3515:
1094:
1051:
1782:
1680:
69:
39:
1565:
1331:
1085:
1042:
1649:
1532:
1415:
1298:
25:
55:
1214:
2669:
3473:
1448:
2880:
2903:
1941:
1228:
2896:
2873:
2554:
1921:
1901:
1579:
1345:
1076:
1033:
3545:
3059:
3035:
2014:
1994:
1462:
1572:
3277:
3263:
3506:
1640:
1406:
1338:
3321:
3307:
3233:
3219:
3054:
3030:
2910:
2887:
3272:
3258:
3011:
3006:
3316:
3302:
3249:
3244:
1974:
3228:
3214:
3293:
3288:
1868:
1523:
1221:
3205:
3200:
1455:
1289:
1852:
1828:
3133:
3109:
2662:
2547:
2319:
2297:
2207:
2196:
2158:
2147:
1812:
3128:
3104:
2286:
2275:
2185:
2174:
2136:
2122:
3590:
3085:
3080:
2655:
2540:
2308:
2264:
2106:
2095:
3565:
3395:
the regular convex dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons. Further, he recognized that these star pentagons are also regular. In this
487:
A Kepler–Poinsot polyhedron covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the figures which have pentagrammic faces, and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as
Platonic
470:
which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true
450:
In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines
3396:
way he constructed the two stellated dodecahedra. Each has the central convex region of each face "hidden" within the interior, with only the triangular arms visible. Kepler's final step was to recognize that these polyhedra fit the definition of regularity, even though they were not
3411:
rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron. Some people call these two the
3761:"These figures are so closely related the one to the dodecahedron the other to the icosahedron that the latter two figures, particularly the dodecahedron, seem somehow truncated or maimed when compared to the figures with spikes." (
402:
329:
211:
2532:
2504:
544:
does not always hold. Schläfli held that all polyhedra must have χ = 2, and he rejected the small stellated dodecahedron and great dodecahedron as proper polyhedra. This view was never widely held.
2762:
2716:
2647:
2601:
1016:
969:
922:
873:
826:
779:
3776:"A small stellated dodecahedron can be constructed by cumulation of a dodecahedron, i.e., building twelve pentagonal pyramids and attaching them to the faces of the original dodecahedron."
3335:
Most, if not all, of the Kepler-Poinsot polyhedra were known of in some form or other before Kepler. A small stellated dodecahedron appears in a marble tarsia (inlay panel) on the floor of
3141:
3117:
3067:
3043:
257:
2443:
3093:
3019:
682:
1059:
451:
intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges.
1102:
539:
715:. Two relationships described in the article below are also easily seen in the images: That the violet edges are the same, and that the green faces lie in the same planes.
1619:
1472:
1385:
1238:
1599:
1589:
1502:
1482:
1365:
1355:
1268:
1248:
1609:
1492:
1375:
1258:
2404:
1614:
1497:
1380:
1263:
1604:
1594:
1487:
1477:
1370:
1360:
1253:
1243:
611:
584:
471:
ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear. Now
213:
times the original icosahedron edge length. The small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron edge lengths are respectively
3808:"Another way to construct a great stellated dodecahedron via cumulation is to make 20 triangular pyramids and attach them to the sides of an icosahedron."
4180:
2245:
The great stellated dodecahedron shares its vertices with the dodecahedron. The other three Kepler–Poinsot polyhedra share theirs with the icosahedron.
2234:
3671:
334:
261:
143:
2831:
This is just a help to visualize the shape of these solids, and not actually a claim that the edge intersections (false vertices) are vertices.
2510:
2482:
3881:
3172:
1755:
changes pentagonal faces into pentagrams. (In this sense stellation is a unique operation, and not to be confused with the more general
3597:
Regular star polyhedra first appear in
Renaissance art. A small stellated dodecahedron is depicted in a marble tarsia on the floor of
4173:
2938:
resemble the icosahedron and its dual in that they have faces and vertices on the 3-fold (yellow) and 5-fold (red) symmetry axes.
4280:
4494:
4069:
3979:
3962:
3865:
3514:
475:
holds: 60 − 90 + 32 = 2. However, this polyhedron is no longer the one described by the
2722:
2676:
2607:
2561:
3974:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
4166:
978:
4398:
4383:
4368:
4285:
4085:
4043:
931:
884:
835:
788:
741:
4413:
4388:
4373:
3380:. It is clear from the general arrangement of the book that he regarded only the five Platonic solids as regular.
617:, and holds both for convex polyhedra (where the correction factors are all 1), and the Kepler–Poinsot polyhedra:
4408:
4403:
3532:
3352:
4363:
3970:
3841:
1781:
1679:
2827:
states that the two star polyhedra can be constructed by adding pyramids to the faces of the
Platonic solids.
216:
4393:
2409:
1093:
1050:
4504:
4330:
4320:
4260:
4250:
4220:
4210:
3527:
3458:
3377:
3365:
3183:
3179:
2995:
2984:
2945:
2935:
2371:
2367:
2323:
2301:
2211:
2200:
2162:
2151:
2070:
1714:
1556:
1322:
926:
736:
455:
421:
417:
74:
44:
479:{5/2, 5}, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.
4335:
4325:
4275:
623:
4458:
4448:
4345:
4340:
4142:
3614:
2815:
In his view the great stellation is related to the icosahedron as the small one is to the dodecahedron.
1564:
1330:
68:
38:
2948:
all faces and vertices are on 5-fold symmetry axes (so there are no yellow elements in these images).
4463:
4453:
3982:
2057:
1084:
1041:
497:
3573:
4499:
4473:
4468:
4255:
3641:
3472:
2775:
This implies that the pentagrams have the same size, and that the cores have the same edge length.
2384:
The pentagon faces of these cores are the invisible parts of the star polyhedra's pentagram faces.
4270:
4265:
3691:
3681:
2959:
1648:
3814:
3782:
3733:
Conway et al. (2008), p.405 Figure 26.1 Relationships among the three-dimensional star-polytopes
4378:
3336:
1531:
1414:
1297:
1149:
472:
4147:
2668:
2389:
127:. They can all be seen as three-dimensional analogues of the pentagram in one way or another.
4057:
3950:
3923:
3622:
3598:
3481:
3373:
2952:
2839:
2222:
If the intersections are treated as new edges and vertices, the figures obtained will not be
1213:
24:
3190:
have the same edge length, namely the side length of a pentagram in the surrounding decagon.
