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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
761:
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
2337:
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
832:
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).
384:
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.
1725:
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
3005:
1314:
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
968:
878:
1737:
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
922:
334:
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
1917:
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.
3463:
2243:
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.)
792:
1307:
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.
1323:. On the other side, past the rhombic dodecahedron, we get a nonconvex equilateral dodecahedron with fish-shaped self-intersecting equilateral pentagonal faces.
782:
563:
928:
256:
3456:
2403:
1340:
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).
838:
4441:
1901:
3449:
1722:
faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular and the figure has no fivefold symmetry axes.
1335:
884:
495:{5/2, 3}. The small stellated dodecahedron and great dodecahedron are dual to each other; the great stellated dodecahedron is dual to the
3876:
3070:
2869:
1666:
676:
638:
628:
618:
610:
600:
2325:
A lower symmetry form of the regular dodecahedron can be constructed as the dual of a polyhedron constructed from two triangular
692:
590:
202:
2861:
1881:
507:; the small stellated dodecahedron and great dodecahedron are different realisations of another abstract regular polyhedron.
3368:
1863:
633:
623:
605:
595:
320:
2658:
520:
187:
1607:
480:
2637:
2533:
1505:
1316:
492:
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467:
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197:
56:
46:
1872:
1099:
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1296:
1141:
273:
4458:
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3258:
2520:
1739:
1626:
769:
Although regular dodecahedra do not exist in crystals, the pyritohedron form occurs in the crystals of the mineral
2785:
1908 Chambers's
Twentieth Century Dictionary of the English Language, 1913 Webster's Revised Unabridged Dictionary
3687:
3682:
2731:
2724:
1140:
Two pyritohedra with swapped nonzero coordinates are in dual positions to each other like the dodecahedra in the
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3252:
2645:
503:
faces. The convex regular dodecahedron and great stellated dodecahedron are different realisations of the same
2801:
1156:
556:
3869:
3793:
3788:
3667:
3573:
3380:
3308:
3228:
3063:
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2576:
2418:
2374:
374:
134:
1174:
4463:
3657:
3598:
3588:
3533:
3314:
2767:
1848:
1847:. (The tetartoid shown here is based on one that is itself created by enlarging 24 of the 48 faces of the
1165:
527:
that are topologically equivalent to the regular dodecahedron but less symmetrical: the pyritohedron with
366:
249:
290:
4413:
4406:
4399:
3677:
3593:
3548:
2752:
2704:
2601:
2414:
746:
681:
528:
343:
2395:
where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces.
2982:, by Paul Stephenson, 1993, The Mathematical Gazette, Vol. 77, No. 479 (Jul., 1993), pp. 220–226
1525:
4473:
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2473:
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1930:
1715:
1711:
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1320:
1304:
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1128:
786:
754:
697:
536:
524:
434:
394:
358:
351:
328:
312:
192:
129:
41:
2796:
2343:
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4074:
3803:
3672:
3647:
3632:
3568:
3516:
2597:
2584:
2555:
2254:
2240:
1844:
763:
362:
2881:
2845:
1547:
can be divided into a pyritohedron by bisecting all the edges, and faces in alternate directions.
1209:
231:
4294:
4244:
4194:
4151:
4121:
4081:
4044:
3862:
3818:
3783:
3642:
3537:
3486:
3425:
3302:
3296:
3056:
3028:
K.J.M. MacLean, A Geometric
Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
3015:
2828:
2810:
2757:
2620:
2384:) and occurs in nature as a crystal form. The rhombic dodecahedron packs together to fill space.
488:
456:
51:
20:
404:
210:
1499:
1310:
It is possible to go past these limiting cases, creating concave or nonconvex pyritohedra. The
462:
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224:
217:
85:
71:
4433:
3798:
3608:
3583:
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2511:
2494:
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1656:
1244:
1200:
710:
666:
504:
496:
1098:(a quarter of the cube edge length) for perfect natural pyrite (also the pyritohedron in the
4437:
4002:
3991:
3980:
3969:
3960:
3951:
3938:
3916:
3904:
3890:
3886:
3737:
3420:
3400:
3222:
2820:
1072:
727:
1334:
Versions with equal absolute values and opposing signs form a honeycomb together. (Compare
1319:
where all edges and angles are equal again, and the faces have been distorted into regular
4027:
4012:
3374:
3286:
3281:
3246:
3201:
3191:
3181:
3176:
2865:
2736:
2708:
2628:
2580:
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2549:
2537:
2507:
2498:
2460:
2430:
1686:
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720:
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656:
646:
582:
516:
411:
2998:
1516:
4377:
3558:
3481:
3430:
3333:
3196:
3186:
2437:
along with the rhombic triacontahedron, the rhombic icosahedron and rhombic hexahedra.
