3043:
804:
1913:
1893:
1875:
1884:
1168:
568:
1186:
1177:
1537:
2355:
1221:
243:
222:
1212:
1511:
474:
452:
236:
229:
97:
83:
1256:
1528:
1815:
1784:
1775:
1002:
1612:
1766:
2367:
1572:
168:
2323:
463:
90:
1832:
1550:
744:
1585:
1263:
1806:
175:
993:
2316:
2309:
2302:
2295:
2288:
2281:
2274:
1563:
440:
154:
76:
161:
2460:
dodecahedra, excluding mirror images—the number of vertices ranges from 8 to 20. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the
772:
faces, with three meeting in each of the 20 vertices (see figure). However, the pentagons are not constrained to be regular, and the underlying atomic arrangement has no true fivefold symmetry axis. Its 30 edges are divided into two sets – containing 24 and 6 edges of the same length. The only axes
2348:
symmetry, order 12. It has 2 sets of 3 identical pentagons on the top and bottom, connected 6 pentagons around the sides which alternate upwards and downwards. This form has a hexagonal cross-section and identical copies can be connected as a partial hexagonal honeycomb, but all vertices will not
843:
is 2·arctan(2) ≈ 126.87° and each pentagonal face has one angle of approximately 121.6° in between two angles of approximately 106.6° and opposite two angles of approximately 102.6°. The following formulas show the measurements for the face of a perfect crystal (which is rarely found in nature).
395:
While the regular dodecahedron shares many features with other
Platonic solids, one unique property of it is that one can start at a corner of the surface and draw an infinite number of straight lines across the figure that return to the original point without crossing over any other corner.
1736:
Although regular dodecahedra do not exist in crystals, the tetartoid form does. The name tetartoid comes from the Greek root for one-fourth because it has one fourth of full octahedral symmetry, and half of pyritohedral symmetry. The mineral
3016:
1325:
is concave and equilateral; it can tessellate space with the convex regular dodecahedron. Continuing from there in that direction, we pass through a degenerate case where twelve vertices coincide in the centre, and on to the regular
979:
889:
1748:
and symmetry can be created from the cube and the tetrahedron. In the cube each face is bisected by a slanted edge. In the tetrahedron each edge is trisected, and each of the new vertices connected to a face center. (In
933:
345:
Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The
1928:
on the right shows a tetartoid created by enlarging the blue faces of the dyakis dodecahedral core. Therefore, the edges between the blue faces are covered by the red skeleton edges.
3474:
2254:
is a degenerate case with 12 zero-length edges. (In terms of the colors used above this means, that the white vertices and green edges are absorbed by the green vertices.)
803:
1318:
as the other limit as 6 edges are degenerated to length zero. The regular dodecahedron represents a special intermediate case where all edges and angles are equal.
1334:. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces.
793:
574:
939:
267:
3467:
2414:
1351:
The ratio shown is that of edge lengths, namely those in a set of 24 (touching cube vertices) to those in a set of 6 (corresponding to cube faces).
849:
4452:
1912:
3460:
1733:
faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.
1346:
895:
506:{5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the
3887:
3081:
2880:
1677:
687:
649:
639:
629:
621:
611:
2336:
A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular
703:
601:
213:
2872:
1892:
518:; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.
3379:
1874:
644:
634:
616:
606:
331:
2669:
531:
198:
1618:
491:
2648:
2544:
1516:
1327:
503:
495:
478:
456:
381:
208:
67:
57:
1883:
1110:
3673:
3614:
1307:
1152:
284:
4469:
3703:
3663:
3269:
2531:
1750:
1637:
780:
Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral
2796:
1908 Chambers's
Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary
3698:
3693:
2742:
2735:
1151:
Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the
4479:
3263:
2656:
514:
faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same
2812:
1167:
567:
3880:
3804:
3799:
3678:
3584:
3391:
3319:
3239:
3074:
2921:
2690:
2587:
2429:
2385:
385:
145:
1185:
4474:
3668:
3609:
3599:
3544:
3325:
2778:
1859:
1858:. (The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the
1176:
538:
that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with
377:
260:
301:
4424:
4417:
4410:
3688:
3604:
3559:
2763:
2715:
2612:
2425:
757:
692:
539:
354:
2406:
where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
2993:, by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226
1536:
4484:
4081:
4028:
3648:
3574:
3522:
2970:
2751:
2616:
2484:
2445:
2418:
2376:
2261:
2247:
1941:
1726:
1722:
1682:
1577:
1331:
1315:
1270:
1139:
797:
765:
708:
547:
535:
445:
405:
369:
362:
339:
323:
203:
140:
52:
2807:
2354:
4436:
4335:
4085:
3814:
3683:
3658:
3643:
3579:
3527:
2608:
2595:
2566:
2265:
2251:
1855:
774:
373:
2892:
2856:
1558:
can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
1220:
242:
4305:
4255:
4205:
4162:
4132:
4092:
4055:
3873:
3829:
3794:
3653:
3548:
3497:
3436:
3313:
3307:
3067:
3039:
K.J.M. MacLean, A Geometric
Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
3026:
2839:
2821:
2768:
2631:
2395:) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space.
499:
467:
62:
31:
415:
221:
1510:
1321:
It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The
473:
451:
235:
228:
96:
82:
4444:
3809:
3619:
3594:
3538:
3426:
3350:
3302:
3275:
3245:
2917:
2773:
2694:
2681:
2660:
2599:
2578:
2535:
2522:
2505:
2488:
2475:
2392:
1667:
1255:
1211:
721:
677:
515:
507:
1109:(a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the
4448:
4013:
4002:
3991:
3980:
3971:
3962:
3949:
3927:
3915:
3901:
3897:
3748:
3431:
3411:
3233:
2831:
1083:
738:
1345:
Versions with equal absolute values and opposing signs form a honeycomb together. (Compare
1330:
where all edges and angles are equal again, and the faces have been distorted into regular
4038:
4023:
3385:
3297:
3292:
3257:
3212:
3202:
3192:
3187:
2876:
2747:
2719:
2639:
2591:
2574:
2560:
2548:
2518:
2509:
2471:
2441:
1697:
1657:
1647:
731:
716:
667:
657:
593:
527:
422:
3009:
1527:
4388:
3569:
3492:
3441:
3344:
3207:
3197:
2448:
along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.