2879:
1940:
1447:
54:
2973:
2951:
The following table shows the solids in pairs of duals. In the top row they are shown with
2924:
2902:
2820:
2375:
2312:
2110:
2063:
1227:
1183:
1173:
589:
562:
489:
3454:
1920:
1900:
1710:
1578:
1344:
8:
4305:
4107:
3686:
3626:
3544:
2969:
2843:
2268:
2099:
2053:
973:
3612:'s interest in geometric forms often led to works based on or including regular solids;
3058:
3034:
2895:
2872:
2553:
4315:
4310:
4245:
4215:
4053:
3946:
3676:
3666:
3523:
3450:
3369:
3276:
3262:
2980:
2941:
2379:
2339:
2279:
2223:
2189:
2140:
1696:
1684:
1639:
1405:
1205:
1188:
783:
548:
467:
432:
92:
30:
3505:
2013:
1993:
1571:
1461:
1337:
1143:
1075:
1032:
476:
4433:
4225:
4104:
4081:
4065:
4039:
3975:
3958:
3861:
3845:
3811:
3779:
3569:
3446:
gave the Kepler–Poinsot polyhedra the names by which they are generally known today.
3397:
3320:
3306:
3187:
2991:
2955:, in the bottom row with icosahedral symmetry (to which the mentioned colors refer).
2931:
2348:
2290:
2178:
2126:
2045:
1439:
879:
436:
106:
60:
3232:
3218:
4443:
4438:
4136:
4031:
3853:
3763:
3748:
3661:
3550:
3361:
3053:
3029:
2909:
2886:
4126:
3850:
Shaping Space: Exploring
Polyhedra in Nature, Art, and the Geometrical Imagination
3271:
3257:
4189:
4027:, Cambridge University Press (1976) - discussion of proof of Euler characteristic
3909:
3555:
3432:
3420:
3388:
3010:
3005:
1198:
830:
693:
120:
95:
3633:
3315:
3301:
3248:
3243:
4297:
3428:
3401:
3227:
3213:
1193:
1178:
700:, or more precisely, Petrie polygons with the same two dimensional projection.
697:
440:
4131:
3857:
3292:
3287:
1973:
1765:
maintains the type of faces, shifting and resizing them into parallel planes.
1683:
Conway's system of relations between the six polyhedra (ordered vertically by
4488:
4010:
4000:
3933:
3602:
3490:
3443:
3408:
3344:
3204:
3199:
2864:
1288:
1167:
704:
614:
556:
444:
124:
3416:. Poinsot did not know if he had discovered all the regular star polyhedra.
1522:
1220:
397:{\displaystyle \phi ^{5}={\tfrac {1}{2}}{\bigl (}11+5{\sqrt {5}}\,{\bigr )}}
4158:
3649:
3609:
3132:
3108:
2661:
2546:
2448:
2318:
2296:
2206:
2195:
2157:
2146:
1454:
712:
466:
part hidden inside the solid. The visible parts of each face comprise five
425:
324:{\displaystyle \phi ^{2}={\tfrac {1}{2}}{\bigl (}3+{\sqrt {5}}\,{\bigr )},}
206:{\displaystyle \phi ^{4}={\tfrac {1}{2}}{\bigl (}7+3{\sqrt {5}}\,{\bigr )}}
109:
1867:
3645:
113:
2962:
from the 5-fold (red), 3-fold (yellow) and 2-fold (blue) symmetry axes.
2799:
i.e. as dodecahedron and icosahedron with pyramids added to their faces.
1851:
1827:
4152:
4121:
3767:, Book II, Proposition XXVI — p. 117 in the translation by E. J. Aiton)
3695:
3424:
3392:
3343:, Italy. It dates from the 15th century and is sometimes attributed to
2248:
2227:
2049:
1756:
1136:
102:
3127:
3103:
2527:{\displaystyle {\frac {\text{core midradius}}{\text{hull midradius}}}}
2499:{\displaystyle {\frac {\text{hull midradius}}{\text{core midradius}}}}
2285:
2274:
2184:
2173:
2135:
2121:
1811:
4112:
3819:
3809:
3787:
3777:
3589:
3162:
2824:
2770:
2456:
1788:
708:
459:
428:
413:
117:
3084:
3079:
2654:
2539:
2307:
2263:
2105:
2094:
3436:
2835:
2252:
2074:
463:
84:
3601:, Venice, Italy, dating from ca. 1430 and sometimes attributed to
3752:, Book V, Chapter III — p. 407 in the translation by E. J. Aiton)
3457:
for stellations in up to four dimensions. Within this scheme the
3166:
3722:
Star polytopes and the Schläfli function f(α,β,γ)
3564:
3383:
The small and great stellated dodecahedra, sometimes called the
2459:
is a common measure to compare the size of different polyhedra.)
3485:
3477:
3340:
4072:(Chapter 26. pp. 404: Regular star-polytopes Dimension 3)
4102:
4007:. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989.
3584:
1674:
4155:: Software used to create many of the images on this page.
3742:"augmented dodecahedron to which I have given the name of
2789:
Traditionally the two star polyhedra have been defined as
3892:
H.S.M. Coxeter, P. Du Val, H.T. Flather and J.F. Petrie;
3698:, 4-dimensional analogues of the Kepler–Poinsot polyhedra
2382:
with their edges and faces extended until they intersect.
3625:
of the great dodecahedron was used for the 1980s puzzle
4080:. California: University of California Press Berkeley.
3965:(Chapter 24, Regular Star-polytopes, pp. 404–408)
1787:
The polyhedra in this section are shown with the same
352:
279:
161:
3914:
Comptes rendus des séances de l'Académie des
Sciences
2777:(Compare the 5-fold orthographic projections below.)
2757:{\displaystyle {\frac {3-{\sqrt {5}}}{2}}=0.38196...}
2725:
2711:{\displaystyle {\frac {3+{\sqrt {5}}}{2}}=2.61803...}
2679:
2642:{\displaystyle {\frac {{\sqrt {5}}-1}{2}}=0.61803...}
2610:
2596:{\displaystyle {\frac {{\sqrt {5}}+1}{2}}=1.61803...}
2564:
2513:
2485:
2412:
2392:
981:
934:
887:
838:
791:
744:
626:
592:
565:
500:
337:
264:
219:
146:
4127:
Free paper models (nets) of Kepler–Poinsot polyhedra
2406:
times bigger than the core, and for the great it is
2079:The three others are facetings of the icosahedron.
1691:
2756:
2710:
2641:
2595:
2526:
2498:
2437:
2398:
1010:
963:
916:
867:
820:
773:
676:
605:
578:
533:
396:
323:
251:
205:
3994:Star Polytopes and the Schlafli Function f(α,β,γ)
3431:, and almost half a century after that, in 1858,
3161:The platonic hulls in these images have the same
2769:The platonic hulls in these images have the same
2386:For the small stellated dodecahedron the hull is
416:(star pentagons) as faces or vertex figures. The
4486:
3844:(2013). "Regular and semiregular polyhedra". In
2062:The three others are all the stellations of the
1011:{\displaystyle \left\{{\frac {10}{3,5}}\right\}}
3912:, Note sur la théorie des polyèdres réguliers,
3644:. The star spans 14 meters, and consists of an
3165:, so all the 5-fold projections below are in a
964:{\displaystyle \left\{{\frac {5}{2}},3\right\}}
917:{\displaystyle \left\{3,{\frac {5}{2}}\right\}}
868:{\displaystyle \left\{{\frac {6}{1,3}}\right\}}
821:{\displaystyle \left\{5,{\frac {5}{2}}\right\}}
774:{\displaystyle \left\{{\frac {5}{2}},5\right\}}
2834:If they were, the two star polyhedra would be
711:. They also show that the Petrie polygons are
687:
4174:
404:times the original dodecahedron edge length.
