2553:
2552:
2546:
2540:
2528:
2523:
2518:
2512:
2510:
2506:
2505:
2503:
2502:
2496:
2490:
2484:
2478:
2472:
2466:
2460:
2453:
2451:
2447:
2446:
2444:
2443:
2438:
2433:
2431:Octadecahedron
2428:
2423:
2421:Hexadecahedron
2418:
2413:
2408:
2403:
2398:
2392:
2390:
2386:
2385:
2383:
2382:
2377:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2337:
2331:
2329:
2325:
2324:
2321:
2318:
2317:
2310:
2309:
2302:
2295:
2287:
2281:
2280:
2274:
2255:
2249:
2237:
2231:
2226:
2221:
2215:
2209:
2185:
2184:External links
2182:
2181:
2180:
2164:
2150:
2122:
2115:
2097:
2096:
2085:
2066:
2023:
2012:
1998:
1978:
1976:
1973:
1972:
1971:
1966:
1961:
1959:Golden rhombus
1954:
1951:
1936:Roger von Oech
1905:
1902:
1901:
1900:
1897:
1890:
1888:
1881:
1874:
1872:
1865:
1858:
1856:
1853:
1846:
1841:
1840:
1835:
1830:
1825:
1820:
1815:
1810:
1805:
1799:
1798:
1791:
1784:
1777:
1770:
1763:
1756:
1749:
1745:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1722:
1721:
1718:
1715:
1710:
1704:
1703:
1681:
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1639:
1638:
1631:
1624:
1617:
1610:
1603:
1596:
1589:
1581:
1580:
1576:
1575:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1534:
1533:
1504:
1475:
1446:
1417:
1388:
1359:
1330:
1300:
1299:
1292:
1285:
1278:
1271:
1264:
1257:
1250:
1242:
1241:
1238:
1228:
1227:
1220:
1217:
1176:
1173:
1170:
1169:
1162:
1155:
1148:
1141:
1135:
1134:
1127:
1120:
1113:
1106:
1102:
1101:
1099:
1097:
1095:
1093:
1079:Penrose tiling
1073:
1070:
1067:
1066:
1057:
1047:
1046:
1043:
1029:
1026:
1006:
1005:
990:
987:
984:
981:
979:
976:
971:
964:
960:
955:
952:
948:
944:
941:
939:
934:
929:
925:
924:
921:
918:
915:
912:
910:
907:
898:
894:
889:
886:
881:
878:
869:
865:
861:
858:
852:
848:
842:
839:
837:
832:
827:
823:
822:
817:
813:
809:
806:
803:
801:
796:
792:
783:
778:
775:
772:
767:
764:
761:
759:
757:
754:
753:
748:
744:
740:
737:
734:
732:
727:
723:
716:
711:
708:
705:
703:
701:
698:
697:
666:
663:
555:
552:
456:symmetry group
454:, meaning the
438:golden rhombus
405:, so that the
381:
380:
300:
299:
288:
270:
269:
252:
248:
247:
244:
242:Dihedral angle
238:
237:
234:
232:Rotation group
228:
227:
223:
218:
214:
212:Symmetry group
208:
207:
204:
200:
199:
196:
192:
191:
188:
184:
183:
180:
176:
175:
163:
157:
156:
153:
147:
146:
117:
111:
110:
105:
101:
100:
88:
87:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3997:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3967:
3965:
3951:
3945:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3906:
3904:
3901:
3899:
3896:
3894:
3891:
3889:
3886:
3885:
3883:
3879:
3869:
3866:
3864:
3861:
3860:
3858:
3854:
3848:
3845:
3843:
3840:
3839:
3836:
3833:
3829:
3823:
3822:
3818:
3816:
3815:
3811:
3810:
3808:
3804:
3798:
3795:
3793:
3790:
3788:
3785:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3734:
3732:
3725:
3720:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3658:cuboctahedron
3656:
3654:
3651:
3650:
3648:
3643:
3639:
3633:
3628:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3604:
3602:
3599:
3598:
3596:
3592:
3587:
3583:
3579:
3571:
3566:
3564:
3559:
3557:
3552:
3551:
3548:
3524:
3520:
3515:
3510:
3504:
3500:
3495:
3490:
3484:
3480:
3475:
3470:
3465:
3462:
3458:
3453:
3449:
3447:
3443:
3438:
3434:
3432:
3428:
3423:
3419:
3417:
3413:
3408:
3404:
3402:
3398:
3393:
3389:
3387:
3383:
3378:
3374:
3373:
3370:
3365:
3359:
3353:
3348:
3347:Cuboctahedron
3342:
3336:
3331:
3325:
3320:
3317:
3313:
3308:
3304:
3302:
3298:
3293:
3289:
3287:
3283:
3278:
3274:
3272:
3268:
3263:
3259:
3257:
3253:
3248:
3244:
3242:
3238:
3233:
3229:
3228:
3225:
3221:
3216:
3212:
3210:
3206:
3201:
3197:
3193:
3189:
3184:
3180:
3178:
3174:
3169:
3165:
3161:
3157:
3152:
3148:
3146:
3142:
3137:
3133:
3132:
3127:
3122:
3117:
3112:
3109:
3105:
3100:
3095:
3089:
3085:
3080:
3075:
3069:
3065:
3060:
3055:
3050:
3047:
3043:
3038:
3034:
3032:
3028:
3023:
3019:
3017:
3013:
3008:
3004:
3002:
2998:
2993:
2989:
2987:
2983:
2978:
2974:
2972:
2968:
2963:
2959:
2958:
2955:
2950:
2944:
2938:
2933:
2927:
2921:
2916:
2910:
2905:
2902:
2898:
2893:
2889:
2887:
2883:
2878:
2874:
2872:
2868:
2863:
2859:
2857:
2853:
2848:
2844:
2842:
2838:
2833:
2829:
2827:
2823:
2818:
2814:
2813:
2810:
2806:
2801:
2797:
2795:
2791:
2786:
2782:
2778:
2774:
2769:
2765:
2763:
2759:
2754:
2750:
2746:
2742:
2737:
2733:
2731:
2727:
2722:
2718:
2717:
2712:
2708:
2701:
2696:
2694:
2689:
2687:
2682:
2681:
2678:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2632:
2630:
2627:
2623:
2616:
2613:
2610:
2607:
2604:
2601:
2600:
2598:
2594:
2587:
2586:Johnson solid
2584:
2581:
2580:Catalan solid
2578:
2575:
2572:
2569:
2566:
2565:
2563:
2559:
2550:
2547:
2544:
2541:
2538:
2535:
2534:
2532:
2529:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2511:
2507:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2475:Hexoctahedron
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2454:
2452:
2448:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2406:Tridecahedron
2404:
2402:
2399:
2397:
2396:Hendecahedron
2394:
2393:
2391:
2387:
2381:
2378:
2376:
2373:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2332:
2330:
2326:
2319:
2315:
2308:
2303:
2301:
2296:
2294:
2289:
2288:
2285:
2278:
2275:
2273:
2269:
2265:
2261:
2260:
2256:
2253:
2250:
2248:
2244:
2241:
2240:Make your own
2238:
2235:
2232:
2230:
2227:
2225:
2222:
2219:
2216:
2213:
2210:
2206:
2205:
2200:
2199:Catalan solid
2196:
2192:
2188:
2187:
2178:
2176:
2172:
2168:
2165:
2161:
2157:
2153:
2147:
2143:
2139:
2135:
2131:
2127:
2123:
2121:(Section 3-9)
2118:
2116:0-486-23729-X
2112:
2108:
2104:
2100:
2099:
2094:
2089:
2081:
2077:
2070:
2062:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2027:
2021:
2016:
2008:
2002:
1995:
1994:Wolfram Alpha
1991:
1988:
1983:
1979:
1970:
1967:
1965:
1962:
1960:
1957:
1956:
1950:
1948:
1944:
1939:
1937:
1933:
1927:
1922:
1917:
1910:
1894:
1889:
1886:
1878:
1873:
1870:
1862:
1857:
1850:
1845:
1844:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1796:
1792:
1789:
1785:
1782:
1778:
1775:
1771:
1768:
1764:
1761:
1757:
1754:
1750:
1747:
1746:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1723:
1716:
1714:
1709:
1705:
1702:
1697:
1694:
1692:
1691:Coxeter group
1688:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1644:
1641:
1640:
1636:
1632:
1629:
1625:
1622:
1618:
1615:
1611:
1608:
1604:
1601:
1597:
1594:
1590:
1587:
1583:
1582:
1577:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1535:
1505:
1476:
1447:
1418:
1389:
1360:
1331:
1302:
1301:
1297:
1293:
1290:
1286:
1283:
1279:
1276:
1272:
1269:
1265:
1262:
1258:
1255:
1251:
1248:
1244:
1243:
1239:
1236:
1233:
1229:
1224:
1216:
1204:
1198:
1189:
1181:
1167:
1163:
1160:
1156:
1153:
1149:
1146:
1142:
1137:
1136:
1132:
1128:
1125:
1121:
1118:
1114:
1111:
1107:
1104:
1103:
1100:
1098:
1096:
1094:
1089:
1082:
1080:
1062:
1058:
1053:
1049:
1044:
1041:
1040:
1037:
1035:
1025:
1022:
1017:
1015:
988:
985:
982:
980:
974:
969:
962:
958:
953:
950:
946:
942:
940:
927:
919:
916:
913:
911:
905:
896:
892:
887:
884:
879:
876:
867:
863:
859:
856:
850:
846:
840:
838:
825:
815:
811:
807:
804:
802:
794:
790:
781:
776:
773:
770:
765:
762:
760:
755:
746:
742:
738:
735:
733:
725:
721:
714:
709:
706:
704:
699:
688:
687:
686:
684:
680:
676:
662:
655:
620:
605:
591:
581:
577:
571:
565:
549:
544:
540:
538:
533:
531:
527:
523:
519:
515:
510:
508:
504:
500:
499:cuboctahedron
496:
492:
487:
477:
473:
470:, there is a
461:
457:
453:
452:
447:
442:
440:
439:
433:) = arctan(2)
428:
414:
408:
400:
392:
387:
378:
370:
365:
362:
360:
356:
352:
348:
347:Catalan solid
344:
340:
336:
333:
329:
325:
321:
317:
307:
298:
293:
289:
286:
281:
276:
272:
271:
268:
264:
260:
257:
253:
250:
249:
245:
243:
240:
239:
235:
233:
230:
229:
221:
215:
213:
210:
209:
205:
202:
201:
197:
194:
193:
189:
186:
185:
181:
178:
177:
169:
164:
162:
159:
158:
154:
152:
149:
148:
118:
116:
113:
112:
109:
108:Catalan solid
106:
103:
102:
99:
94:
89:
84:
81:
73:
70:
62:
59:December 2010