1262:
840:
785:
411:
327:
3038:
3032:
1814:
1783:
1611:
4463:
4405:
4293:
4286:
4279:
4243:
4236:
4229:
4193:
4186:
3910:
3774:
3630:
3564:
3362:
3356:
3251:
3182:
3172:
2843:
2754:
2499:
2388:
1925:
824:
808:
2998:
1774:
1001:
974:{\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}
4345:
2935:
2869:
2711:
2433:
2403:
2366:
1571:
1284:
1230:
1131:
836:
789:
167:
130:
2904:
2835:
2322:
1765:
462:
89:
4354:
4315:
4265:
4215:
4172:
4142:
4074:
4060:
3507:
3217:
3151:
3141:
3131:
3126:
3054:
2444:
in 1960. This figure is another spacefiller, and can also occur in non-periodic
1831:
1549:
1311:
743:
426:
2956:
1584:
4340:
4324:
4274:
4224:
4181:
4151:
4065:
3839:
3727:
3517:
3484:
3402:
3156:
3146:
3136:
3121:
3111:
3090:
3021:
3002:
2635:
2437:
2410:
2381:
2337:
1693:
884:{\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}
487:
335:
319:
174:
4396:
4310:
4260:
4210:
4167:
4137:
4106:
3834:
3824:
3769:
3753:
3589:
3416:
2652:
2464:
Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)
1901:
1835:
1805:
1738:
1520:
543:
511:
135:
17:
2315:
2308:
2301:
2294:
2287:
2280:
992:
510:{3, 5/2}. All of these regular star dodecahedra have regular pentagonal or
30:
This article is about the three-dimensional shape. Not to be confused with
2273:
1562:
439:
153:
75:
4370:
4125:
4121:
4048:
3720:
3116:
2806:
Athreya, Jayadev S.; Aulicino, David; Hooper, W. Patrick (May 27, 2020).
2731:
2570:
1745:
1730:
1630:
769:
586:
490:, all of which are regular star dodecahedra. They form three of the four
252:
1567:
A regular dodecahedron is an intermediate case with equal edge lengths.
4379:
4349:
4116:
4111:
4102:
4043:
3844:
3819:
3421:
3048:
2677:
777:
are three mutually perpendicular twofold axes and four threefold axes.
160:
4319:
4269:
4219:
4176:
4146:
4097:
4033:
2384:
with twelve rhombic faces and octahedral symmetry. It is dual to the
928:{\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}
828:
781:
350:
307:
290:
273:
2994:
1273:
of alternating convex and concave pyritohedra with heights between ±
3452:
2826:
1580:
is a degenerate case with the 6 crossedges reduced to length zero.
2757:
1911:
3017:
Editable printable net of a dodecahedron with interactive 3D view
4069:
3512:
1940:
The following points are vertices of a tetartoid pentagon under
1854:
A tetartoid can be created by enlarging 12 of the 24 faces of a
1555:
1086:-shaped "roof" above the faces of that cube with edge length 2.
1025:
The eight vertices of a cube have the coordinates (±1, ±1, ±1).
832:
784:, and it may be an inspiration for the discovery of the regular
3456:
3063:
530:, two important dodecahedra can occur as crystal forms in some
372:
can be seen as a limiting case of the pyritohedron, and it has
2738:(4D polytope) whose surface consists of 120 dodecahedral cells
429:{3, 5}, having five equilateral triangles around each vertex.
322:
with twelve flat faces. The most familiar dodecahedron is the
788:
form. The true regular dodecahedron can occur as a shape for
3059:
2700:, order 20, topologically equivalent to regular dodecahedron
2808:"Platonic Solids and High Genus Covers of Lattice Surfaces"
410:
The convex regular dodecahedron is one of the five regular
3051:: Software used to create some of the images on this page.
2250:
is a tetartoid with more than the required symmetry. The
1904:
tetartoids based on the dyakis dodecahedron in the middle
1306:
The pyritohedron has a geometric degree of freedom with
2959:. Numericana.com (2001-12-31). Retrieved on 2016-12-02.
431:
2934:
Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000).
2907:. Demonstrations.wolfram.com. Retrieved on 2016-12-02.
2714:
used a dodecahedron as the "globe" equivalent for his
36:
942:
898:
852:
384:
variations, along with the rhombic dodecahedra, are
3787:
3762:
3737:
3712:
3628:
3536:
3491:
3401:
3372:
3337:
3285:
3226:
3165:
3104:
3001:VRML models and animations of Pyritohedron and its
1194:Orthographic projections of the pyritohedron with
1028:The coordinates of the 12 additional vertices are
973:
927:
883:
2508:– 5 triangles, 5 squares, 1 pentagon, 1 decagon,
3055:How to make a dodecahedron from a Styrofoam cube
2936:"Tilings, coverings, clusters and quasicrystals"
1792:Orthographic projections from 2- and 3-fold axes
1541:The concave equilateral dodecahedron, called an
486:The convex regular dodecahedron also has three
1010:Natural pyrite (with face angles on the right)
800:, which includes true fivefold rotation axes.
3881:
3468:
3075:
2432:, i.e. the diagonals are in the ratio of the
2428:, has twelve faces congruent to those of the
8:
2424:Another important rhombic dodecahedron, the
835:). In pyritohedral pyrite, the faces have a
326:with regular pentagons as faces, which is a
2991:Plato's Fourth Solid and the "Pyritohedron"
2456:There are 6,384,634 topologically distinct
1589:Self-intersecting equilateral dodecahedron
3888:
3874:
3866:
3741:
3475:
3461:
3453:
3082:
3068:
3060:
2256:
1844:
1599:
1336:
1245:
557:
3010:"3D convex uniform polyhedra o3o5x – doe"
2895:. Galleries.com. Retrieved on 2016-12-02.
2859:. Galleries.com. Retrieved on 2016-12-02.
2825:
2487:– 10 equilateral triangles, 2 pentagons,
966:
951:
943:
941:
920:
907:
899:
897:
876:
861:
853:
851:
2918:Introduction to golden rhombic polyhedra
2365:
1848:Relationship to the dyakis dodecahedron
1830:
802:
4453:List of regular polytopes and compounds
2789:
2634:– 11 isosceles triangles and 1 regular
1314:at one limit of collinear edges, and a
1532:Degenerate, 12 vertices in the center
1144:
389:
338:of the convex form. All of these have
2920:. Faculty of Electrical Engineering,
2409:The rhombic dodecahedron has several
7:
2332:Dual of triangular gyrobianticupola
1605:Tetragonal pentagonal dodecahedron
794:holmium–magnesium–zinc quasicrystal
3099:Listed by number of faces and type
3029:– Models made with Modular Origami
2716:Digital Dome planetarium projector
2691:Truncated pentagonal trapezohedron
2611:– 12 isosceles triangles, dual of
1711:tetragonal pentagonal dodecahedron
1340:Special cases of the pyritohedron
433:Four kinds of regular dodecahedra
25:
2881:University of Wisconsin-Green Bay
2340:connected base-to-base, called a
1744:Abstractions sharing the solid's
1719:tetrahedric pentagon dodecahedron
823:comes from one of the two common
346:
2971:"Forerunners of the Planetarium"
2879:. Natural and Applied Sciences,
2547:– 10 triangles and 2 pentagons,
2461:angles between edges or faces.)
2353:
2321:
2314:
2307:
2300:
2293:
2286:
2279:
2272:
2126:under the following conditions:
1891:
1882:
1873:
1813:
1804:
1782:
1773:
1764:
1721:) is a dodecahedron with chiral
1610:
1583:
1570:
1561:
1548:
1535:
1526:
1509:
1261:
1254:
1219:
1210:
1184:
1175:
1166:
1000:
991:
742:
647:
642:
637:
632:
627:
619:
614:
609:
604:
599:
566:
472:
461:
450:
438:
241:
234:
227:
220:
173:
166:
159:
152:
95:
88:
81:
74:
2718:, based upon a suggestion from
839:of (210), which means that the
414:and can be represented by its
358:
1:
3310:(two infinite groups and 75)
3044:Dodecahedron 3D Visualization
3035:The Encyclopedia of Polyhedra
2836:10.1080/10586458.2020.1712564
2676:– 8 rhombi and 4 equilateral
2670:Rhombo-hexagonal dodecahedron
2534:– 8 triangles and 4 squares,
1753:this is a gyro tetrahedron.)