389:
365:
313:
292:
198:
174:
4188:
4075:
3882:File:Perspectiva Corporum Regularium 27e.jpg
3618:is based on a small stellated dodecahedron.
2356:
2240:
2039:
4143:VRML models of the Kepler–Poinsot polyhedra
3930:J. de l'École Polytechnique 9, 68–86, 1813.
2466:Hull and core of the stellated dodecahedra
4181:
4167:
4148:Stellation and facetting - a brief history
4013:, Memoire sur les polygones et polyèdres.
3852:(2nd ed.). Springer. pp. 41–52.
547:A modified form of Euler's formula, using
116:, and differ from these in having regular
4030:
3585:Regular star polyhedra in art and culture
3360:), a book of woodcuts published in 1568,
1746:great stellated dodecahedron (sgD = gsD)
1675:Relationships among the regular polyhedra
482:
386:
310:
195:
4122:Paper models of Kepler–Poinsot polyhedra
3936:, On Poinsot's Four New Regular Solids.
3588:
3563:
3543:
3471:
2374:stellated dodecahedron can be seen as a
2012:
1992:
1972:
1939:
1919:
1899:
1699:defines the Kepler–Poinsot polyhedra as
1678:
252:{\displaystyle \phi ^{3}=2+{\sqrt {5}},}
3840:
3652:inside a great stellated dodecahedron.
2923:All Kepler–Poinsot polyhedra have full
2853:Stellated dodecahedra as augmentations
2438:{\displaystyle \varphi +1=\varphi ^{2}}
4487:
4281:nonconvex great rhombicosidodecahedron
3724:p. 121 1. The Kepler–Poinsot polyhedra
3387:, were first recognized as regular by
692:The Kepler–Poinsot polyhedra exist in
4162:
4103:
3810:
3778:
2803:Kepler calls the small stellation an
2335:
140:The great icosahedron edge length is
3969:Kaleidoscopes: Selected Writings of
3372:(both shown below). There is also a
2251:of the solids sharing vertices are
2235:List of Wenninger polyhedron models
2226:, but they can still be considered
677:{\displaystyle d_{v}V-E+d_{f}F=2D.}
13:
4092:Chapter 8: Kepler Poisot polyhedra
3896:, 3rd Edition, Tarquin, 1999. p.11
3358:Perspectives of the regular solids
703:The following images show the two
488:solids are, and in particular the
443:polygonal faces, but pentagrammic
130:
14:
4516:
4399:great stellapentakis dodecahedron
4384:medial pentagonal hexecontahedron
4369:small stellapentakis dodecahedron
4286:great truncated icosidodecahedron
4096:
3943:, pp. 123–127 and 209, 1859.
3435:provided a more elegant proof by
3391:around 1619. He obtained them by
2823:definitions are still used. E.g.
4414:great pentagonal hexecontahedron
4389:medial disdyakis triacontahedron
4374:medial deltoidal hexecontahedron
3513:
3504:
3319:
3314:
3305:
3300:
3291:
3286:
3275:
3270:
3261:
3256:
3247:
3242:
3231:
3226:
3217:
3212:
3203:
3198:
3131:
3126:
3107:
3102:
3083:
3078:
3057:
3052:
3033:
3028:
3009:
3004:
2927:, just like their convex hulls.
2908:
2901:
2894:
2885:
2878:
2871:
2784:
2667:
2660:
2653:
2552:
2545:
2538:
2361:
2317:
2306:
2295:
2284:
2273:
2262:
2205:
2194:
2183:
2172:
2156:
2145:
2134:
2120:
2104:
2093:
1866:
1850:
1826:
1810:
1780:
1692:Conway's operational terminology
1647:
1638:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1577:
1570:
1563:
1530:
1521:
1500:
1495:
1490:
1485:
1480:
1475:
1470:
1460:
1453:
1446:
1413:
1404:
1383:
1378:
1373:
1368:
1363:
1358:
1353:
1343:
1336:
1329:
1296:
1287:
1266:
1261:
1256:
1251:
1246:
1241:
1236:
1226:
1219:
1212:
1092:
1083:
1074:
1049:
1040:
1031:
407:
67:
53:
37:
23:
4409:great disdyakis triacontahedron
4404:great deltoidal hexecontahedron
3903:
3533:Perspectiva Corporum Regularium
3353:Perspectiva corporum regularium
2347:share vertices, skeletons form
534:{\displaystyle \chi =V-E+F=2\ }
16:Any of 4 regular star polyhedra
4364:medial rhombic triacontahedron
4038:. Cambridge University Press.
4003:, (The Kepler–Poinsot Solids)
3886:
3875:
3834:
3815:"Great Stellated Dodecahedron"
3802:
3783:"Small Stellated Dodecahedron"
3770:
3755:
3736:
3727:
3714:
2994:) and {5/2, 3} (
2983:) and {5/2, 5} (
2215:(the one with yellow vertices)
1:
4394:great rhombic triacontahedron
3928:Recherches sur les polyèdres.
3702:
1771:Conway relations illustrated
1551:great stellated dodecahedron
1317:small stellated dodecahedron
4331:great dodecahemidodecahedron
4321:small dodecahemidodecahedron
4261:truncated dodecadodecahedron
4251:truncated great dodecahedron
4221:great stellated dodecahedron
4211:small stellated dodecahedron
4153:Stella: Polyhedron Navigator
4078:Polyhedra: A Visual Approach
3920:(1858), pp. 79–82, 117.
3608:In the 20th century, artist
3528:great stellated dodecahedron
3459:small stellated dodecahedron
3423:proved the list complete by
3378:small stellated dodecahedron
3366:great stellated dodecahedron
2972:) and {5, 3} (
2324:great stellated dodecahedron
2302:small stellated dodecahedron
2071:great stellated dodecahedron
1738:stellated dodecahedron (sD)
1715:small stellated dodecahedron
1557:great stellated dodecahedron
1554:
1437:
1323:small stellated dodecahedron
1320:
1203:
927:great stellated dodecahedron
737:small stellated dodecahedron
456:small stellated dodecahedron
422:great stellated dodecahedron
75:Great stellated dodecahedron
45:Small stellated dodecahedron
7:
4336:great icosihemidodecahedron
4326:small icosihemidodecahedron
4276:truncated great icosahedron
4015:J. de l'École Polytechnique
3992:(Paper 10) H.S.M. Coxeter,
3655:
2918:
2130:(the one with yellow faces)
696:pairs. Duals have the same
688:Duality and Petrie polygons
10:
4521:
4459:great dodecahemidodecacron
4449:small dodecahemidodecacron
4346:small dodecahemicosahedron
4341:great dodecahemicosahedron
3986:(Paper 1) H.S.M. Coxeter,
3330:
3173:projection of the compound
2085:Stellations and facetings
1115:
4464:great icosihemidodecacron
4454:small icosihemidodecacron
4422:
4354:
4294:
4234:
4196:
3894:The Fifty-Nine Icosahedra
3858:10.1007/978-0-387-92714-5
3672:List of regular polytopes
3159:
3155:orthographic projections
3154:
2852:
2767:
2465:
2357:The stellated dodecahedra
2346:
2333:share vertices and edges
2332:
2330:share vertices and edges
2329:
2241:Shared vertices and edges
2204:
2119:
2103:
2092:
2084:
2058:The Fifty-Nine Icosahedra
2040:Stellations and facetings
1770:
89:Kepler–Poinsot polyhedron
4495:Kepler–Poinsot polyhedra
4474:small dodecahemicosacron
4469:great dodecahemicosacron
4256:rhombidodecadodecahedron
4190:Star-polyhedra navigator
4062:The Symmetries of Things
3872:See in particular p. 42.