1251:
829:
774:
400:
316:
3027:
3021:
1803:
1772:
1600:
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3351:
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3240:
3171:
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2743:
2488:
2377:
1914:
813:
797:
2987:
1763:
990:
963:{\displaystyle {\text{Short sides}}={\sqrt {\frac {7}{12}}}\cdot {\text{Long side}}}
4334:
2924:
2858:
2700:
2422:
2392:
2355:
1560:
1273:
1219:
1120:
825:
778:
156:
119:
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78:
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3496:
3206:
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in 1960. This figure is another spacefiller, and can also occur in non-periodic
1820:
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1300:
732:
415:
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2624:
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1682:
873:{\displaystyle {\text{Height}}={\frac {\sqrt {5}}{2}}\cdot {\text{Long side}}}
476:
324:
308:
163:
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2641:
2453:
Topologically distinct dodecahedra (excluding pentagonal and rhombic forms)
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1727:
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532:
500:
124:
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2297:
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2283:
2276:
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981:
499:{3, 5/2}. All of these regular star dodecahedra have regular pentagonal or
19:
This article is about the three-dimensional shape. Not to be confused with
2262:
1551:
428:
142:
64:
4359:
4114:
4110:
4037:
3709:
3105:
2795:
Athreya, Jayadev S.; Aulicino, David; Hooper, W. Patrick (May 27, 2020).
2720:
2559:
1734:
1719:
1619:
758:
575:
479:, all of which are regular star dodecahedra. They form three of the four
241:
1556:
A regular dodecahedron is an intermediate case with equal edge lengths.
4368:
4338:
4105:
4100:
4091:
4032:
3833:
3808:
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3037:
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766:
are three mutually perpendicular twofold axes and four threefold axes.
149:
4308:
4258:
4208:
4165:
4135:
4086:
4022:
2373:
with twelve rhombic faces and octahedral symmetry. It is dual to the
917:{\displaystyle {\text{Width}}={\frac {4}{3}}\cdot {\text{Long side}}}
817:
770:
339:
296:
279:
262:
2983:
1262:
of alternating convex and concave pyritohedra with heights between ±
3441:
2815:
1569:
is a degenerate case with the 6 crossedges reduced to length zero.
2746:
1900:
3006:
Editable printable net of a dodecahedron with interactive 3D view
4058:
3501:
1929:
The following points are vertices of a tetartoid pentagon under
1843:
A tetartoid can be created by enlarging 12 of the 24 faces of a
1544:
1075:-shaped "roof" above the faces of that cube with edge length 2.
1014:
The eight vertices of a cube have the coordinates (±1, ±1, ±1).
821:
773:, and it may be an inspiration for the discovery of the regular
3445:
3052:
519:, two important dodecahedra can occur as crystal forms in some
361:
can be seen as a limiting case of the pyritohedron, and it has
2727:(4D polytope) whose surface consists of 120 dodecahedral cells
418:{3, 5}, having five equilateral triangles around each vertex.
311:
with twelve flat faces. The most familiar dodecahedron is the
777:
form. The true regular dodecahedron can occur as a shape for
3048:
2689:, order 20, topologically equivalent to regular dodecahedron
2797:"Platonic Solids and High Genus Covers of Lattice Surfaces"
399:
The convex regular dodecahedron is one of the five regular
3040:: Software used to create some of the images on this page.
2239:
is a tetartoid with more than the required symmetry. The
1893:
tetartoids based on the dyakis dodecahedron in the middle
1295:
The pyritohedron has a geometric degree of freedom with
2948:. Numericana.com (2001-12-31). Retrieved on 2016-12-02.
420:
2923:
Lord, E. A.; Ranganathan, S.; Kulkarni, U. D. (2000).
2896:. Demonstrations.wolfram.com. Retrieved on 2016-12-02.