1741:can have this symmetry form.
1295:and 1 (rhombic dodecahedron)
314: 'base, seat, face') or
3855:Degenerate polyhedra are in
3328:(two infinite groups and 50)
3049:Stella: Polyhedron Navigator
2969:Ley, Willy (February 1965).
2870:The 48 Special Crystal Forms
2649:Trapezo-rhombic dodecahedron
2559:Congruent irregular faced: (
2545:Metabidiminished icosahedron
2402:can be seen as a degenerate
2342:triangular gyrobianticupola.
1517:great stellated dodecahedron
1328:great stellated dodecahedron
736:
522:Other pentagonal dodecahedra
504:great stellated dodecahedron
496:small stellated dodecahedron
479:Great stellated dodecahedron
457:Small stellated dodecahedron
436:
382:trapezo-rhombic dodecahedron
218:
196:
180:
3674:pentagonal icositetrahedron
3615:truncated icosidodecahedron
2419:parallelohedral spacefiller
1153:compound of two dodecahedra
516:abstract regular polyhedron
349:, a common crystal form in
334:, which are constructed as
297: 'twelve' and
4501:
4442:
3869:
3704:pentagonal hexecontahedron
3664:deltoidal icositetrahedron
2999:Stellation of Pyritohedron
2627:Other less regular faced:
2532:Elongated square dipyramid
2474:– 10 squares, 2 decagons,
2260:Tetartoid variations from
1823:Cubic and tetrahedral form
1751:Conway polyhedron notation
1729:, it has twelve identical
768:, it has twelve identical
403:
300:
283:
266:
29:
3853:
3744:
3699:disdyakis triacontahedron
3694:deltoidal hexecontahedron
3380:Kepler–Poinsot polyhedron
3097:
3033:Virtual Reality Polyhedra
2916:Hafner, I. and Zitko, T.
2743:Braarudosphaera bigelowii
2259:
1609:
1602:
1463:
1437:
1410:
1381:
1344:
1339:
1248:
831:(the other one being the
807:Dual positions in pyrite
737:
565:
560:
188:
181:
42:
2973:. For Your Information.
2813:Experimental Mathematics
2746:– a dodecahedron shaped
2657:triangular orthobicupola
1293:Heights between 0 (cube)
492:Kepler–Poinsot polyhedra
332:regular star dodecahedra
27:Polyhedron with 12 faces
3805:gyroelongated bipyramid
3679:rhombic triacontahedron
3585:truncated cuboctahedron
3392:Uniform star polyhedron
3320:quasiregular polyhedron
2922:University of Ljubljana
2588:Hexagonal trapezohedron
2430:rhombic triacontahedron
1715:pentagon-tritetrahedron
1111:Weaire–Phelan structure
756:is a dodecahedron with
330:. There are also three
3800:truncated trapezohedra
3669:disdyakis dodecahedron
3635:(duals of Archimedean)
3610:rhombicosidodecahedron
3600:truncated dodecahedron
3326:semiregular polyhedron
2975:Galaxy Science Fiction
2779:Truncated dodecahedron
2674:elongated Dodecahedron
2371:
1917:
1860:disdyakis dodecahedron
1838:
1621:for a rotating model.)
975:
929:
885:
811:
577:for a rotating model.)
378:elongated dodecahedron
3689:pentakis dodecahedron
3605:truncated icosahedron
3560:truncated tetrahedron
3373:non-convex polyhedron
3022:The Uniform Polyhedra
2764:Pentakis dodecahedron
2613:truncated tetrahedron
2440:and was described by
2426:Bilinski dodecahedron
2369:
1936:Cartesian coordinates
1915:
1834:
1089:An important case is
1082:is the height of the
1021:Cartesian coordinates
976:
930:
886:
806:
764:) symmetry. Like the
540:pyritohedral symmetry
388:. There are numerous
355:pyritohedral symmetry
3649:rhombic dodecahedron
3575:truncated octahedron
2485:Pentagonal antiprism
2400:rhombic dodecahedron
2377:rhombic dodecahedron
2370:Rhombic dodecahedron
2362:Rhombic dodecahedron
2262:regular dodecahedron
2248:regular dodecahedron
1942:tetrahedral symmetry
1727:regular dodecahedron
1723:tetrahedral symmetry
1685:, , (332), order 12
1578:rhombic dodecahedron
1316:rhombic dodecahedron
1140:regular dodecahedron
940:
896:
850:
798:icosahedral symmetry
766:regular dodecahedron
711:, , (332), order 12
698:, , (3*2), order 24
548:tetrahedral symmetry
536:cubic crystal system
446:regular dodecahedron
406:Regular dodecahedron
400:Regular dodecahedron
370:rhombic dodecahedron
363:tetrahedral symmetry
340:icosahedral symmetry
324:regular dodecahedron
4437:pentagonal polytope
4336:Uniform 10-polytope
3896:Fundamental convex
3684:triakis icosahedron
3659:tetrakis hexahedron
3644:triakis tetrahedron
3580:rhombicuboctahedron
3008:Klitzing, Richard.
2609:Triakis tetrahedron
2596:hexagonal antiprism
2567:Hexagonal bipyramid
2481:symmetry, order 40.
2468:Uniform polyhedra:
2266:triakis tetrahedron
2252:triakis tetrahedron
1856:dyakis dodecahedron
1138:= 0.618... for the
775:rotational symmetry
434:
374:octahedral symmetry
39:
38:Common dodecahedra
4306:Uniform 9-polytope
4256:Uniform 8-polytope
4206:Uniform 7-polytope
4163:Uniform 6-polytope
4133:Uniform 5-polytope
4093:Uniform polychoron
4056:Uniform polyhedron
3904:in dimensions 2–10
3654:triakis octahedron
3539:Archimedean solids
3314:regular polyhedron
3308:uniform polyhedron
3270:Hectotriadiohedron
2957:Counting polyhedra
2875:2013-09-18 at the
2769:Roman dodecahedron
2736:regular polychoron
2687:symmetry, order 16
2666:symmetry, order 12
2622:symmetry, order 24
2605:symmetry, order 24
2584:symmetry, order 24
2541:symmetry, order 16
2515:symmetry, order 10
2494:symmetry, order 20
2372:
1918:
1839:
1631:irregular pentagon
1229:Heights 1/2 and 1/
971:
925:
881:
812:
587:isosceles pentagon
502:{5, 5/2}, and the
500:great dodecahedron
468:Great dodecahedron
432:
214:Rhombo-triangular-
37:
32:Roman dodecahedron
4470:Individual graphs
4458:
4457:
4445:Polytope families
3902:uniform polytopes
3864:
3863:
3783:
3782:
3620:snub dodecahedron
3595:icosidodecahedron
3450:
3449:
3351:Archimedean solid
3338:convex polyhedron
3246:Icosidodecahedron
3027:Origami Polyhedra
2977:. pp. 87–98.