3707:
3642:Oslo Airport, Gardermoen
2960:orthographic projections
2399:{\displaystyle \varphi }
1735:great dodecahedron (gD)
1125:(Conway's abbreviation)
722:horizontal edge in front
135:
101:They may be obtained by
4271:great icosidodecahedron
4266:snub dodecadodecahedron
4020:, pp. 16–48, 1810.
3988:The Nine Regular Solids
3694:– the ten regular star
3692:Regular star 4-polytope
3682:Uniform star polyhedron
3548:Stellated dodecahedra,
3449:A hundred years later,
1743:great icosahedron (gI)
462:faces with the central
4425:uniform polyhedra with
4379:small rhombidodecacron
4108:"Kepler–Poinsot solid"
4025:Proofs and Refutations
4005:The Joy of Mathematics
3955:The Symmetry of Things
3594:
3577:
3559:
3495:
3463:stellated dodecahedron
3455:systematic terminology
2958:The table below shows
2805:augmented dodecahedron
2758:
2712:
2643:
2597:
2528:
2500:
2439:
2400:
2028:
2008:
1988:
1955:
1935:
1915:
1719:stellated dodecahedron
1688:
1103:Compound of gI and gsD
1012:
965:
918:
869:
822:
775:
727:vertical edge in front
678:
607:
580:
535:
483:Euler characteristic χ
398:
325:
253:
207:
4137:Kepler-Poinsot Solids
4132:The Uniform Polyhedra
4076:Anthony Pugh (1976).
4058:Chaim Goodman-Strauss
3951:Chaim Goodman-Strauss
3924:Augustin-Louis Cauchy
3592:
3568:Cardboard model of a
3567:
3547:
3475:
3400:, as the traditional
2953:pyritohedral symmetry
2840:pentakis dodecahedron
2759:
2713:
2644:
2598:
2529:
2501:
2440:
2401:
2016:
1996:
1976:
1943:
1923:
1903:
1707:of the convex solids.
1682:
1105:with Petrie decagrams
1060:Compound of sD and gD
1013:
966:
919:
870:
823:
776:
679:
608:
606:{\displaystyle d_{f}}
581:
579:{\displaystyle d_{v}}
536:
399:
326:
254:
208:
4427:infinite stellations
4235:Uniform truncations
3442:The following year,
2925:icosahedral symmetry
2807:(then nicknaming it
2723:
2677:
2608:
2562:
2511:
2483:
2410:
2390:
2077:of the dodecahedron.
1062:with Petrie hexagons
979:
932:
885:
836:
789:
742:
624:
590:
563:
498:
335:
262:
217:
144:
4505:Nonconvex polyhedra
4355:Duals of nonconvex
4306:tetrahemihexahedron
4139:in Visual Polyhedra
3687:Polyhedral compound
3599:St. Mark's Basilica
3574:Tübingen University
3419:Three years later,
3337:St. Mark's Basilica
2844:triakis icosahedron
1434:great dodecahedron
468:isosceles triangles
412:These figures have
4423:Duals of nonconvex
4316:octahemioctahedron
4311:cubohemioctahedron
4295:Nonconvex uniform
4246:dodecadodecahedron
4237:of Kepler-Poinsot
4216:great dodecahedron
4204:regular polyhedra)
4105:Weisstein, Eric W.
3846:Senechal, Marjorie
3812:Weisstein, Eric W.
3780:Weisstein, Eric W.
3677:Uniform polyhedron
3667:Regular polyhedron
3640:is displayed near
3595:
3578:
3560:
3524:Great dodecahedron
3496:
3370:great dodecahedron
3178:This implies that
3169:of the same size.
2942:great dodecahedron
2838:equivalent to the
2754:
2708:
2639:
2593:
2524:
2496:
2435:
2396:
2380:great dodecahedron
2340:dodecahedral graph
2280:great dodecahedron
2029:
2009:
1989:
1956:
1936:
1916:
1759:described below.)
1689:
1668:great icosahedron
1206:great dodecahedron
1008:
961:
914:
865:
818:
784:great dodecahedron
771:
674:
603:
576:
531:
433:great dodecahedron
394:
361:
321:
288:
249:
203:
170:
31:Great dodecahedron
4482:
4481:
4434:tetrahemihexacron
4357:uniform polyhedra
4226:great icosahedron
4070:978-1-56881-220-5
4056:, Heidi Burgiel,
4050:, pp. 39–41.
4032:Wenninger, Magnus
3980:978-0-471-01003-6
3963:978-1-56881-220-5
3949:, Heidi Burgiel,
3867:978-0-387-92713-8
3842:Coxeter, H. S. M.
