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1111:
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310:
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1427:
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1219:
1226:
1212:
1759:
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1191:
1259:
40:
1945:
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1137:
1130:
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1966:
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1158:
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1123:
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712:
1973:
1745:
1340:
1717:
505:
746:
the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of squares as five
762:
featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the
734:
the right amount, without changing the orientation or size of the faces, and patch the square holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of
2496:
Unus igitur
Trigonicus cum duobus Tetragonicis & uno Pentagonico, minus efficiunt 4 rectis, & congruunt 20 Trigonicum 30 Tetragonis & 12 Pentagonis, in unum Hexacontadyhedron, quod appello Rhombicoſidodecaëdron, ſeu ſectum Rhombum
707:{\displaystyle {\begin{aligned}A&=\left(30+5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&&\approx 59.305\,982\,844\,9a^{2}\\V&={\frac {60+29{\sqrt {5}}}{3}}a^{3}&&\approx 41.615\,323\,782\,5a^{3}\end{aligned}}}
785:
and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are
510:
153:
2982:
2665:: features this solid as the answer to an impromptu science quiz the main four characters have in Leonard and Sheldon's apartment, and is also illustrated in
2773:
2167:
2156:
418:
2189:
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2975:
2079:
2068:
2312:
2290:
2240:
2217:
767:
1033:, centered, on a vertex, on two types of edges, and three types of faces: triangles, squares, and pentagons. The last two correspond to the A
436:
2968:
2393:
2090:
2301:
2236:
2132:
2121:
1265:
810:
806:
278:
2486:]. Linz, Austria: Sumptibus Godofredi Tampachii bibl. Francof. excudebat Ioannes Plancus printed by Johann Planck]. p. 64.
2650:
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2057:
1866:
1362:
1277:
214:
1321:
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2143:
1660:
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1631:
1621:
1611:
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1544:
1534:
1505:
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814:
234:
206:
186:
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1457:
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196:
44:
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1568:
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1529:
1510:
1500:
1481:
1471:
1452:
1442:
900:. Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely
201:
191:
817:. Eight more can be constructed by removing up to three cupolae, sometimes also rotating one or more of the other cupolae.
2110:
802:
112:
2266:
2221:
1251:, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
3384:
3181:
3122:
2913:
2759:
2213:
2020:
1698:
1103:
2172:
2161:
3389:
3211:
3171:
2614:
2010:
1808:
1096:
992:
94:
2194:
2183:
1110:
3206:
3201:
2015:
2005:
1803:
1798:
331:
259:
2435:
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1307:
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476:
2084:
2073:
3312:
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1822:
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1302:
468:
326:
309:
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3052:
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3112:
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2801:
2422:
2345:
2137:
2126:
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1030:
825:
2473:
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2897:
2710:
2062:
343:
300:
3156:
3082:
3030:
2823:
1980:
1366:
1324:
1017:
828:
for the vertices of a rhombicosidodecahedron with an edge length of 2 centered at the origin are all
743:
739:
731:
723:
239:
219:
65:
2148:
3322:
3191:
3166:
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3087:
3035:
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2660:
2355:
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1995:
1778:
2937:
3337:
3302:
3161:
3056:
3005:
2688:
2365:
2209:
771:
264:
58:
2924:
2115:
104:
2919:
2334:
2261:
2250:
1765:
3317:
3127:
3102:
3046:
2947:
2869:
2840:
2829:
2796:
2782:
2707:
2684:
2646:
2622:
2567:
2542:
2524:
2487:
2429:
2418:
2382:
2307:
2046:
1834:
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1703:
1678:
798:
727:
480:
398:
390:
369:
365:
292:
54:
3256:
2818:
2807:
2478:
2440:
2296:
1985:
1958:
1923:
1826:
1426:
1419:
1344:
1240:
829:
787:
451:
348:
17:
1951:
1412:
1398:
1384:
1204:
1197:
479:(left), the one that creates the uniform solid (center), and the rectification of the dual
2272:
1758:
1751:
1405:
1391:
1377:
1225:
1218:
1211:
751:
446:
402:
379:
376:
372:
336:
178:
2728:
1190:
3077:
3000:
2812:
1944:
1937:
1930:
1730:
1258:
247:
168:
39:
2745:
2545:
3378:
3282:
3138:
3072:
2856:
2692:
2426:
2386:
2378:
2034:
1990:
1248:
1041:
797:
are derived from the rhombicosidodecahedron, four of them by rotation of one or more
794:
782:
484:
317:
2864:
1737:
1136:
1129:
2406:
1793:
1668:
1328:
1157:
1150:
1143:
897:
2851:
1965:
1723:
1122:
3015:
2666:
2402:
2051:
1773:
1688:
1332:
2476:[Book II. On the Congruence of Harmonic Figures. Proposition XXVIII.].