2774:Snub dodecahedron
2554:symmetry, order 4
2506:Pentagonal cupola
2502:(regular faced):
2452:Other dodecahedra
2393:Archimedean solid
2329:
2328:
2242:Geometric freedom
1933:
1932:
1922:
1921:
1843:
1842:
1703:
1702:
1622:
1593:
1592:
1543:endo-dodecahedron
1323:endo-dodecahedron
1302:Geometric freedom
1299:
1298:
1244:
1243:
1148:for other cases.
1145:Geometric freedom
1018:
1017:
969:
961:
960:
946:
923:
915:
902:
879:
871:
867:
856:
750:
749:
722:Pseudoicosahedron
578:
508:great icosahedron
484:
483:
390:other dodecahedra
249:
248:
199:Rhombo-hexagonal-
16:(Redirected from
4492:
4449:Regular polytope
4010:
3999:
3988:
3947:
3890:
3883:
3876:
3867:
3742:
3738:Dihedral uniform
3713:Dihedral regular
3636:
3552:
3501:
3477:
3470:
3463:
3454:
3286:elemental things
3264:Enneacontahedron
3234:Icositetrahedron
3084:
3077:
3070:
3061:
3013:
2979:
2978:
2966:
2960:
2954:
2948:
2947:
2931:
2925:
2914:
2908:
2902:
2896:
2890:
2884:
2866:
2860:
2854:
2848:
2847:
2829:
2803:
2797:
2794:
2521:– 12 triangles,
2357:
2325:
2318:
2311:
2304:
2297:
2290:
2283:
2276:
2257:
2236:
2143:
2121:
2119:
2118:
2110:
2107:
2098:
2096:
2095:
2087:
2084:
2075:
2073:
2072:
2064:
2061:
2040:
2038:
2037:
2029:
2026:
2017:
2015:
2014:
2006:
2003:
1994:
1992:
1991:
1983:
1980:
1895:
1886:
1877:
1865:
1864:
1845:
1817:
1808:
1786:
1777:
1768:
1756:
1755:
1616:
1614:
1600:
1587:
1574:
1565:
1552:
1539:
1530:
1513:
1487:
1485:
1484:
1481:
1478:
1476:
1475:
1461:
1459:
1458:
1455:
1452:
1450:
1449:
1435:
1433:
1432:
1429:
1426:
1424:
1423:
1408:
1406:
1405:
1402:
1399:
1397:
1396:
1337:
1290:
1288:
1287:
1282:
1279:
1265:
1258:
1246:
1223:
1214:
1188:
1179:
1170:
1158:
1157:
1137:
1135:
1134:
1129:
1126:
1108:
1106:
1105:
1102:
1099:
1004:
995:
983:
982:
980:
978:
977:
972:
970:
967:
962:
953:
952:
947:
944:
934:
932:
931:
926:
924:
921:
916:
908:
903:
900:
890:
888:
887:
882:
880:
877:
872:
863:
862:
857:
854:
746:
652:
651:
650:
646:
645:
641:
640:
636:
635:
631:
630:
624:
623:
622:
618:
617:
613:
612:
608:
607:
603:
602:
594:Coxeter diagrams
572:
570:
558:
532:symmetry classes
476:
465:
454:
442:
435:
311:
304:
294:
287:
277:
270:
245:
238:
231:
224:
209:Trapezo-rhombic-
177:
170:
163:
156:
99:
92:
85:
78:
68:Great stellated-
58:Small stellated-
40:
21:
4500:
4499:
4495:
4494:
4493:
4491:
4490:
4489:
4480:Platonic solids
4460:
4459:
4428:
4421:
4414:
4297:
4290:
4283:
4247:
4240:
4233:
4197:
4190:
4024:Regular polygon
4017:
4008:
4001:
3997:
3990:
3986:
3977:
3968:
3961:
3957:
3945:
3939:
3935:
3923:
3905:
3894:
3865:
3860:
3849:
3788:Dihedral others
3779:
3758:
3733:
3708:
3637:
3634:
3633:
3624:
3553:
3542:
3541:
3532:
3495:
3493:Platonic solids
3487:
3481:
3451:
3446:
3397:
3386:Star polyhedron
3368:
3333:
3281:
3258:Hexecontahedron
3240:Triacontahedron
3222:
3213:Enneadecahedron
3203:Heptadecahedron
3193:Pentadecahedron
3188:Tetradecahedron
3161:
3100:
3093:
3088:
3007:
2987:
2982:
2968:
2967:
2963:
2955:
2951:
2933:
2932:
2928:
2915:
2911:
2903:
2899:
2891:
2887:
2877:Wayback Machine
2867:
2863:
2855:
2851:
2805:
2804:
2800:
2795:
2791:
2787:
2748:coccolithophore
2728:
2720:Albert Einstein
2709:
2707:Practical usage
2698:
2685:
2664:
2643:
2620:
2603:
2582:
2575:hexagonal prism
2569:– 12 isosceles
2561:face-transitive
2552:
2539:
2526:
2519:Snub disphenoid
2513:
2492:
2479:
2472:Decagonal prism
2454:
2436:. It is also a
2364:
2347:
2334:
2244:
2234:
2228:
2222:
2198:
2169:
2130:
2117:
2111:
2108:
2103:
2102:
2100:
2094:
2088:
2085:
2080:
2079:
2077:
2071:
2065:
2062:
2057:
2056:
2054:
2036:
2030:
2027:
2022:
2021:
2019:
2013:
2007:
2004:
1999:
1998:
1996:
1990:
1984:
1981:
1976:
1975:
1973:
1938:
1908:
1907:
1906:
1905:
1898:
1897:
1896:
1888:
1887:
1879:
1878:
1827:
1826:
1825:
1824:
1820:
1819:
1818:
1810:
1809:
1796:
1795:
1794:
1793:
1789:
1788:
1787:
1779:
1778:
1770:
1769:
1698:face transitive
1638:Conway notation
1615:
1604:
1598:
1588:
1575:
1566:
1553:
1540:
1531:
1519:, with regular
1514:
1482:
1479:
1473:
1471:
1470:
1469:
1467:
1456:
1453:
1447:
1445:
1444:
1443:
1441:
1430:
1427:
1421:
1419:
1417:
1416:
1414:
1403:
1400:
1394:
1392:
1391:
1390:
1388:
1350:
1304:
1294:
1283:
1280:
1277:
1276:
1274:
1238:
1237:
1236:
1235:
1226:
1225:
1224:
1216:
1215:
1202:
1201:
1200:
1199:
1191:
1190:
1189:
1181:
1180:
1172:
1171:
1130:
1127:
1124:
1123:
1121:
1116:Another one is
1103:
1100:
1097:
1096:
1094:
1075:
1063:
1059:
1047:
1043:
1031:
1023:
1014:
1013:
1012:
1011:
1007:
1006:
1005:
997:
996:
938:
937:
894:
893:
848:
847:
817:
763:
741:
732:face transitive
717:Dual polyhedron
696:
648:
643:
638:
633:
628:
626:
625:
620:
615:
610:
605:
600:
598:
571:
556:
528:crystallography
524:
494:. They are the
477:
466:
455:
443:
425:is the regular
423:dual polyhedron
416:Schläfli symbol
412:Platonic solids
408:
402:
192:
185:
124:
117:
107:
46:
35:
28:
23:
22:
15:
12:
11:
5:
4498:
4496:
4488:
4487:
4482:
4477:
4472:
4462:
4461:
4456:
4455:
4440:
4439:
4430:
4426:
4419:
4412:
4408:
4399:
4382:
4373:
4362:
4361:
4359:
4357:
4352:
4343:
4338:
4332:
4331:
4329:
4327:
4322:
4313:
4308:
4302:
4301:
4299:
4295:
4288:
4281:
4277:
4272:
4263:
4258:
4252:
4251:
4249:
4245:
4238:
4231:
4227:
4222:
4213:
4208:
4202:
4201:
4199:
4195:
4188:
4184:
4179:
4170:
4165:
4159:
4158:
4156:
4154:
4149:
4140:
4135:
4129:
4128:
4119:
4114:
4109:
4100:
4095:
4089:
4088:
4079:
4077:
4072:
4063:
4058:
4052:
4051:
4046:
4041:
4036:
4031:
4026:
4020:
4019:
4015:
4011:
4006:
3995:
3984:
3975:
3966:
3959:
3953:
3943:
3937:
3931:
3925:
3919:
3913:
3907:
3906:
3895:
3893:
3892:
3885:
3878:
3870:
3862:
3861:
3854:
3851:
3850:
3848:
3847:
3842:
3837:
3832:
3827:
3822:
3817:
3812:
3807:
3802:
3797:
3791:
3789:
3785:
3784:
3781:
3780:
3778:
3777:
3772:
3766:
3764:
3760:
3759:
3757:
3756:
3751:
3745:
3739:
3735:
3734:
3732:
3731:
3724:
3716:
3714:
3710:
3709:
3707:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3656:
3651:
3646:
3640:
3638:
3631:Catalan solids
3629:
3626:
3625:
3623:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3577:
3572:
3570:truncated cube
3567:
3562:
3556:
3554:
3537:
3534:
3533:
3531:
3530:
3525:
3520:
3515:
3510:
3504:
3502:
3489:
3488:
3482:
3480:
3479:
3472:
3465:
3457:
3448:
3447:
3445:
3444:
3442:parallelepiped
3439:
3434:
3429:
3424:
3419:
3414:
3408:
3406:
3399:
3398:
3396:
3395:
3389:
3383:
3376:
3374:
3370:
3369:
3367:
3366:
3360:
3354:
3348:
3345:Platonic solid
3341:
3339:
3335:
3334:
3332:
3331:
3330:
3329:
3323:
3317:
3305:
3300:
3295:
3289:
3287:
3283:
3282:
3280:
3279:
3273:
3267:
3261:
3255:
3249:
3243:
3237:
3230:
3228:
3224:
3223:
3221:
3220:
3215:
3210:
3208:Octadecahedron
3205:
3200:
3198:Hexadecahedron
3195:
3190:
3185:
3180:
3175:
3169:
3167:
3163:
3162:
3160:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3124:
3119:
3114:
3108:
3106:
3102:
3101:
3098:
3095:
3094:
3089:
3087:
3086:
3079:
3072:
3064:
3058:
3057:
3052:
3046:
3041:
3036:
3030:
3024:
3019:
3014:
3005:
2996:
2986:
2985:External links
2983:
2981:
2980:
2961:
2949:
2926:
2909:
2897:
2885:
2868:Dutch, Steve.
2861:
2849:
2820:(3): 847–877.
2798:
2788:
2786:
2783:
2782:
2781:
2776:
2771:
2766:
2761:
2745:
2739:
2727:
2724:
2708:
2705:
2704:
2703:
2702:
2701:
2696:
2688:
2683:
2667:
2662:
2651:– 6 rhombi, 6
2646:
2641:
2625:
2624:
2623:
2618:
2606:
2601:
2585:
2580:
2557:
2556:
2555:
2550:
2542:
2537:
2529:
2524:
2516:
2511:
2500:Johnson solids
2497:
2496:
2495:
2490:
2482:
2477:
2453:
2450:
2415:first of which
2363:
2360:
2359:
2358:
2345:
2333:
2330:
2327:
2326:
2319:
2312:
2305:
2298:
2291:
2284:
2277:
2269:
2268:
2243:
2240:
2239:
2238:
2232:
2226:
2220:
2196:
2191:
2167:
2162:
2145:
2124:
2123:
2115:
2092:
2069:
2034:
2011:
1988:
1937:
1934:
1931:
1930:
1920:
1919:
1909:
1900:
1899:
1890:
1889:
1881:
1880:
1872:
1871:
1870:
1869:
1868:
1850:
1849:
1841:
1840:
1828:
1822:
1821:
1812:
1811:
1803:
1802:
1801:
1800:
1799:
1797:
1791:
1790:
1781:
1780:
1772:
1771:
1763:
1762:
1761:
1760:
1759:
1725:(T). Like the
1701:
1700:
1691:
1687:
1686:
1680:
1678:Symmetry group
1674:
1673:
1670:
1664:
1663:
1660:
1654:
1653:
1650:
1644:
1643:
1640:
1634:
1633:
1628:
1624:
1623:
1607:
1606:
1597:
1594:
1591:
1590:
1581:
1568:
1559:
1546:
1533:
1524:
1515:Regular star,
1507:
1503:
1502:
1499:
1496:
1493:
1489:
1488:
1465:
1462:
1439:
1436:
1412:
1409:
1385:
1379:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1353:
1352:
1347:this animation
1342:
1341:
1308:limiting cases
1303:
1300:
1297:
1296:
1291:
1267:
1266:
1259:
1251:
1250:
1242:
1241:
1239:
1228:
1227:
1218:
1217:
1209:
1208:
1207:
1206:
1205:
1203:
1193:
1192:
1183:
1182:
1174:
1173:
1165:
1164:
1163:
1162:
1161:
1142:. See section
1073:
1061:
1057:
1045:
1041:
1029:
1022:
1019:
1016:
1015:
1009:
1008:
999:
998:
990:
989:
988:
987:
986:
965:
959:
956:
950:
919:
914:
911:
906:
875:
870:
866:
860:
841:dihedral angle
825:crystal habits
821:crystal pyrite
816:
815:Crystal pyrite
813:
809:crystal models
786:Platonic solid
761:
748:
747:
735:
734:
729:
725:
724:
719:
713:
712:
706:
704:Rotation group
700:
699:
694:
690:
688:Symmetry group
684:
683:
680:
674:
673:
670:
664:
663:
660:
654:
653:
596:
590:
589:
584:
580:
579:
563:
562:
555:
552:
523:
520:
498:{5/2, 5}, the
482:
481:
470:
459:
448:
404:Main article:
401:
398:
328:Platonic solid
247:
246:
239:
232:
225:
217:
216:
211:
206:
204:Rhombo-square-
201:
195:
194:
190:
187:
183:
179:
178:
171:
164:
157:
149:
148:
143:
138:
133:
127:
126:
122:
119:
115:
112:
109:
105:
101:
100:
93:
86:
79:
71:
70:
65:
60:
55:
49:
48:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4497:
4486:
4483:
4481:
4478:
4476:
4475:Planar graphs
4473:
4471:
4468:
4467:
4465:
4454:
4450:
4446:
4441:
4438:
4434:
4431:
4429:
4422:
4415:
4409:
4407:
4403:
4400:
4398:
4394:
4390:
4386:
4383:
4381:
4377:
4374:
4372:
4368:
4364:
4363:
4360:
4358:
4356:
4353:
4351:
4347:
4344:
4342:
4339:
4337:
4334:
4333:
4330:
4328:
4326:
4323:
4321:
4317:
4314:
4312:
4309:
4307:
4304:
4303:
4300:
4298:
4291:
4284:
4278:
4276:
4273:
4271:
4267:
4264:
4262:
4259:
4257:
4254:
4253:
4250:
4248:
4241:
4234:
4228:
4226:
4223:
4221:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4200:
4198:
4191:
4185:
4183:
4180:
4178:
4174:
4171:
4169:
4166:
4164:
4161:
4160:
4157:
4155:
4153:
4150:
4148:
4144:
4141:
4139:
4136:
4134:
4131:
4130:
4127:
4123:
4120:
4118:
4115:
4113:
4112:Demitesseract
4110:
4108:
4104:
4101:
4099:
4096:
4094:
4091:
4090:
4087:
4083:
4080:
4078:
4076:
4073:
4071:
4067:
4064:
4062:
4059:
4057:
4054:
4053:
4050:
4047:
4045:
4042:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4021:
4018:
4012:
4009:
4005:
3998:
3994:
3987:
3983:
3978:
3974:
3969:
3965:
3960:
3958:
3956:
3952:
3942:
3938:
3936:
3934:
3930:
3926:
3924:
3922:
3918:
3914:
3912:
3909:
3908:
3903:
3899:
3891:
3886:
3884:
3879:
3877:
3872:
3871:
3868:
3858:
3852:
3846:
3843:
3841:
3838:
3836:
3833:
3831:
3828:
3826:
3823:
3821:
3818:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3792:
3790:
3786:
3776:
3773:
3771:
3768:
3767:
3765:
3761:
3755:
3752:
3750:
3747:
3746:
3743:
3740:
3736:
3730:
3729:
3725:
3723:
3722:
3718:
3717:
3715:
3711:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3641:
3639:
3632:
3627:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3578:
3576:
3573:
3571:
3568:
3566:
3565:cuboctahedron
3563:
3561:
3558:
3557:
3555:
3550:
3546:
3540:
3535:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3505:
3503:
3499:
3494:
3490:
3486:
3478:
3473:
3471:
3466:
3464:
3459:
3458:
3455:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3409:
3407:
3404:
3400:
3393:
3390:
3387:
3384:
3381:
3378:
3377:
3375:
3371:
3364:
3363:Johnson solid
3361:
3358:
3357:Catalan solid
3355:
3352:
3349:
3346:
3343:
3342:
3340:
3336:
3327:
3324:
3321:
3318:
3315:
3312:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3290:
3288:
3284:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3252:Hexoctahedron
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3231:
3229:
3225:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3183:Tridecahedron
3181:
3179:
3176:
3174:
3173:Hendecahedron
3171:
3170:
3168:
3164:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3109:
3107:
3103:
3096:
3092:
3085:
3080:
3078:
3073:
3071:
3066:
3065:
3062:
3056:
3053:
3050:
3047:
3045:
3042:
3040:
3037:
3034:
3031:
3028:
3025:
3023:
3020:
3018:
3015:
3011:
3006:
3004:
3000:
2997:
2995:
2992:
2989:
2988:
2984:
2976:
2972:
2965:
2962:
2958:
2953:
2950:
2945:
2941:
2937:
2930:
2927:
2923:
2919:
2913:
2910:
2906:
2905:The Tetartoid
2901:
2898:
2894:
2893:Crystal Habit
2889:
2886:
2882:
2878:
2874:
2871:
2865:
2862:
2858:
2857:Crystal Habit
2853:
2850:
2845:
2841:
2837:
2833:
2828:
2823:
2819:
2815:
2814:
2809:
2802:
2799:
2793:
2790:
2784:
2780:
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2759:
2756:
2755:phytoplankton
2753:
2749:
2744:
2741:
2740:
2737:
2733:
2730:
2729:
2725:
2723:
2721:
2717:
2713:
2706:
2699:
2692:
2689:
2686:
2679:
2675:
2671:
2668:
2665:
2658:
2654:
2650:
2647:
2644:
2637:
2633:
2630:Hendecagonal
2629:
2628:
2626:
2621:
2614:
2610:
2607:
2604:
2597:
2593:
2589:
2586:
2583:
2576:
2572:
2568:
2565:
2564:
2562:
2558:
2553:
2546:
2543:
2540:
2533:
2530:
2527:
2520:
2517:
2514:
2507:
2504:
2503:
2501:
2498:
2493:
2486:
2483:
2480:
2473:
2470:
2469:
2467:
2466:
2465:
2462:
2459:
2451:
2449:
2447:
2446:spacefillings
2443:
2439:
2435:
2431:
2427:
2422:
2420:
2416:
2412:
2407:
2405:
2401:
2396:
2394:
2390:
2389:cuboctahedron
2387:
2383:
2379:
2378:
2368:
2361:
2356:
2352:
2351:
2350:
2343:
2339:
2331:
2324:
2320:
2317:
2313:
2310:
2306:
2303:
2299:
2296:
2292:
2289:
2285:
2282:
2278:
2275:
2271:
2270:
2267:
2263:
2258:
2255:
2253:
2249:
2241:
2231:
2225:
2221:
2218:
2214:
2210:
2206:
2202:
2195:
2192:
2189:
2185:
2181:
2177:
2173:
2166:
2163:
2160:
2156:
2153:
2149:
2146:
2142:
2138:
2134:
2129:
2128:
2127:
2114:
2106:
2091:
2083:
2068:
2060:
2052:
2048:
2044:
2033:
2025:
2010:
2002:
1987:
1979:
1971:
1967:
1963:
1959:
1955:
1951:
1947:
1946:
1945:
1943:
1935:
1929:
1927:
1926:crystal model
1916:Crystal model
1914:
1910:
1903:
1894:
1885:
1876:
1867:
1866:
1863:
1861:
1857:
1852:
1851:
1847:
1846:
1837:
1833:
1829:
1816:
1807:
1798:
1785:
1776:
1767:
1758:
1757:
1754:
1752:
1747:
1742:
1740:
1734:
1732:
1728:
1724:
1720:
1716:
1712:
1708:
1699:
1695:
1692:
1689:
1688:
1684:
1681:
1679:
1676:
1675:
1671:
1669:
1666:
1665:
1662:30 (6+12+12)
1661:
1659:
1656:
1655:
1651:
1649:
1646:
1645:
1641:
1639:
1636:
1635:
1632:
1629:
1626:
1625:
1620:
1613:
1608:
1601:
1595:
1586:
1582:
1579:
1573:
1569:
1564:
1560:
1557:
1551:
1547:
1544:
1538:
1534:
1529:
1525:
1522:
1518:
1512:
1508:
1505:
1504:
1500:
1497:
1494:
1491:
1490:
1466:
1440:
1413:
1386:
1384:
1380:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1354:
1348:
1343:
1338:
1335:
1333:
1329:
1324:
1319:
1317:
1313:
1309:
1301:
1292:
1286:
1272:
1269:
1268:
1264:
1260:
1257:
1253:
1252:
1247:
1240:
1234:
1233:
1222:
1213:
1204:
1197:
1187:
1178:
1169:
1160:
1159:
1156:
1154:
1149:
1147:
1146:
1141:
1133:
1119:
1114:
1112:
1092:
1087:
1085:
1081:
1077:
1071:
1067:
1055:
1051:
1039:
1035:
1026:
1020:
1003:
994:
985:
984:
981:
963:
957:
954:
948:
935:
917:
912:
909:
904:
891:
873:
868:
864:
858:
845:
842:
838:
834:
830:
826:
822:
814:
810:
805:
801:
799:
795:
791:
790:quasicrystals
787:
783:
778:
776:
771:
767:
759:
755:
745:
740:
733:
730:
727:
726:
723:
720:
718:
715:
714:
710:
707:
705:
702:
701:
697:
691:
689:
686:
685:
681:
679:
676:
675:
671:
669:
666:
665:
661:
659:
656:
655:
597:
595:
592:
591:
588:
585:
582:
581:
576:
569:
564:
561:Pyritohedron
559:
553:
551:
549:
545:
541:
537:
533:
529:
521:
519:
517:
513:
509:
505:
501:
497:
493:
489:
480:
475:
471:
469:
464:
460:
458:
453:
449:
447:
441:
437:
430:
428:
424:
419:
417:
413:
407:
399:
397:
393:
391:
387:
386:space-filling
383:
379:
375:
371:
366:
364:
360:
356:
352:
348:
343:
342:, order 120.