2703:
used a dodecahedron as the "globe" equivalent for his
25:
931:
887:
841:
373:
variations, along with the rhombic dodecahedra, are
3776:
3751:
3726:
3701:
3617:
3525:
3480:
3390:
3361:
3326:
3274:
3215:
3154:
3093:
2990:VRML models and animations of Pyritohedron and its
1183:Orthographic projections of the pyritohedron with
1017:The coordinates of the 12 additional vertices are
962:
916:
872:
2497:– 5 triangles, 5 squares, 1 pentagon, 1 decagon,
3044:How to make a dodecahedron from a Styrofoam cube
2925:"Tilings, coverings, clusters and quasicrystals"
1781:Orthographic projections from 2- and 3-fold axes
1530:The concave equilateral dodecahedron, called an
475:The convex regular dodecahedron also has three
999:Natural pyrite (with face angles on the right)
789:, which includes true fivefold rotation axes.
3870:
3457:
3064:
2421:, i.e. the diagonals are in the ratio of the
2417:, has twelve faces congruent to those of the
8:
2413:Another important rhombic dodecahedron, the
824:). In pyritohedral pyrite, the faces have a
315:with regular pentagons as faces, which is a
2980:Plato's Fourth Solid and the "Pyritohedron"
2445:There are 6,384,634 topologically distinct
1578:Self-intersecting equilateral dodecahedron
3877:
3863:
3855:
3730:
3464:
3450:
3442:
3071:
3057:
3049:
2245:
1833:
1588:
1325:
1234:
546:
2999:"3D convex uniform polyhedra o3o5x – doe"
2884:. Galleries.com. Retrieved on 2016-12-02.
2848:. Galleries.com. Retrieved on 2016-12-02.
2814:
2476:– 10 equilateral triangles, 2 pentagons,
955:
940:
932:
930:
909:
896:
888:
886:
865:
850:
842:
840:
2907:Introduction to golden rhombic polyhedra
2354:
1837:Relationship to the dyakis dodecahedron
1819:
791:
4442:List of regular polytopes and compounds
2778:
2623:– 11 isosceles triangles and 1 regular
1303:at one limit of collinear edges, and a
1521:Degenerate, 12 vertices in the center
1133:
378:
327:of the convex form. All of these have
2909:. Faculty of Electrical Engineering,
2398:The rhombic dodecahedron has several
7:
2321:Dual of triangular gyrobianticupola
1594:Tetragonal pentagonal dodecahedron
783:holmium–magnesium–zinc quasicrystal
3088:Listed by number of faces and type
3018:– Models made with Modular Origami
2705:Digital Dome planetarium projector
2680:Truncated pentagonal trapezohedron
2600:– 12 isosceles triangles, dual of
1700:tetragonal pentagonal dodecahedron
1329:Special cases of the pyritohedron
422:Four kinds of regular dodecahedra
14:
2870:University of Wisconsin-Green Bay
2329:connected base-to-base, called a
1733:Abstractions sharing the solid's
1708:tetrahedric pentagon dodecahedron
812:comes from one of the two common
335:
2960:"Forerunners of the Planetarium"
2868:. Natural and Applied Sciences,
2536:– 10 triangles and 2 pentagons,
2450:angles between edges or faces.)
2342:
2310:
2303:
2296:
2289:
2282:
2275:
2268:
2261:
2115:under the following conditions:
1880:
1871:
1862:
1802:
1793:
1771:
1762:
1753:
1710:) is a dodecahedron with chiral
1599:
1572:
1559:
1550:
1537:
1524:
1515:
1498:
1250:
1243:
1208:
1199:
1173:
1164:
1155:
989:
980:
731:
636:
631:
626:
621:
616:
608:
603:
598:
593:
588:
555:
461:
450:
439:
427:
230:
223:
216:
209:
162:
155:
148:
141:
84:
77:
70:
63:
2707:, based upon a suggestion from
828:of (210), which means that the
403:and can be represented by its
347:
1:
3299:(two infinite groups and 75)
3033:Dodecahedron 3D Visualization
3024:The Encyclopedia of Polyhedra
2825:10.1080/10586458.2020.1712564
2665:– 8 rhombi and 4 equilateral
2659:Rhombo-hexagonal dodecahedron
2523:– 8 triangles and 4 squares,
1742:this is a gyro tetrahedron.)
1730:can have this symmetry form.