3347:
3235:
3025:
2992:
2740:
2511:
by
Johannes Kepler, Translated into English with an introduction and notes by
2414:
1972:
386:
2491:
3342:
3332:
3277:
3261:
3097:
2929:
2715:
2697:
2575:
2550:
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1339:
2570:
1825:
polyhedra with vertex figure (3.4.n.4), which continues as tilings of the
1744:
3228:
1347:
1282:
1270:
778:
472:
394:
357:
2619:
The
Geometrical Foundation of Natural Structure: A Source Book of Design
1716:
3352:
3327:
2474:"Liber II. De Congruentia Figurarum Harmonicarum. XXVIII. Propositio."
1320:
471:. There are different truncations of a rhombic triacontahedron into a
2751:
1294:
1821:
This polyhedron is topologically related as a part of a sequence of
2960:
919:
for edge length 2. For unit edge length, R must be halved, giving
2434:
1343:
A version with golden rectangles is used as vertex element of the
1319:
766:
The rhombicosidodecahedron shares the vertex arrangement with the
3020:
2224:(having the triangular and pentagonal faces in common), and the
2964:
2755:
1239:
The rhombicosidodecahedron can also be represented as a
375:
nonprismatic solids constructed of two or more types of
124:
742:
each of five cubes by moving the faces away from the
508:
115:
1046:
29:
3295:
3270:
3245:
3220:
3136:
3044:
2999:
2374:
2364:
2354:
2344:
2327:
148:{\displaystyle r{\begin{Bmatrix}5\\3\end{Bmatrix}}}
2663:Series 8 Episode 2 - The Junior Professor Solution
770:, and with the uniform compounds of six or twelve
706:
147:
2037:, 5 by diminishment, and 8 including gyrations:
2231:It also shares its vertex arrangement with the
1847:32 symmetry mutation of expanded tilings: 3.4.
495:For a rhombicosidodecahedron with edge length
2976:
2767:
8:
2641:. United Kingdom: Cambridge. pp. 79–86
763:very center of the rhombicosidodecahedron.
3249:
2983:
2969:
2961:
2774:
2760:
2752:
2729:"3D convex uniform polyhedra x3o5x - srid"
2417:of the rhombicosidodecahedron, one of the
1839:
1353:
1255:
694:
686:
682:
678:
661:
644:
632:
612:
604:
600:
596:
579:
561:
550:
537:
509:
507:
119:
114:
2245:
2103:
2039:
1357:Family of uniform icosahedral polyhedra
1338:
1253:
475:rhombicosidodecahedron: Prominently its
2464:
2484:The Harmony of the World in Five Books
2324:
2313:Compound of twelve pentagrammic prisms
2291:Small stellated truncated dodecahedron
2218:small stellated truncated dodecahedron
2208:The rhombicosidodecahedron shares its
768:small stellated truncated dodecahedron
2228:(having the square faces in common).
1243:, and projected onto the plane via a
7:
2711:"Small rhombicosidodecahedron graph"
2591:Read, R. C.; Wilson, R. J. (1998),
2302:Compound of six pentagrammic prisms
499:, its surface area and volume are:
461:truncated icosidodecahedral rhombus
2339:Pentagon centered Schlegel diagram
730:by moving the faces away from the
483:(right), which is the core of the
25:
1833:figures have (*n32) reflectional
1335:creates a rhombicosidodecahedron.
2941:
2936:
2923:
2918:
2907:
2896:
2885:
2874:
2863:
2850:
2839:
2828:
2817:
2806:
2795:
2453:Truncated rhombicosidodecahedron
2333:
2306:
2295:
2284:
2271:
2260:
2249:
2193:
2182:
2171:
2160:
2147:
2136:
2125:
2114:
2094:
2083:
2072:
2061:
2050:
1971:
1964:
1957:
1950:
1943:
1936:
1929:
1922:
1764:
1757:
1750:
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1729:
1722:
1715:
1658:
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1629:
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1614:
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1600:
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1551:
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1537:
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1527:
1522:
1513:
1508:
1503:
1498:
1493:
1484:
1479:
1474:
1469:
1464:
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1445:
1440:
1435:
1425:
1418:
1411:
1404:
1397:
1390:
1383:
1376:
1288:
1276:
1264:
1257:
1224:
1217:
1210:
1203:
1196:
1189:
1156:
1149:
1142:
1135:
1128:
1121:
1109:
1102:
1095:
1002:
991:
815:trigyrate rhombicosidodecahedron
435:
426:
417:
342:
325:
308:
299:
204:
199:
194:
189:
184:
38:
455:(1618) named this polyhedron a
45:(Click here for rotating model)
2394:Table of graphs and parameters
1:
2748:The Encyclopedia of Polyhedra
750:Two clusters of faces of the
3363:Degenerate polyhedra are in
2689:Small Rhombicosidodecahedron
2411:rhombicosidodecahedral graph
2328:Rhombicosidodecahedral graph
2321:Rhombicosidodecahedral graph
2267:Small dodecicosidodecahedron
2222:small dodecicosidodecahedron
2154:
2108:
2044:
1163:
18:Small rhombicosidodecahedron
3182:pentagonal icositetrahedron
3123:truncated icosidodecahedron
2914:Truncated icosidodecahedron
2673:at the end of that episode.