52:
48:
42:
41:
35:
30:
21:
20:
3985:Golden ratio
3949:
3868:trapezohedra
3819:
3812:
3771:
3616:dodecahedron
3205:Dodecahedron
2948:
2805:Dodecahedron
2499:Apeirohedron
2462:
2450:>20 faces
2401:Dodecahedron
2257:
2202:
2166:
2129:
2106:
2088:
2079:
2075:
2069:
2036:
2032:
2026:
2015:
2001:
1982:
1940:
1934:
1931:
1915:
1838:V(3.∞)
1817:
1700:
1684:
1652:
1200:
1075:
1065:Obtuse form
1031:
1018:
1014:golden ratio
1007:
668:
653:
618:
603:
589:
584:(±1, ±1, ±1)
569:
564:golden ratio
557:
534:
526:dodecahedron
511:
488:
460:transitively
449:
443:
436:
426:
407:acute angles
399:golden ratio
396:
377:golden ratio
337:. It has 60
319:
315:
313:
206:20{3}+12{5}
80:
65:
56:
37:
3638:semiregular
3621:icosahedron
3601:tetrahedron
3220:Icosahedron
3156:Tetrahedron
3141:Tetrahedron
3121:Archimedean
2790:Icosahedron
2741:Tetrahedron
2726:Tetrahedron
2441:Icosahedron
2389:11–20 faces
2375:Enneahedron
2365:Heptahedron
2355:Pentahedron
2350:Tetrahedron
2130:Dual Models
1947:roleplaying
1743:*∞32
1720:Hyperbolic
1175:Stellations
1056:Acute form
578:. Its dual
522:icosahedron
236:I, , (532)
226:, , (*532)
51:introducing
3964:Categories
3933:prismatoid
3863:bipyramids
3847:antiprisms
3821:hosohedron
3611:octahedron
3301:(Truncate)
3271:(Truncate)
3241:(Truncate)
3188:Octahedron
2758:Octahedron
2626:prismatoid
2611:(infinite)
2380:Decahedron
2370:Octahedron
2360:Hexahedron
2335:Monohedron
2328:1–10 faces
1975:References
1717:Euclidean
1678:V3.3.3.3.5
1663:V3.3.3.3.3
1090:Projective
1028:Dissection
665:Dimensions
514:tetrahedra
505:, and the
476:reflection
359:zonohedron
357:. It is a
349:, and the
324:polyhedron
267:zonohedron
251:Properties
34:references
3980:Zonohedra
3928:birotunda
3918:bifrustum
3683:snub cube
3578:polyhedra
3499:Snub cube
2640:antiprism
2345:Trihedron
2314:Polyhedra
2204:MathWorld
2061:123506315
2053:1572-9168
1926:STL model
1713:Spherical
1237:, (*532)
1092:symmetry
983:≈
914:≈
864:φ
847:φ
805:≈
736:≈
568:(0, ±1, ±
530:octahedra
259:isohedral
161:Face type
3908:bicupola
3888:pyramids
3814:dihedron
3446:(Expand)
3416:(Expand)
3386:(Expand)
2886:(Needle)
2856:(Needle)
2826:(Needle)
2340:Dihedron
2243:Archived
2234:IQ-light
2128:(1983),
2105:(1979).
2082:: 25–46.
1953:See also
1740:*832...
1668:V3.4.5.4
1653:V3.5.3.5
1648:V3.10.10
1240:, (532)
1232:Symmetry
1021:insphere
482:to face
472:rotation
343:vertices
330:with 30
263:isotoxal
254:convex,
195:Vertices
174:rhombus
165:V3.5.3.5
3950:italics
3938:scutoid
3923:rotunda
3913:frustum
3642:uniform
3591:regular
3576:Convex
3461:(Bevel)
3431:(Bevel)
3401:(Bevel)
3031:(Ortho)
3001:(Ortho)
2971:(Ortho)
2660:pyramid
2645:frustum
2160:0730208