341:
337:
333:
329:
325:
321:
317:
316:duodecahedron
313:
310:
303:
299:
296:
293:
286:
282:
279:
276:
269:
265:
262:
261:Ancient Greek
258:
254:
244:
240:
237:
233:
230:
226:
223:
219:
215:
212:
210:
207:
205:
202:
200:
197:
176:
172:
169:
165:
162:
158:
155:
151:
150:
147:
144:
142:
139:
137:
134:
132:
129:
128:
120:
113:
110:
103:
102:
98:
94:
91:
87:
84:
80:
77:
73:
72:
69:
66:
64:
61:
59:
56:
54:
51:
50:
41:
33:
19:
4432:
4401:
4392:
4384:
4375:
4366:
4346:10-orthoplex
4082:Dodecahedron
4003:
3992:
3981:
3972:
3963:
3954:
3950:
3940:
3932:
3928:
3920:
3916:
3856:
3775:trapezohedra
3726:
3719:
3523:dodecahedron
3276:Apeirohedron
3227:>20 faces
3178:Dodecahedron
3177:
2990:
2974:
2964:
2952:
2943:
2939:
2929:
2912:
2900:
2888:
2864:
2852:
2817:
2811:
2801:
2792:
2712:Armand Spitz
2710:
2673:
2463:
2457:
2455:
2434:golden ratio
2423:
2408:
2404:pyritohedron
2399:
2397:
2386:quasiregular
2375:
2373:
2341:
2335:
2245:
2229:
2223:
2216:
2212:
2208:
2204:
2200:
2193:
2187:
2183:
2179:
2175:
2171:
2164:
2158:
2154:
2151:
2147:
2140:
2136:
2132:
2125:
2112:
2104:
2089:
2081:
2066:
2058:
2050:
2046:
2042:
2031:
2023:
2008:
2000:
1985:
1977:
1969:
1965:
1961:
1957:
1953:
1949:
1939:
1923:
1853:
1743:
1735:
1718:
1714:
1710:
1706:
1704:
1672:20 (4+4+12)
1627:Face polygon
1542:
1382:
1322:
1320:
1305:
1231:
1195:
1150:
1143:
1117:
1115:
1090:
1088:
1079:
1078:
1069:
1068:), 0, ±(1 +
1065:
1053:
1049:
1037:
1033:
1027:
1024:
936:
892:
846:
837:Miller index
820:
818:
779:
758:pyritohedral
754:pyritohedron
753:
751:
682:20 (8 + 12)
672:30 (6 + 24)
583:Face polygon
554:Pyritohedron
525:
512:pentagrammic
485:
420:
409:
394:
367:
357:, while the
347:pyritohedron
344:
315:
308:
305:
298:
291:
288:
281:
274:
271:
264:
257:dodecahedron
256:
250:
131:Pyritohedron
111:T, order 12
47:, order 120
18:Dodecahedral
4485:12 (number)
4355:10-demicube
4316:9-orthoplex
4266:8-orthoplex
4216:7-orthoplex
4173:6-orthoplex
4143:5-orthoplex
4098:Pentachoron
4086:Icosahedron
4061:Tetrahedron
3545:semiregular
3528:icosahedron
3508:tetrahedron
3218:Icosahedron
3166:11–20 faces
3152:Enneahedron
3142:Heptahedron
3132:Pentahedron
3127:Tetrahedron
3003:stellations
2924:, Slovenia.
2752:unicellular
2411:stellations
1377:1 : 1
1374:0 : 1
1371:1 : 1
1368:2 : 1
1365:1 : 1
1362:0 : 1
1359:1 : 1
1312:convex hull
1310:of a cubic
1249:Animations
945:Short sides
488:stellations
427:icosahedron
336:stellations
280:; from
275:dōdekáedron
268:δωδεκάεδρον
193:, order 12
186:, order 16
146:Triangular-
118:, order 48
108:, order 24
4464:Categories
4341:10-simplex
4325:9-demicube
4275:8-demicube
4225:7-demicube
4182:6-demicube
4152:5-demicube
4066:Octahedron
3840:prismatoid
3770:bipyramids
3754:antiprisms
3728:hosohedron
3518:octahedron
3403:prismatoid
3388:(infinite)
3157:Decahedron
3147:Octahedron
3137:Hexahedron
3112:Monohedron
3105:1–10 faces
2827:1811.04131
2785:References
2655:– dual of
2653:trapezoids
2645:, order 11
2636:hendecagon
2594:, dual of
2573:, dual of
2438:zonohedron
2417:is also a
2382:zonohedron
2338:anticupola
1731:pentagonal
1690:Properties
1495:−0.618...
1492:−1.618...
1332:pentagrams
770:pentagonal
728:Properties
542:, and the
320:polyhedron
259:(from
121:Johnson (J
4389:orthoplex
4311:9-simplex
4261:8-simplex
4211:7-simplex
4168:6-simplex
4138:5-simplex
4107:Tesseract
3835:birotunda
3825:bifrustum
3590:snub cube
3485:polyhedra
3417:antiprism
3122:Trihedron
3091:Polyhedra
2940:Curr. Sci
2844:119318080
2571:triangles
2528:, order 8
1836:Cobaltite
1739:cobaltite
1707:tetartoid
1603:Tetartoid
1596:Tetartoid
1521:pentagram
1501:1.618...
1498:0.618...
1271:Honeycomb
1052:), ±(1 −
1036:), ±(1 −
1032:0, ±(1 +
968:Long side
964:⋅
922:Long side
918:⋅
878:Long side
874:⋅
827:shown by
819:The name
792:(such as
544:tetartoid
359:tetartoid
136:Tetartoid
4443:Topics:
4406:demicube
4371:polytope
4365:Uniform
4126:600-cell
4122:120-cell
4075:Demicube
4049:Pentagon
4029:Triangle
3815:bicupola
3795:pyramids
3721:dihedron
3117:Dihedron
2946:: 64–72.