2621:. Dover Publications, Inc.
2415:graph of vertices and edges
2214:nonconvex uniform polyhedra
1710:Duals to uniform polyhedra
3406:
3212:pentagonal hexecontahedron
3172:deltoidal icositetrahedron
2241:twelve pentagrammic prisms
2105:Gyrated and/or diminished
1012:Orthogonal projections in
254:4-5: 148°16′57″ (148.28°)
3361:
3252:
3207:disdyakis triacontahedron
3202:deltoidal hexecontahedron
2789:
2746:Virtual Reality Polyhedra
2472:Ioannis Keppler (1619).
2392:
2332:
1875:
1865:
1855:
1842:
1709:
1361:
1356:
1308:Stereographic projections
1306:
465:icosidodecahedral rhombus
332:Deltoidal hexecontahedron
252:3-4: 159°05′41″ (159.09°)
103:
50:
37:
32:
2479:Harmonices Mundi Libri V
2425:and 120 edges, and is a
2278:Small rhombidodecahedron
2226:small rhombidodecahedron
1245:stereographic projection
3313:gyroelongated bipyramid
3187:rhombic triacontahedron
3093:truncated cuboctahedron
2892:Truncated cuboctahedron
2597:Oxford University Press
1303:Orthographic projection
1048:Orthogonal projections
469:rhombic triacontahedron
33:Rhombicosidodecahedron
3308:truncated trapezohedra
3177:disdyakis dodecahedron
3143:(duals of Archimedean)
3118:rhombicosidodecahedron
3108:truncated dodecahedron
2903:Rhombicosidodecahedron
2835:Truncated dodecahedron
2509:Harmonies Of The World
2443:
2256:Rhombicosidodecahedron
1351:
1336:
1031:orthogonal projections
1027:rhombicosidodecahedron
983:Orthogonal projections
738:Alternatively, if you
708:
457:rhombicosidodecahedron
362:rhombicosidodecahedron
229:, , (*532), order 120
149:
3197:pentakis dodecahedron
3113:truncated icosahedron
3068:truncated tetrahedron
2846:Truncated icosahedron
2802:Truncated tetrahedron
2741:The Uniform Polyhedra
2637:Cromwell, P. (1997).
2497:Icoſidododecaëdricum.
2438:
2033:There are 12 related
1342:
1323:
1247:. This projection is
826:Cartesian coordinates
821:Cartesian coordinates
709:
467:being his name for a
150:
3157:rhombic dodecahedron
3083:truncated octahedron
2824:Truncated octahedron
1018:Augustin Hirschvogel
506:
242:, , (532), order 60
113:
3192:triakis icosahedron
3167:tetrakis hexahedron
3152:triakis tetrahedron
3088:rhombicuboctahedron
2881:Rhombicuboctahedron
2727:Klitzing, Richard.
2661:The Big Bang Theory
2546:"Icosahedral group"
2106:
2042:
1049:
772:pentagrammic prisms
718:Geometric relations
3385:Archimedean solids
3162:triakis octahedron
3047:Archimedean solids
2783:Archimedean solids
2708:Weisstein, Eric W.
2685:Weisstein, Eric W.
2643:Archimedean solids
2593:An Atlas of Graphs
2568:Weisstein, Eric W.
2543:Weisstein, Eric W.