3632:Norwegian artist
3582:
3581:
3570:great icosahedron
3494:
3414:Poinsot polyhedra
3328:
3327:
3150:
3149:
3145:
3121:
3097:
3071:
3047:
3023:
2932:great icosahedron
2916:
2915:
2782:
2781:
2746:
2740:
2700:
2694:
2631:
2619:
2585:
2573:
2522:
2521:
2518:
2494:
2493:
2490:
2460:
2354:
2353:
2349:icosahedral graph
2291:great icosahedron
2220:
2219:
2216:
2131:
2046:great icosahedron
2037:
2036:
2033:
2032:
1960:
1959:
1887:
1886:
1792:
1750:
1749:
1730:dodecahedron (D)
1711:naming convention
1672:
1671:
1440:great icosahedron
1113:
1112:
1021:
1020:
1002:
948:
907:
880:great icosahedron
859:
811:
758:
530:
454:For example, the
437:great icosahedron
426:nonconvex regular
384:
360:
308:
287:
244:
193:
169:
61:Great icosahedron
4512:
4444:octahemioctacron
4439:hexahemioctacron
4183:
4176:
4169:
4160:
4159:
4118:
4117:
4091:
4049:
3971:H. S. M. Coxeter
3897:
3890:
3884:
3879:
3873:
3871:
3838:
3832:
3831:
3830:
3828:
3827:
3806:
3800:
3799:
3798:
3796:
3795:
3774:
3768:
3764:Harmonices Mundi
3759:
3753:
3749:Harmonices Mundi
3740:
3734:
3731:
3725:
3718:
3662:Regular polytope
3627:Alexander's Star
3593:Alexander's Star
3551:Harmonices Mundi
3517:
3508:
3488:
3468:
3467:
3385:Kepler polyhedra
3362:Wenzel Jamnitzer
3324:
3323:
3318:
3310:
3309:
3304:
3296:
3295:
3290:
3280:
3279:
3274:
3266:
3265:
3260:
3252:
3251:
3246:
3236:
3235:
3230:
3222:
3221:
3216:
3208:
3207:
3202:
3191:
3176:
3152:
3151:
3139:
3136:
3135:
3130:
3115:
3112:
3111:
3106:
3091:
3088:
3087:
3082:
3065:
3062:
3061:
3056:
3041:
3038:
3037:
3032:
3017:
3014:
3013:
3008:
2965:
2964:
2912:
2905:
2898:
2889:
2882:
2875:
2861:Star polyhedron
2850:
2849:
2847:
2832:
2816:
2800:
2763:
2761:
2760:
2755:
2747:
2742:
2741:
2736:
2727:
2717:
2715:
2714:
2709:
2701:
2696:
2695:
2690:
2681:
2671:
2664:
2657:
2648:
2646:
2645:
2640:
2632:
2627:
2620:
2615:
2612:
2602:
2600:
2599:
2594:
2586:
2581:
2574:
2569:
2566:
2556:
2549:
2542:
2533:
2531:
2530:
2525:
2523:
2519:
2516:
2515:
2505:
2503:
2502:
2497:
2495:
2491:
2488:
2487:
2474:Star polyhedron
2463:
2462:
2454:
2452:
2444:
2442:
2441:
2436:
2434:
2433:
2405:
2403:
2402:
2397:
2342:
2336:share vertices,
2321:
2310:
2299:
2288:
2277:
2266:
2259:
2258:
2256:
2214:
2209:
2198:
2187:
2176:
2160:
2149:
2138:
2129:
2124:
2108:
2097:
2082:
2081:
2027:
2021:
2007:
2001:
1987:
1981:
1969:
1968:
1954:
1948:
1934:
1928:
1914:
1908:
1896:
1895:
1878:
1870:
1861:
1854:
1837:
1830:
1821:
1814:
1802:
1801:
1785:
1784:
1768:
1767:
1727:icosahedron (I)
1724:
1723:
1651:
1642:
1622:
1621:
1620:
1616:
1615:
1611:
1610:
1606:
1605:
1601:
1600:
1596:
1595:
1591:
1590:
1581:
1574:
1567:
1534:
1525:
1505:
1504:
1503:
1499:
1498:
1494:
1493:
1489:
1488:
1484:
1483:
1479:
1478:
1474:
1473:
1464:
1457:
1450:
1417:
1408:
1388:
1387:
1386:
1382:
1381:
1377:
1376:
1372:
1371:
1367:
1366:
1362:
1361:
1357:
1356:
1347:
1340:
1333:
1300:
1291:
1271:
1270:
1269:
1265:
1264:
1260:
1259:
1255:
1254:
1250:
1249:
1245:
1244:
1240:
1239:
1230:
1223:
1216:
1120:
1119:
1096:
1087:
1078:
1053:
1044:
1035:
1023:
1022:
1017:
1015:
1014:
1009:
1007:
1003:
1001:
987:
970:
968:
967:
962:
960:
956:
949:
941:
923:
921:
920:
915:
913:
909:
908:
900:
874:
872:
871:
866:
864:
860:
858:
844:
827:
825:
824:
819:
817:
813:
812:
804:
780:
778:
777:
772:
770:
766:
759:
751:
728:
723:
718:
717:
683:
681:
680:
675:
658:
657:
636:
635:
612:
610:
609:
604:
602:
601:
585:
583:
582:
577:
575:
574:
540:
538:
537:
532:
528:
403:
401:
400:
395:
393:
392:
385:
380:
369:
368:
362:
353:
347:
346:
330:
328:
327:
322:
317:
316:
309:
304:
296:
295:
289:
280:
274:
273:
258:
256:
255:
250:
245:
240:
229:
228:
212:
210:
209:
204:
202:
201:
194:
189:
178:
177:
171:
162:
156:
155:
71:
57:
41:
27:
4520:
4519:
4515:
4514:
4513:
4511:
4510:
4509:
4500:Johannes Kepler
4485:
4484:
4483:
4478:
4426:
4424:
4418:
4356:
4350:
4296:
4290:
4238:
4236:
4230:
4203:
4199:
4198:Kepler-Poinsot
4192:
4187:
4099:
4088:
4046:
4023:Lakatos, Imre;
3906:
3901:
3900:
3891:
3887:
3880:
3876:
3868:
3839:
3835:
3825:
3823:
3807:
3803:
3793:
3791:
3775:
3771:
3760:
3756:
3741:
3737:
3732:
3728:
3719:
3715:
3710:
3705:
3658:
3638:The Kepler Star
3587:
3556:Johannes Kepler
3540:
3539:
3538:
3537:
3520:
3519:
3518:
3510:
3509:
3429:Platonic solids
3421:Augustin Cauchy
3402:Platonic solids
3389:Johannes Kepler
3376:version of the
3333:
3313:
3299:
3285:
3269:
3255:
3241:
3225:
3211:
3197:
3177:
3170:
3137:
3125:
3113:
3101:
3089:
3077:
3063:
3051:
3039:
3027:
3015:
3003:
2939:
2921:
2833:
2830:
2814:
2798:
2787:
2776:
2774:
2735:
2728:
2726:
2724:
2721:
2720:
2689:
2682:
2680:
2678:
2675:
2674:
2614:
2613:
2611:
2609:
2606:
2605:
2568:
2567:
2565:
2563:
2560:
2559:
2514:
2512:
2509:
2508:
2486:
2484:
2481:
2480:
2453:
2446:
2429:
2425:
2411:
2408:
2407:
2391:
2388:
2387:
2385:
2383:
2364:
2359:
2338:skeletons form
2337:
2322:
2311:
2300:
2289:
2278:
2267:
2246:
2243:
2210:
2199:
2188:
2177:
2161:
2150:
2139:
2125:
2109:
2098:
2078:
2061:
2042:
2023:
2017:
2003:
1997:
1983:
1977:
1950:
1944:
1930:
1924:
1910:
1904:
1883:
1882:
1881:
1880:
1879:
1873:
1871:
1863:
1862:
1857:
1855:
1842:
1841:
1840:
1839:
1838:
1833:
1831:
1823:
1822:
1817:
1815:
1786:
1708:
1694:
1677:
1665:
1652:
1643:
1634:
1626:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1586:
1585:
1559:
1548:
1535:
1526:
1517:
1509:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1469:
1468:
1442:
1431:
1418:
1409:
1400:
1392:
1384:
1379:
1374:
1369:
1364:
1359:
1354:
1352:
1351:
1325:
1314:
1301:
1292:
1283:
1275:
1267:
1262:
1257:
1252:
1247:
1242:
1237:
1235:
1234:
1208:
1172:
1169:
1163:
1155:
1148:
1147:{p, q} and
1146:
1139:
1132:
1124:
1118:
1109:
1108:
1107:
1106:
1099:
1098:
1097:
1089:
1088:
1080:
1079:
1066:
1065:
1064:
1063:
1056:
1055:
1054:
1046:
1045:
1037:
1036:
991:
986:
982:
980:
977:
976:
940:
939:
935:
933:
930:
929:
899:
892:
888:
886:
883:
882:
848:
843:
839:
837:
834:
833:
803:
796:
792:
790:
787:
786:
750:
749:
745:
743:
740:
739:
731:Petrie polygon
726:
721:
690:
653:
649:
631:
627:
625:
622:
621:
613:) was given by
597:
593:
591:
588:
587:
570:
566:
564:
561:
560:
499:
496:
495:
485:
477:Schläfli symbol
473:Euler's formula
410:
388:
387:
379:
364:
363:
351:
342:
338:
336:
333:
332:
312:
311:
303:
291:
290:
278:
269:
265:
263:
260:
259:
239:
224:
220:
218:
215:
214:
197:
196:
188:
173:
172:
160:
151:
147:
145:
142:
141:
138:
133:
131:Characteristics
91:is any of four
81:
80:
79:
78:
77:
72:
64:
63:
58:
49:
48:
47:
42:
34:
33:
28:
17:
12:
11:
5:
4518:
4508:
4507:
4502:
4497:
4480:
4479:
4477:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4441:
4436:
4430:
4428:
4420:
4419:
4417:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4376:
4371:
4366:
4360:
4358:
4352:
4351:
4349:
4348:
4343:
4338:
4333:
4328:
4323:
4318:
4313:
4308:
4302:
4300:
4292:
4291:
4289:
4288:
4283:
4278:
4273:
4268:
4263:
4258:
4253:
4248:
4242:
4240:
4232:
4231:
4229:
4228:
4223:
4218:
4213:
4207:
4205:
4194:
4193:
4186:
4185:
4178:
4171:
4163:
4157:
4156:
4150:
4145:
4140:
4134:
4129:
4124:
4119:
4098:
4097:External links
4095:
4094:
4093:
4086:
4073:
4054:John H. Conway
4051:
4044:
4028:
4021:
4008:
3998:
3997:
3996:
3990:
3966:
3947:John H. Conway
3944:
3931:
3921:
3905:
3902:
3899:
3898:
3885:
3874:
3866:
3833:
3801:
3769:
3754:
3735:
3726:
3712:
3711:
3709:
3706:
3704:
3701:
3700:
3699:
3689:
3684:
3679:
3674:
3669:
3664:
3657:
3654:
3586:
3583:
3580:
3579:
3561:
3541:
3522:
3521:
3512:
3511:
3503:
3502:
3501:
3500:
3499:
3497:
3332:
3329:
3326:
3325:
3311:
3297:
3282:
3281:
3267:
3253:
3238:
3237:
3223:
3209:
3194:
3193:
3157:
3156:
3148:
3147:
3123:
3099:
3074:
3073:
3049:
3025:
3000:
2999:
2988:
2977:
2920:
2917:
2914:
2913:
2906:
2899:
2891:
2890:
2883:
2876:
2868:
2867:
2862:
2859:
2855:
2854:
2786:
2783:
2780:
2779:
2765:
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2745:
2739:
2734:
2731:
2718:
2707:
2704:
2699:
2693:
2688:
2685:
2672:
2665:
2658:
2650:
2649:
2638:
2635:
2630:
2626:
2623:
2618:
2603:
2592:
2589:
2584:
2580:
2577:
2572:
2557:
2550:
2543:
2535:
2534:
2520:hull midradius
2517:core midradius
2506:
2492:core midradius
2489:hull midradius
2478:
2475:
2472:
2468:
2467:
2445:times bigger.
2432:
2428:
2424:
2421:
2418:
2415:
2395:
2363:
2360:
2358:
2355:
2352:
2351:
2344:
2343:
2334:
2331:
2327:
2326:
2315:
2304:
2293:
2282:
2271:
2242:
2239:
2218:
2217:
2203:
2192:
2181:
2170:
2166:
2165:
2154:
2143:
2132:
2118:
2114:
2113:
2102:
2091:
2087:
2086:
2048:is one of the
2041:
2038:
2035:
2034:
2031:
2030:
2010:
1990:
1966:
1962:
1961:
1958:
1957:
1937:
1917:
1893:
1889:
1888:
1885:
1884:
1872:
1865:
1864:
1856:
1849:
1848:
1847:
1846:
1845:
1843:
1832:
1825:
1824:
1816:
1809:
1808:
1807:
1806:
1805:
1799:
1795:
1794:
1777:
1773:
1772:
1748:
1747:
1744:
1740:
1739:
1736:
1732:
1731:
1728:
1693:
1690:
1676:
1673:
1670:
1669:
1666:
1663:
1660:
1657:
1654:
1645:
1636:
1631:
1628:
1623:
1582:
1575:
1568:
1561:
1553:
1552:
1549:
1546:
1543:
1540:
1537:
1528:
1519:
1514:
1511:
1506:
1465:
1458:
1451:
1444:
1436:
1435:
1432:
1429:
1426:
1423:
1420:
1411:
1402:
1397:
1394:
1389:
1348:
1341:
1334:
1327:
1319:
1318:
1315:
1312:
1309:
1306:
1303:
1294:
1285:
1280:
1277:
1272:
1231:
1224:
1217:
1210:
1202:
1201:
1196:
1191:
1186:
1181:
1179:Petrie polygon
1176:
1165:
1160:
1157:
1152:
1150:Coxeter-Dynkin
1141:
1134:
1129:
1126:
1117:
1114:
1111:
1110:
1101:
1100:
1091:
1090:
1082:
1081:
1073:
1072:
1071:
1070:
1069:
1067:
1058:
1057:
1048:
1047:
1039:
1038:
1030:
1029:
1028:
1027:
1026:
1019:
1018:
1006:
1000:
997:
994:
990:
985:
971:
959:
955:
952:
947:
944:
938:
924:
912:
906:
903:
898:
895:
891:
876:
875:
863:
857:
854:
851:
847:
842:
828:
816:
810:
807:
802:
799:
795:
781:
769:
765:
762:
757:
754:
748:
733:
732:
729:
724:
707:with the same
705:dual compounds
698:Petrie polygon
689:
686:
685:
684:
673:
670:
667:
664:
661:
656:
652:
648:
645:
642:
639:
634:
630:
600:
596:
573:
569:
557:vertex figures
542:
541:
527:
524:
521:
518:
515:
512:
509:
506:
503:
490:Euler relation
484:
481:
445:vertex figures
409:
406:
391:
383:
378:
375:
372:
367:
359:
356:
350:
345:
341:
320:
315:
307:
302:
299:
294:
286:
283:
277:
272:
268:
248:
243:
238:
235:
232:
227:
223:
200:
192:
187:
184:
181:
176:
168:
165:
159:
154:
150:
137:
134:
132:
129:
125:vertex figures
96:star polyhedra
73:
66:
65:
59:
52:
51:
50:
43:
36:
35:
29:
22:
21:
20:
19:
18:
15:
9:
6:
4:
3:
2:
4517:
4506:
4503:
4501:
4498:
4496:
4493:
4492:
4490:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4431:
4429:
4421:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4361:
4359:
4353:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4303:
4301:
4299:
4298:hemipolyhedra
4293:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4257:
4254:
4252:
4249:
4247:
4244:
4243:
4241:
4233:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4208:
4206:
4201:
4195:
4191:
4184:
4179:
4177:
4172:
4170:
4165:
4164:
4161:
4154:
4151:
4149:
4146:
4144:
4141:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4115:
4114:
4109:
4106:
4101:
4100:
4089:
4087:0-520-03056-7
4083:
4079:
4074:
4071:
4067:
4063:
4059:
4055:
4052:
4047:
4045:0-521-54325-8
4041:
4037:
4033:
4029:
4026:
4022:
4019:
4016:
4012:
4011:Louis Poinsot
4009:
4006:
4002:
4001:Theoni Pappas
3999:
3995:
3991:
3989:
3985:
3984:
3983:
3981:
3977:
3973:
3972:
3967:
3964:
3960:
3956:
3952:
3948:
3945:
3942:
3939:
3935:
3934:Arthur Cayley
3932:
3929:
3925:
3922:
3919:
3915:
3911:
3908:
3907:
3895:
3889:
3883:
3878:
3869:
3863:
3859:
3855:
3851:
3847:
3843:
3837:
3822:
3821:
3816:
3813:
3805:
3790:
3789:
3784:
3781:
3773:
3766:
3765:
3758:
3751:
3750:
3745:
3739:
3730:
3723:
3717:
3713:
3697:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3659:
3653:
3651:
3647:
3643:
3639:
3636:'s sculpture
3635:
3630:
3628:
3624:
3619:
3617:
3616:
3611:
3606:
3604:
3603:Paulo Uccello
3600:
3591:
3576:(around 1860)
3575:
3571:
3566:
3562:
3557:
3553:
3552:
3546:
3542:
3535:
3534:
3529:
3525:
3516:
3507:
3498:
3492:
3491:Paolo Uccello
3489:(possibly by
3487:
3483:
3479:
3474:
3470:
3469:
3466:
3464:
3460:
3456:
3452:
3447:
3445:
3444:Arthur Cayley
3440:
3438:
3434:
3430:
3426:
3422:
3417:
3415:
3410:
3409:Louis Poinsot
3405:
3403:
3399:
3394:
3390:
3386:
3381:
3379:
3375:
3371:
3367:
3363:
3359:
3355:
3354:
3348:
3346:
3345:Paolo Uccello
3342:
3338:
3322:
3317:
3312:
3308:
3303:
3298:
3294:
3289:
3284:
3283:
3278:
3273:
3268:
3264:
3259:
3254:
3250:
3245:
3240:
3239:
3234:
3229:
3224:
3220:
3215:
3210:
3206:
3201:
3196:
3195:
3192:
3189:
3185:
3181:
3174:
3168:
3164:
3158:
3153:
3146:
3143:
3134:
3129:
3124:
3122:
3119:
3110:
3105:
3100:
3098:
3095:
3086:
3081:
3076:
3075:
3072:
3069:
3060:
3055:
3050:
3048:
3045:
3036:
3031:
3026:
3024:
3021:
3012:
3007:
3002:
3001:
2997:
2993:
2989:
2986:
2982:
2978:
2975:
2971:
2967:
2966:
2963:
2961:
2956:
2954:
2949:
2947:
2943:
2937:
2933:
2928:
2926:
2911:
2907:
2904:
2900:
2897:
2893:
2892:
2888:
2884:
2881:
2877:
2874:
2870:
2869:
2866:
2865:Catalan solid
2863:
2860:
2857:
2856:
2851:
2848:
2845:
2841:
2837:
2836:topologically
2828:
2826:
2822:
2817:
2812:
2810:
2806:
2801:
2796:
2792:
2791:augmentations
2785:Augmentations
2778:
2772:
2766:
2751:
2748:
2743:
2737:
2732:
2729:
2719:
2705:
2702:
2697:
2691:
2686:
2683:
2673:
2670:
2666:
2663:
2659:
2656:
2652:
2651:
2636:
2633:
2628:
2624:
2621:
2616:
2604:
2590:
2587:
2582:
2578:
2575:
2570:
2558:
2555:
2551:
2548:
2544:
2541:
2537:
2536:
2507:
2479:
2476:
2473:
2470:
2469:
2464:
2461:
2458:
2450:
2430:
2426:
2422:
2419:
2416:
2413:
2393:
2381:
2377:
2373:
2369:
2362:Hull and core
2350:
2345:
2341:
2328:
2325:
2320:
2316:
2314:
2309:
2305:
2303:
2298:
2294:
2292:
2287:
2283:
2281:
2276:
2272:
2270:
2265:
2261:
2260:
2257:
2254:
2253:topologically
2250:
2238:
2236:
2231:
2229:
2225:
2213:
2208:
2202:
2197:
2193:
2191:
2186:
2182:
2180:
2175:
2171:
2168:
2167:
2164:
2159:
2155:
2153:
2148:
2144:
2142:
2137:
2133:
2128:
2123:
2116:
2115:
2112:
2107:
2101:
2096:
2089:
2088:
2083:
2080:
2076:
2072:
2067:
2065:
2059:
2055:
2051:
2047:
2026:
2020:
2015:
2011:
2006:
2000:
1995:
1991:
1986:
1980:
1975:
1971:
1970:
1967:
1964:
1963:
1953:
1947:
1942:
1938:
1933:
1927:
1922:
1918:
1913:
1907:
1902:
1898:
1897:
1894:
1891:
1890:
1877:
1869:
1860:
1853:
1844:
1836:
1829:
1820:
1813:
1804:
1803:
1800:
1797:
1796:
1793:
1790:
1783:
1778:
1775:
1774:
1769:
1766:
1764:
1760:
1758:
1754:
1745:
1742:
1741:
1737:
1734:
1733:
1729:
1726:
1725:
1722:
1720:
1716:
1712:
1706:
1702:
1698:
1686:
1681:
1667:
1661:
1658:
1655:
1650:
1646:
1641:
1637:
1632:
1629:
1624:
1584:{5/2, 3}
1583:
1580:
1576:
1573:
1569:
1566:
1562:
1558:
1555:
1550:
1544:
1541:
1538:
1533:
1529:
1524:
1520:
1515:
1512:
1507:
1467:{3, 5/2}
1466:
1463:
1459:
1456:
1452:
1449:
1445:
1441:
1438:
1433:
1427:
1424:
1421:
1416:
1412:
1407:
1403:
1398:
1395:
1390:
1350:{5/2, 5}
1349:
1346:
1342:
1339:
1335:
1332:
1328:
1324:
1321:
1316:
1310:
1307:
1304:
1299:
1295:
1290:
1286:
1281:
1278:
1273:
1233:{5, 5/2}
1232:
1229:
1225:
1222:
1218:
1215:
1211:
1207:
1204:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1177:
1175:
1171:
1166:
1161:
1158:
1153:
1151:
1145:
1142:
1138:
1135:
1130:
1127:
1122:
1121:
1104:
1095:
1086:
1077:
1068:
1061:
1052:
1043:
1034:
1025:
1024:
1004:
998:
995:
992:
988:
983:
975:
972:
957:
953:
950:
945:
942:
936:
928:
925:
910:
904:
901:
896:
893:
889:
881:
878:
877:
861:
855:
852:
849:
845:
840:
832:
829:
814:
808:
805:
800:
797:
793:
785:
782:
767:
763:
760:
755:
752:
746:
738:
735:
734:
730:
725:
720:
719:
716:
714:
710:
706:
701:
699:
695:
671:
668:
665:
662:
659:
654:
650:
646:
643:
640:
637:
632:
628:
620:
619:
618:
616:
615:Arthur Cayley
598:
594:
586:) and faces (
571:
567:
558:
554:
550:
545:
525:
522:
519:
516:
513:
510:
507:
504:
501:
494:
493:
492:
491:
480:
478:
474:
469:
465:
461:
457:
452:
448:
446:
442:
438:
434:
430:
427:
423:
419:
415:
408:Non-convexity
405:
381:
376:
373:
370:
357:
354:
348:
343:
339:
318:
305:
300:
297:
284:
281:
275:
270:
266:
246:
241:
236:
233:
230:
225:
221:
190:
185:
182:
179:
166:
163:
157:
152:
148:
128:
126:
122:
119:
115:
111:
108:
104:
99:
97:
94:
90:
86:
76:
70:
62:
56:
46:
40:
32:
26:
4197:
4111:
4077:
4061:
4035:
4024:
4017:
4014:
4004:
3993:
3987:
3968:
3954:
3940:
3937:
3927:
3917:
3913:
3904:Bibliography
3893:
3888:
3877:
3849:
3836:
3824:. Retrieved
3818:
3804:
3792:. Retrieved
3786:
3772:
3762:
3757:
3747:
3743:
3738:
3729:
3721:
3716:
3650:dodecahedron
3637:
3634:Vebjørn Sand
3631:
3620:
3613:
3610:M. C. Escher
3607:
3596:
3549:
3531:
3462:
3461:is just the
3453:developed a
3448:
3441:
3418:
3413:
3406:
3384:
3382:
3364:depicts the
3357:
3351:
3349:
3334:
3160:
3138:
3114:
3090:
3064:
3040:
3016:
2957:
2950:
2929:
2922:
2829:
2818:
2813:
2808:
2804:
2802:
2794:
2790:
2788:
2768:
2449:Golden ratio
2365:
2313:dodecahedron
2244:
2232:
2221:
2117:Stellations
2111:dodecahedron
2068:
2064:dodecahedron
2043:
2024:
2018:
2004:
1998:
1984:
1978:
1951:
1945:
1931:
1925:
1911:
1905:
1875:
1858:
1834:
1818:
1779:
1762:
1761:
1752:
1751:
1718:
1717:is just the
1704:
1700:
1695:
1560:(sgD = gsD)
702:
691:
552:
546:
543:
486:
453:
449:
411:
139:
118:pentagrammic
110:dodecahedron
105:the regular
100:
88:
82:
4202:(nonconvex
4036:Dual Models
3910:J. Bertrand
3696:4-polytopes
3646:icosahedron
3615:Gravitation
3451:John Conway
2795:cumulations
2269:icosahedron
2255:equivalent.
2228:stellations
2100:icosahedron
2054:icosahedron
2050:stellations
1892:greatening
1798:stellation
1705:stellations
1701:greatenings
1697:John Conway
709:edge radius
431:faces. The
114:icosahedron
4489:Categories
3938:Phil. Mag.
3826:2018-09-21
3794:2018-09-21
3703:References
3623:dissection
3425:stellating
3393:stellating
3142:animations
3118:animations
3094:animations
3068:animations
3044:animations
3020:animations
2990:{3, 5/2} (
2979:{5, 5/2} (
2752:0.38196...
2706:2.61803...
2637:0.61803...
2591:1.61803...
2233:(See also
2169:Facetings
1763:Greatening
1757:stellation
1753:Stellation
1137:Stellation
464:pentagonal
414:pentagrams
103:stellating
4239:polyhedra
4200:polyhedra
4113:MathWorld
3820:MathWorld
3788:MathWorld
3720:Coxeter,
3482:St Mark's
3407:In 1809,
3374:truncated
3171:(Compare
3163:midradius
2825:MathWorld
2771:midradius
2733:−
2622:−
2457:midradius
2427:φ
2414:φ
2394:φ
2249:skeletons
1789:midradius
1174:(config.)
1131:Spherical
641:−
555:) of the
511:−
502:χ
460:pentagram
429:pentagram
340:ϕ
267:ϕ
222:ϕ
149:ϕ
4034:(1983).
3656:See also
3437:faceting
3433:Bertrand
2968:{3, 5} (
2946:its dual
2936:its dual
2919:Symmetry
2842:and the
2809:hedgehog
2075:faceting
1965:duality
1776:diagram
1194:Symmetry
1162:Vertices
1144:Schläfli
1140:diagram
1128:Picture
974:decagram
85:geometry
3848:(ed.).
3744:Echinus
3350:In his
3331:History
3167:decagon
2940:In the
2376:regular
2224:regular
2090:Convex
2056:. (See
2052:of the
1709:In his
1685:density
1653:{10/3}
1536:{10/3}
1189:Density
1133:tiling
1116:Summary
831:hexagon
549:density
458:has 12
93:regular
4084:
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4064:2008,
4042:
3978:
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3864:
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3558:(1619)
3536:(1568)
3486:Venice
3478:mosaic
3476:Floor
3439:them.
3404:were.
3398:convex
3368:and a
3341:Venice
2819:These
2378:and a
1874:sgD =
1644:(5/2)
1627:{5/2}
1527:(3)/2
1410:(5/2)
1393:{5/2}
1293:(5)/2
1170:figure
1168:Vertex
1159:Edges
529:
441:convex
107:convex
3708:Notes
3572:from
2858:Core
2821:naïve
2477:Core
2471:Hull
2455:(The
2447:(See
2372:great
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2073:is a
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1154:Faces
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136:Sizes
121:faces
4082:ISBN
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1123:Name
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435:and
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