2873:Archived
2732:120-cell
2726:See also
2678:hexagons
2442:Bilinski
2344:It has D
1746:topology
1668:Vertices
678:Vertices
418:{5, 3}.
253:geometry
141:Rhombic-
53:Regular-
4380:simplex
4350:10-cube
4117:24-cell
4103:16-cell
4044:Hexagon
3898:regular
3857:italics
3845:scutoid
3830:rotunda
3820:frustum
3549:uniform
3498:regular
3483:Convex
3437:pyramid
3422:frustum
2632:pyramid
2349:match.
2120:
2101:
2097:
2078:
2074:
2055:
2039:
2020:
2016:
1997:
1993:
1974:
1486:
1472:√
1468:
1460:
1446:√
1442:
1434:
1420:√
1415:
1407:
1393:√
1389:
1289:
1275:
1136:
1122:
1107:
1095:
796:) with
534:of the
444:Convex
318:is any
4320:9-cube
4270:8-cube
4220:7-cube
4177:6-cube
4147:5-cube
4034:Square
3911:Family
3810:cupola
3763:duals:
3749:prisms
3427:cupola
3303:vertex
2883:, U.S.
2842:
2458:convex
2413:, the
1902:Chiral
1717:, and
1709:(also
1694:convex
1523:faces
1506:Image
1356:Ratio
1064:±(1 −
1048:±(1 +
855:Height
829:pyrite
782:pyrite
376:. The
353:, has
351:pyrite
292:dṓdeka
285:δώδεκα
63:Great-
4039:p-gon
3432:wedge
3412:prism
3272:(132)
2840:S2CID
2822:arXiv
2758:algae
2592:kites
2590:– 12
2380:is a
2053:); (−
2041:); (−
1972:); (−
1960:); (−
1658:Edges
1648:Faces
1617:(See
1198:= 1/2
1084:wedge
901:Width
668:Edges
658:Faces
573:(See
546:with
309:hédra
263:
4397:cube
4070:Cube
3900:and
3513:cube
3394:(57)
3365:(92)
3359:(13)
3353:(13)
3322:(16)
3298:edge
3293:face
3266:(90)
3260:(60)
3254:(48)
3248:(32)
3242:(30)
3236:(24)
2734:– a
2398:The
2391:(an
2374:The
2246:The
2131:0 ≤
1924:The
1619:here
1556:cube
1060:and
1056:), 0
833:cube
575:here
421:The
380:and
368:The
361:has
302:ἕδρα
255:, a
3946:(p)
3547:or
3382:(4)
3347:(5)
3316:(9)
3278:(∞)
2832:doi
2750:(a
2672:or
2642:11v
2478:10h
2264:to
2235:≠ 0
2215:− 2
2186:− 2
2045:, −
1995:, −
1964:, −
1862:.)
1652:12
1642:gT
1545:.
1477:+ 1
1451:− 1
1425:+ 1
1411:−1
1398:+ 1
1113:).
1044:,
773:of
739:Net
662:12
526:In
251:In
4466::
4451:•
4447:•
4427:21
4423:•
4420:k1
4416:•
4413:k2
4391:•
4348:•
4318:•
4296:21
4292:•
4289:41
4285:•
4282:42
4268:•
4246:21
4242:•
4239:31
4235:•
4232:32
4218:•
4196:21
4192:•
4189:22
4175:•
4145:•
4124:•
4105:•
4084:•
4068:•
4000:/
3989:/
3979:/
3970:/
3948:/
3405:s
2944:78
2942:.
2938:.
2838:.
2830:.
2818:31
2816:.
2810:.
2722:.
2697:5d
2693:,
2684:4h
2680:,
2663:3h
2659:,
2638:,
2615:,
2602:6d
2598:,
2581:6h
2577:,
2563:)
2551:2v
2538:4h
2525:2d
2512:5v
2491:5d
2421:.
2346:3d
2224:nd
2217:bc
2213:ac
2211:−
2207:+
2205:ab
2203:+
2199:=
2188:bc
2184:ac
2182:+
2178:+
2176:ab
2174:−
2170:=
2159:bc
2157:−
2150:=
2139:≤
2135:≤
2122:),
2099:,
2076:,
2049:,
2018:,
1968:,
1956:,
1952:,
1944::
1713:,
1705:A
1696:,
1576:A
1554:A
1464:1
1438:0
1349:.)
1155:.
1120:=
1093:=
1076:.
958:12
760:(T
752:A
550::
392:.
365:.
191:3h
184:4h
125:)
123:84
4435:-
4433:n
4425:k
4418:2
4411:1
4404:-
4402:n
4395:-
4393:n
4387:-
4385:n
4378:-
4376:n
4369:-
4367:n
4294:4
4287:2
4280:1
4244:3
4237:2
4230:1
4194:2
4187:1
4016:n
4014:H
4007:2
4004:G
3996:4
3993:F
3985:8
3982:E
3976:7
3973:E
3967:6
3964:E
3955:n
3951:D
3944:2
3941:I
3933:n
3929:B
3921:n
3917:A
3889:e
3882:t
3875:v
3859:.
3551:)
3543:(
3500:)
3496:(
3476:e
3469:t
3462:v
3083:e
3076:t
3069:v
3012:.
2846:.
2834::
2824::
2760:)
2695:D
2682:D
2661:D
2640:C
2619:d
2617:T
2600:D
2579:D
2549:C
2536:D
2523:D
2510:C
2489:D
2476:D
2237:.
2233:2
2230:d
2227:1
2219:,
2209:b
2201:a
2197:2
2194:d
2190:,
2180:b
2172:a
2168:1
2165:d
2161:,
2155:c
2152:a
2148:n
2144:,
2141:c
2137:b
2133:a
2116:2
2113:d
2109:/
2105:n
2093:2
2090:d
2086:/
2082:n
2070:2
2067:d
2063:/
2059:n
2051:b
2047:a
2043:c
2035:1
2032:d
2028:/
2024:n
2012:1
2009:d
2005:/
2001:n
1989:1
1986:d
1982:/
1978:n
1970:c
1966:b
1962:a
1958:c
1954:b
1950:a
1948:(
1683:T
1483:2
1480:/
1474:5
1457:2
1454:/
1448:5
1431:2
1428:/
1422:5
1418:−
1404:2
1401:/
1395:5
1387:−
1383:h
1285:φ
1281:/
1278:1
1232:φ
1196:h
1132:φ
1128:/
1125:1
1118:h
1104:2
1101:/
1098:1
1091:h
1080:h
1074:)
1072:)
1070:h
1066:h
1062:(
1058:)
1054:h
1050:h
1046:(
1042:)
1040:)
1038:h
1034:h
1030:(
955:7
949:=
913:3
910:4
905:=
869:2
865:5
859:=
762:h
709:T
695:h
693:T
312:)
306:(
295:)
289:(
278:)
272:(
189:D
182:D
116:h
114:O
106:h
104:T
45:h
43:I
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