2444:
2419:Archimedean solids
2210:vertex arrangement
2204:Vertex arrangement
2104:
2040:
1817:Symmetry mutations
1352:
1337:
1047:
799:pentagonal cupolae
704:
702:
459:, being short for
393:faces, 12 regular
385:It has 20 regular
368:, one of thirteen
145:
139:
89:20{3}+30{4}+12{5}
59:Uniform polyhedron
3390:Uniform polyhedra
3372:
3371:
3291:
3290:
3128:snub dodecahedron
3103:icosidodecahedron
2958:
2957:
2953:
2952:
2948:Snub dodecahedron
2870:Icosidodecahedron
2693:Archimedean solid
2430:Archimedean graph
2399:
2398:
2318:
2317:
2233:uniform compounds
2201:
2200:
2102:
2101:
2026:
2025:
1831:vertex-transitive
1814:
1813:
1316:Related polyhedra
1313:
1312:
1232:
1231:
830:even permutations
793:Twelve of the 92
788:golden rectangles
728:icosidodecahedron
655:
649:
568:
566:
542:
481:icosidodecahedron
366:Archimedean solid
354:
353:
55:Archimedean solid
27:Archimedean solid
16:(Redirected from
3397:
3250:
3246:Dihedral uniform
3221:Dihedral regular
3144:
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3009:
2985:
2978:
2971:
2962:
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2927:
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2854:
2843:
2832:
2821:
2810:
2799:
2792:
2791:
2776:
2769:
2762:
2753:
2732:
2721:
2720:
2702:
2671:Vanity Card #461
2656:
2632:
2615:Williams, Robert
2601:
2600:
2588:
2582:
2581:
2580:
2563:
2557:
2556:
2555:
2538:
2532:
2506:
2500:
2499:
2469:
2441:Schlegel diagram
2439:Square centered
2337:
2325:
2310:
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2197:
2186:
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2087:
2076:
2065:
2054:
2043:
1975:
1968:
1961:
1954:
1947:
1940:
1933:
1926:
1876:Compact hyperb.
1840:
1827:hyperbolic plane
1768:
1761:
1754:
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1733:
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1719:
1663:
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1443:
1439:
1438:
1429:
1422:
1415:
1408:
1401:
1394:
1387:
1380:
1354:
1345:construction set
1292:
1280:
1268:
1261:
1254:
1241:spherical tiling
1235:Spherical tiling
1228:
1221:
1214:
1207:
1200:
1193:
1160:
1153:
1146:
1139:
1132:
1125:
1113:
1106:
1099:
1050:
1029:has six special
1006:
995:
977:
975:
974:
971:
968:
967:
966:
965:
964:
950:
948:
947:
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940:
939:
918:
917:
909:
908:
895:
893:
892:
889:
886:
885:
884:
781:kits for making
713:
711:
710:
705:
703:
699:
698:
668:
666:
665:
656:
651:
650:
645:
633:
617:
616:
586:
584:
583:
574:
570:
569:
567:
562:
551:
543:
538:
452:Harmonices Mundi
439:
430:
421:
346:
329:
312:
303:
209:
208:
207:
203:
202:
198:
197:
193:
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188:
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154:
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146:
144:
143:
105:Schläfli symbols
42:
30:
21:
3405:
3404:
3400:
3399:
3398:
3396:
3395:
3394:
3375:
3374:
3373:
3368:
3357:
3296:Dihedral others
3287:
3266:
3241:
3216:
3145:
3142:
3141:
3132:
3061:
3050:
3049:
3040:
3003:
3001:Platonic solids
2995:
2989:
2959:
2954:
2946:
2928:
2912:
2901:
2890:
2879:
2868:
2855:
2844:
2833:
2822:
2811:
2800:
2785:
2780:
2726:
2706:
2705:
2683:
2680:
2653:
2636:
2629:
2613:
2610:
2605:
2604:
2590:
2589:
2585:
2566:
2565:
2564:
2560:
2541:
2540:
2539:
2535:
2507:
2503:
2471:
2470:
2466:
2461:
2449:
2340:
2323:
2311:
2300:
2289:
2276:
2265:
2254:
2206:
2192:
2181:
2170:
2159:
2146:
2135:
2124:
2113:
2093:
2082:
2071:
2060:
2049:
2031:
1914:
1909:
1905:
1901:
1897:
1893:
1889:
1885:
1863:
1857:
1819:
1659:
1654:
1649:
1644:
1639:
1637:
1630:
1625:
1620:
1615:
1610:
1608:
1601:
1596:
1591:
1586:
1581:
1579:
1572:
1567:
1562:
1557:
1552:
1550:
1543:
1538:
1533:
1528:
1523:
1521:
1514:
1509:
1504:
1499:
1494:
1492:
1485:
1480:
1475:
1470:
1465:
1463:
1456:
1451:
1446:
1441:
1436:
1434:
1318:
1293:
1281:
1269:
1237:
1185:
1166:
1080:
1075:
1070:
1065:
1060:
1040:
1036:
1023:
1022:
1021:
1020:
1009:
1008:
1007:
998:
997:
996:
985:
972:
969:
962:
960:
958:
956:
955:
954:
952:
945:
942:
933:
931:
930:
929:
927:
913:
911:
903:
901:
890:
887:
882:
880:
878:
877:
875:
823:
752:bilunabirotunda
720:
701:
700:
690:
667:
657:
634:
625:
619:
618:
608:
585:
575:
527:
523:
516:
504:
503:
493:
447:Johannes Kepler
444:
443:
442:
441:
440:
432:
431:
423:
422:
411:
377:regular polygon
347:
337:dual polyhedron
334:
330:
315:
313:
304:
283:
276:
269:
253:
228:
223:
205:
200:
195:
190:
185:
183:
179:Coxeter diagram
162:
138:
137:
131:
130:
120:
111:
110:
95:Conway notation
57:
43:
28:
23:
22:
15:
12:
11:
5:
3403:
3401:
3393:
3392:
3387:
3377:
3376:
3370:
3369:
3362:
3359:
3358:
3356:
3355:
3350:
3345:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3299:
3297:
3293:
3292:
3289:
3288:
3286:
3285:
3280:
3274:
3272:
3268:
3267:
3265:
3264:
3259:
3253:
3247:
3243:
3242:
3240:
3239:
3232:
3224:
3222:
3218:
3217:
3215:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3179:
3174:
3169:
3164:
3159:
3154:
3148:
3146:
3139:Catalan solids
3137:
3134:
3133:
3131:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3085:
3080:
3078:truncated cube
3075:
3070:
3064:
3062:
3045:
3042:
3041:
3039:
3038:
3033:
3028:
3023:
3018:
3012:
3010:
2997:
2996:
2990:
2988:
2987:
2980:
2973:
2965:
2956:
2955:
2951:
2950:
2933:
2932:
2916:
2905:
2894:
2883:
2872:
2860:
2859:
2848:
2837:
2826:
2815:
2813:Truncated cube
2804:
2790:
2787:
2786:
2781:
2779:
2778:
2771:
2764:
2756:
2750:
2749:
2743:
2738:
2733:
2724:
2723:
2722:
2679:
2678:External links
2676:
2675:
2674:
2657:
2651:
2634:
2627:
2609:
2606:
2603:
2602:
2583:
2558:
2533:
2501:
2463:
2462:
2460:
2457:
2456:
2455:
2448:
2445:
2397:
2396:
2390:
2389:
2376:
2372:
2371:
2368:
2362:
2361:
2358:
2352:
2351:
2348:
2342:
2341:
2338:
2330:
2329:
2322:
2319:
2316:
2315:
2304:
2293:
2281:
2280:
2269:
2258:
2205:
2202:
2199:
2198:
2187:
2176:
2165:
2153:
2152:
2141:
2130:
2119:
2100:
2099:
2088:
2077:
2066:
2055:
2035:Johnson solids
2030:
2029:Johnson solids
2027:
2024:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1977:
1976:
1969:
1962:
1955:
1948:
1941:
1934:
1927:
1920:
1916:
1915:
1911:
1906:
1902:
1898:
1894:
1890:
1886:
1881:
1880:
1877:
1874:
1869:
1864:
1853:
1852:
1818:
1815:
1812:
1811:
1806:
1801:
1796:
1791:
1786:
1781:
1776:
1770:
1769:
1762:
1755:
1748:
1741:
1734:
1727:
1720:
1712:
1711:
1707:
1706:
1701:
1696:
1691:
1686:
1681:
1676:
1671:
1665:
1664:
1635:
1606:
1577:
1548:
1519:
1490:
1461:
1431:
1430:
1423:
1416:
1409:
1402:
1395:
1388:
1381:
1373:
1372:
1369:
1359:
1358:
1317:
1314:
1311:
1310:
1305:
1299:
1298:
1286:
1274:
1262:
1236:
1233:
1230:
1229:
1222:
1215:
1208:
1201:
1194:
1187:
1181:
1180:
1178:
1176:
1174:
1172:
1170:
1168:
1162:
1161:
1154:
1147:
1140:
1133:
1126:
1119:
1115:
1114:
1107:
1100:
1093:
1091:
1089:
1087:
1083:
1082:
1077:
1072:
1067:
1062:
1057:
1054:
1042:Coxeter planes
1038:
1034:
1011:
1010:
1001:
1000:
999:
990:
989:
988:
987:
986:
984:
981:
980:
979:
978:≈ 2.233.