1992:" from
1748:Tiling
1673:V4.6.10
1573:sr{5,3}
1568:tr{5,3}
1563:rr{5,3}
1012:is the
986:1.44721
917:1.37638
808:12.3107
739:26.8328
683:tangent
658:
644:
615:√
608:
594:
562:be the
516:, five
431:
417:
353:of the
341:and 32
332:rhombic
326:, is a
47:improve
3903:cupola
3856:duals:
3842:prisms
3523:(Snub)
3503:(Snub)
3483:(Snub)
3369:(Ambo)
3352:(Ambo)
3335:(Ambo)
3224:(Dual)
3209:(Seed)
3192:(Dual)
3177:(Seed)
3160:(Dual)
3145:(Seed)
3108:(Gyro)
3088:(Gyro)
3068:(Gyro)
3046:(Meta)
3016:(Meta)
2986:(Meta)
2954:(Join)
2937:(Join)
2920:(Join)
2809:(Seed)
2794:(Dual)
2777:(Seed)
2762:(Dual)
2745:(Seed)
2730:(Dual)
2650:cupola
2526:vertex
2266:, and
2201:") at
2173:
2158:
2148:
2113:
2059:
2051:
1833:V(3.8)
1828:V(3.7)
1823:V(3.6)
1818:V(3.5)
1813:V(3.4)
1808:V(3.3)
1701:V(3.n)
1658:V5.6.6
1643:V5.5.5
1553:t{3,5}
1548:r{5,3}
1543:t{5,3}
1140:image
1105:Image
1008:where
677:of an
675:radius
524:and a
501:, the
497:, the
413:arctan
36:, but
3316:(Zip)
3286:(Zip)
3256:(Zip)
3123:duals
2901:(Kis)
2871:(Kis)
2841:(Kis)
2655:wedge
2635:prism
2495:(132)
2057:S2CID
1803:Conf.
1737:*732
1734:*632
1731:*532
1728:*432
1725:*332
1558:{3,5}
1538:{5,3}
625:1.175
588:(0, ±
520:, an
518:cubes
339:edges
335:faces
246:144°
187:Edges
179:Faces
3606:cube
3173:Cube
2773:Cube
2617:(57)
2588:(92)
2582:(13)
2576:(13)
2545:(16)
2521:edge
2516:face
2489:(90)
2483:(60)
2477:(48)
2471:(32)
2465:(30)
2459:(24)
2197:" ("
2171:ISBN
2146:ISBN
2111:ISBN
2049:ISSN
1904:Uses
1708:*n32
1138:Dual
1019:The
641:and
617:3 –
558:Let
466:and
391:cube
314:The
104:Type
3640:or
2605:(4)
2570:(5)
2539:(9)
2501:(∞)
2193:, "
2138:doi
2041:doi
1989:, "
1943:d30
1212:097
1209:833
1207:358
1045:10
630:504
627:570
592:, ±
474:or
297:Net
222:, H
198:32
190:60
182:30
155:jD
3966::
2628:s
2262:,
2156:MR
2154:,
2144:,
2136:,
2132:,
2080:21
2078:.
2055:.
2047:.
2035:.
1234::
1215:.
1081:.
1042:10
1016:.
710:12
661:.
633:58
623:≈
532:.
509:.
486:.
441:.
411:2
401:,
379:.
361:.
287:)
265:,
261:,
3952:.
3644:)
3636:(
3593:)
3589:(
3569:e
3562:t
3555:v
2699:e
2692:t
2685:v
2306:e
2299:t
2292:v
2279:.
2207:.
2140::
2119:.
2063:.
2043::
2037:4
2009:.
1010:φ
989:a
975:a
970:)
963:5
959:1
954:+
951:1
947:(
943:=
933:m
928:r
920:a
906:a
897:5
893:2
888:+
885:1
880:=
877:a
868:2
860:+
857:1
851:2
841:=
831:i
826:r
816:3
812:a
795:3
791:a
782:5
777:2
774:+
771:5
766:4
763:=
756:V
747:2
743:a
726:2
722:a
715:5
707:=
700:S
681:(
671:a
654:φ
650:/
647:2
639:2
619:φ
610:)
604:φ
600:/
597:1
590:φ
572:)
570:φ
560:φ
484:B
480:A
468:B
464:A
427:φ
423:/
420:1
415:(
403:φ
283:(
224:3
219:h
217:I
72:)
66:(
61:)
57:(
43:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