868:
867:
856:
841:
822:
819:
795:Johnson solids
783:geodesic domes
719:
716:
715:
714:
697:
693:
689:
685:
681:
677:
674:
671:
669:
664:
660:
654:
648:
643:
640:
637:
631:
628:
626:
624:
621:
620:
615:
611:
607:
603:
599:
595:
592:
589:
587:
582:
578:
573:
565:
560:
557:
554:
549:
546:
541:
536:
533:
530:
526:
522:
519:
517:
515:
512:
511:
492:
489:
434:
433:
425:
424:
416:
415:
414:
413:
412:
410:
407:
352:
351:
340:
322:
321:
306:
305:Colored faces
296:
295:
289:
285:
284:
281:
274:
267:
262:
256:
255:
250:
248:Dihedral angle
244:
243:
237:
235:Rotation group
231:
230:
226:
221:
217:
215:Symmetry group
211:
210:
181:
175:
174:
171:
169:Wythoff symbol
165:
164:
160:
156:
155:
142:
136:
133:
132:
129:
126:
125:
123:
118:
107:
101:
100:
97:
91:
90:
87:
86:Faces by sides
83:
82:
68:
62:
61:
52:
48:
47:
35:
34:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3402:
3391:
3388:
3386:
3383:
3382:
3380:
3366:
3360:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3298:
3294:
3284:
3281:
3279:
3276:
3275:
3273:
3269:
3263:
3260:
3258:
3255:
3254:
3251:
3248:
3244:
3238:
3237:
3233:
3231:
3230:
3226:
3225:
3223:
3219:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3149:
3147:
3140:
3135:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3074:
3073:cuboctahedron
3071:
3069:
3066:
3065:
3063:
3058:
3054:
3048:
3043:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3013:
3011:
3007:
3002:
2998:
2994:
2986:
2981:
2979:
2974:
2972:
2967:
2966:
2963:
2949:
2944:
2939:
2935:
2934:
2931:
2926:
2921:
2917:
2915:
2910:
2906:
2904:
2899:
2895:
2893:
2888:
2884:
2882:
2877:
2873:
2871:
2866:
2862:
2861:
2858:
2857:Cuboctahedron
2853:
2849:
2847:
2842:
2838:
2836:
2831:
2827:
2825:
2820:
2816:
2814:
2809:
2805:
2803:
2798:
2794:
2793:
2788:
2784:
2777:
2772:
2770:
2765:
2763:
2758:
2757:
2754:
2747:
2744:
2742:
2739:
2737:
2734:
2730:
2725:
2718:
2717:
2712:
2709:
2704:
2703:
2700:
2699:
2694:
2690:
2686:
2682:
2681:
2677:
2672:
2668:
2664:
2662:
2658:
2654:
2652:0-521-55432-2
2648:
2644:
2640:
2635:
2633:(Section 3-9)
2630:
2628:0-486-23729-X
2624:
2620:
2616:
2612:
2611:
2607:
2599:, p. 269
2598:
2594:
2587:
2584:
2578:
2577:
2572:
2569:
2562:
2559:
2553:
2552:
2547:
2544:
2537:
2534:
2530:
2529:0-87169-209-0
2526:
2522:
2518:
2514:
2510:
2505:
2502:
2498:
2493:
2489:
2485:
2481:
2480:
2475:
2468:
2465:
2458:
2454:
2451:
2450:
2446:
2442:
2437:
2433:
2431:
2428:
2427:quartic graph
2424:
2420:
2416:
2412:
2408:
2404:
2395:
2391:
2388:
2384:
2380:
2379:Quartic graph
2377:
2373:
2369:
2367:
2366:Automorphisms
2363:
2359:
2357:
2353:
2349:
2347:
2343:
2336:
2331:
2326:
2320:
2314:
2309:
2305:
2303:
2298:
2294:
2292:
2287:
2283:
2282:
2279:
2274:
2270:
2268:
2263:
2259:
2257:
2252:
2248:
2247:
2244:
2242:
2238:
2234:
2229:
2227:
2223:
2219:
2215:
2211:
2203:
2196:
2191:
2188:
2185:
2180:
2177:
2174:
2169:
2166:
2163:
2158:
2155:
2150:
2145:
2142:
2139:
2134:
2131:
2128:
2123:
2120:
2117:
2112:
2109:
2097:
2092:
2089:
2086:
2081:
2078:
2075:
2070:
2067:
2064:
2059:
2056:
2053:
2048:
2045:
2038:
2036:
2028:
2022:
2021:3.4.∞.4
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1978:
1974:
1970:
1967:
1963:
1960:
1956:
1953:
1949:
1946:
1942:
1939:
1935:
1932:
1928:
1925:
1921:
1918:
1917:
1912:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1882:
1878:
1873:
1870:
1868:
1861:
1854:
1850:
1846:
1841:
1838:
1836:
1832:
1828:
1824:
1816:
1810:
1807:
1805:
1802:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1780:
1777:
1775:
1772:
1771:
1767:
1763:
1760:
1756:
1753:
1749:
1746:
1742:
1739:
1735:
1732:
1728:
1725:
1721:
1718:
1714:
1713:
1708:
1705:
1702:
1700:
1697:
1695:
1692:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1670:
1667:
1666:
1636:
1607:
1578:
1549:
1520:
1491:
1462:
1433:
1432:
1428:
1424:
1421:
1417:
1414:
1410:
1407:
1403:
1400:
1396:
1393:
1389:
1386:
1382:
1379:
1375:
1374:
1370:
1367:
1364:
1360:
1355:
1349:
1346:
1341:
1334:
1330:
1326:
1322:
1315:
1309:
1304:
1301:
1300:
1296:
1291:
1287:
1284:
1279:
1275:
1272:
1267:
1263:
1260:
1256:
1252:
1250:
1246:
1242:
1234:
1227:
1223:
1220:
1216:
1213:
1209:
1206:
1202:
1199:
1195:
1192:
1188:
1183:
1182:
1179:
1177:
1175:
1173:
1171:
1169:
1164:
1159:
1155:
1152:
1148:
1145:
1141:
1138:
1134:
1131:
1127:
1124:
1120:
1117:
1116:
1112:
1108:
1105:
1101:
1098:
1094:
1092:
1090:
1088:
1085:
1084:
1078:
1073:
1068:
1063:
1058:
1055:
1052:
1051:
1045:
1043:
1032:
1028:
1019:
1015:
1005:
994:
982:
937:
925:
922:
921:
920:
916:
906:
899:
874: =
873:
865:
861:
857:
854:
850:
846:
842:
839:
835:
834:
833:
831:
827:
820:
818:
816:
812:
808:
804:
800:
796:
791:
789:
784:
780:
775:
773:
769:
764:
761:
757:
753:
748:
745:
741:
736:
733:
729:
725:
717:
695:
691:
687:
683:
679:
675:
672:
670:
662:
658:
652:
646:
641:
638:
635:
629:
627:
622:
613:
609:
605:
601:
597:
593:
590:
588:
580:
576:
571:
563:
558:
555:
552:
547:
544:
539:
534:
531:
528:
524:
520:
518:
513:
502:
501:
500:
498:
490:
488:
486:
485:dual compound
482:
478:
477:rectification
474:
470:
466:
462:
458:
454:
453:
448:
438:
429:
420:
408:
406:
404:
400:
396:
392:
388:
383:
381:
378:
374:
371:
367:
363:
359:
350:
345:
341:
338:
333:
328:
324:
323:
319:
318:Vertex figure
311:
307:
302:
298:
297:
294:
290:
287:
286:
280:
273:
266:
263:
261:
258:
257:
251:
249:
246:
245:
241:
238:
236:
233:
232:
224:
218:
216:
213:
212:
182:
180:
177:
176:
173:3 5 | 2
172:
170:
167:
166:
158:
157:
140:
134:
127:
121:
116:
108:
106:
102:
98:
96:
93:
92:
88:
85:
84:
81:= 60 (χ = 2)
80:
76:
72:
69:
67:
64:
63:
60:
56:
53:
49:
46:
41:
36:
31:
19:
3364:
3283:trapezohedra
3234:
3227:
3117:
3031:dodecahedron
2902:
2714:
2696:
2670:
2659:
2642:
2638:
2618:
2592:
2586:
2574:
2561:
2549:
2536:
2520:
2517:A. M. Duncan
2516:
2512:
2508:
2504:
2495:
2483:
2477:
2467:
2421:. It has 60
2410:
2407:graph theory
2403:mathematical
2400:
2255:
2230:
2207:
2032:
2000:
1859:
1848:
1844:
1820:
1693:
1329:dodecahedron
1327:of either a
1238:
1053:Centered by
1026:
1024:
1013:
935:
923:
914:
904:
898:golden ratio
871:
869:
863:
859:
852:
848:
844:
837:
824:
811:metabigyrate
807:parabigyrate
792:
776:
765:
759:
755:
749:
737:
721:
496:
494:
464:
460:
456:
450:
445:
384:
361:
355:
291:Semiregular
78:
74:
70:
3053:semiregular
3036:icosahedron
3016:tetrahedron
2667:Chuck Lorre
2521:J. V. Field
2513:E. J. Aiton
2383:Hamiltonian
2212:with three
2041:Diminished
1823:cantellated
1333:icosahedron
473:topological
109:rr{5,3} or
3379:Categories
3348:prismatoid
3278:bipyramids
3262:antiprisms
3236:hosohedron
3026:octahedron
2608:References
2531:(page 123)
2375:Properties
1913:*∞32
1879:Paracomp.
1809:V3.3.3.3.5
1794:V3.3.3.3.3
1297:-centered
1285:-centered
1273:-centered
1165:Projective
1118:Wireframe
1016:(1543) by
836:(±1, ±1, ±
491:Dimensions
401:, and 120
397:faces, 60
395:pentagonal
389:faces, 30
387:triangular
288:Properties
260:References
99:eD or aaD
3343:birotunda
3333:bifrustum
3098:snub cube
2993:polyhedra
2930:Snub cube
2716:MathWorld
2698:MathWorld
2639:Polyhedra
2576:MathWorld
2551:MathWorld
2492:863358134
2405:field of
1867:Spherical
1368:, (*532)
1325:Expansion
1249:conformal
1167:symmetry
1081:Pentagon
1076:Triangle
1014:Geometria
673:≈
591:≈
3323:bicupola
3303:pyramids
3229:dihedron
2617:(1979).
2523:, 1997,
2447:See also
2423:vertices
2346:Vertices
1856:Symmetry
1835:symmetry
1829:. These
1799:V3.4.5.4
1784:V3.5.3.5
1779:V3.10.10
1371:, (532)
1363:Symmetry
1348:Zometool
1283:Triangle
1271:Pentagon
779:Zometool
735:either.
399:vertices
373:isogonal
358:geometry
66:Elements
3365:italics
3353:scutoid
3338:rotunda
3328:frustum
3057:uniform
3006:regular
2991:Convex
2413:is the
2401:In the
2387:regular
2016:3.4.8.4
2011:3.4.7.4
2006:3.4.6.4
2001:3.4.5.4
1996:3.4.4.4
1991:3.4.3.4
1986:3.4.2.4
1981:Config.
1919:Figure
1872:Euclid.
1804:V4.6.10
1704:sr{5,3}
1699:tr{5,3}
1694:rr{5,3}
1071:Square
1056:Vertex
976:
961:√
957:√
953:
949:
932:√
928:
912:√
902:√
896:is the
894:
881:√
876:
862:), 0, ±
747:cubes.
722:If you
463:, with
314:3.4.5.4
77:= 120,
3318:cupola
3271:duals:
3257:prisms
2695:") at
2649:
2625:
2571:"Zome"
2527:
2490:
2220:, the
2216:: the
1789:V5.6.6
1774:V5.5.5
1684:t{3,5}
1679:r{5,3}
1674:t{5,3}
1331:or an
1295:Square
1186:image
1086:Solid
870:where
813:, and
803:gyrate
801:: the
758:(each
754:, the
744:origin
740:expand
732:origin
724:expand
676:41.615
594:59.305
391:square
370:convex
364:is an
360:, the
293:convex
163:{5,3}
73:= 62,
2482:[
2459:Notes
2356:Edges
1689:{3,5}
1669:{5,3}
1037:and H
858:(±(2+
756:lunes
409:Names
403:edges
380:faces
3021:cube
2691:" ("
2647:ISBN
2623:ISBN
2525:ISBN
2488:OCLC
2409:, a
1910:...
1908:*832
1904:*732
1900:*632
1896:*532
1892:*432
1888:*332
1884:*232
1184:Dual
1079:Face
1074:Face
1069:Face
1066:5-4
1064:Edge
1061:3-4
1059:Edge
1025:The
959:11+4
915:8φ+7
879:1 +
851:, ±2
832:of:
777:The
760:lune
51:Type
3055:or
2687:, "
2669:'s
2370:120
2360:120
2239:or
2237:six
2235:of
1851:.4
847:, ±
726:an
684:782
680:323
602:844
598:982
449:in
356:In
349:Net
225:, H
161:0,2
3381::
2713:.
2645:.
2595:,
2573:.
2548:.
2519:,
2515:,
2494:.
2432:.
2385:,
2381:,
2350:60
2243:.
2190:82
2179:79
2168:78
2157:77
2144:75
2133:74
2122:73
2111:72
2091:83
2080:81
2069:80
2058:76
2047:J5
1862:32
1837:.
1365::
1044:.
951:=
938:+7
926:=
910:=
907:+2
866:),
855:),
843:(±
840:),
809:,
805:,
790:.
774:.
642:29
636:60
559:10
553:25
529:30
487:.
405:.
382:.
339:)
320:)
282:14
277:,
275:30
270:,
268:27
3367:.
3059:)
3051:(
3008:)
3004:(
2984:e
2977:t
2970:v
2775:e
2768:t
2761:v
2731:.
2719:.
2701:.
2655:.
2631:.
2579:.
2554:.
1860:n
1858:*
1849:n
1845:n
1843:*
1350:.
1039:2
1035:2
973:2
970:/
963:5
946:2
943:/
936:φ
934:8
924:R
905:φ
891:2
888:/
883:5
872:φ
864:φ
860:φ
853:φ
849:φ
845:φ
838:φ
696:3
692:a
688:5
663:3
659:a
653:3
647:5
639:+
630:=
623:V
614:2
610:a
606:9
581:2
577:a
572:)
564:5
556:+
548:3
545:+
540:3
535:5
532:+
525:(
521:=
514:A
497:a
335:(
316:(
279:W
272:C
265:U
240:I
227:3
222:h
220:I
159:t
141:}
135:3
128:5
122:{
117:r
79:V
75:E
71:F
20:)
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