1284:and 1 (rhombic dodecahedron)
303: 'base, seat, face') or
3844:Degenerate polyhedra are in
3317:(two infinite groups and 50)
3038:Stella: Polyhedron Navigator
2958:Ley, Willy (February 1965).
2859:The 48 Special Crystal Forms
2638:Trapezo-rhombic dodecahedron
2548:Congruent irregular faced: (
2534:Metabidiminished icosahedron
2391:can be seen as a degenerate
2331:triangular gyrobianticupola.
1506:great stellated dodecahedron
1317:great stellated dodecahedron
725:
511:Other pentagonal dodecahedra
493:great stellated dodecahedron
485:small stellated dodecahedron
468:Great stellated dodecahedron
446:Small stellated dodecahedron
425:
371:trapezo-rhombic dodecahedron
207:
185:
169:
3663:pentagonal icositetrahedron
3604:truncated icosidodecahedron
2408:parallelohedral spacefiller
1142:compound of two dodecahedra
505:abstract regular polyhedron
338:, a common crystal form in
323:, which are constructed as
286: 'twelve' and
4490:
4431:
3858:
3693:pentagonal hexecontahedron
3653:deltoidal icositetrahedron
2988:Stellation of Pyritohedron
2616:Other less regular faced:
2521:Elongated square dipyramid
2463:– 10 squares, 2 decagons,
2249:Tetartoid variations from
1812:Cubic and tetrahedral form
1740:Conway polyhedron notation
1718:, it has twelve identical
757:, it has twelve identical
392:
289:
272:
255:
18:
3842:
3733:
3688:disdyakis triacontahedron
3683:deltoidal hexecontahedron
3369:Kepler–Poinsot polyhedron
3086:
3022:Virtual Reality Polyhedra
2905:Hafner, I. and Zitko, T.
2732:Braarudosphaera bigelowii
2248:
1598:
1591:
1452:
1426:
1399:
1370:
1333:
1328:
1237:
820:(the other one being the
796:Dual positions in pyrite
726:
554:
549:
177:
170:
31:
2962:. For Your Information.
2802:Experimental Mathematics
2735:– a dodecahedron shaped
2646:triangular orthobicupola
1282:Heights between 0 (cube)
481:Kepler–Poinsot polyhedra
321:regular star dodecahedra
16:Polyhedron with 12 faces
3794:gyroelongated bipyramid
3668:rhombic triacontahedron
3574:truncated cuboctahedron
3381:Uniform star polyhedron
3309:quasiregular polyhedron
2911:University of Ljubljana
2577:Hexagonal trapezohedron
2419:rhombic triacontahedron
1704:pentagon-tritetrahedron
1100:Weaire–Phelan structure
745:is a dodecahedron with
319:. There are also three
3789:truncated trapezohedra
3658:disdyakis dodecahedron
3624:(duals of Archimedean)
3599:rhombicosidodecahedron
3589:truncated dodecahedron
3315:semiregular polyhedron
2964:Galaxy Science Fiction
2768:Truncated dodecahedron
2663:elongated Dodecahedron
2360:
1906:
1849:disdyakis dodecahedron
1827:
1610:for a rotating model.)
964:
918:
874:
800:
566:for a rotating model.)
367:elongated dodecahedron
3678:pentakis dodecahedron
3594:truncated icosahedron
3549:truncated tetrahedron
3362:non-convex polyhedron
3011:The Uniform Polyhedra
2753:Pentakis dodecahedron
2602:truncated tetrahedron
2429:and was described by
2415:Bilinski dodecahedron
2358:
1925:Cartesian coordinates
1904:
1823:
1078:An important case is
1071:is the height of the
1010:Cartesian coordinates
965:
919:
875:
795:
753:) symmetry. Like the
529:pyritohedral symmetry
377:. There are numerous
344:pyritohedral symmetry
3638:rhombic dodecahedron
3564:truncated octahedron
2474:Pentagonal antiprism
2389:rhombic dodecahedron
2366:rhombic dodecahedron
2359:Rhombic dodecahedron
2351:Rhombic dodecahedron
2251:regular dodecahedron
2237:regular dodecahedron
1931:tetrahedral symmetry
1716:regular dodecahedron
1712:tetrahedral symmetry
1674:, , (332), order 12
1567:rhombic dodecahedron
1305:rhombic dodecahedron
1129:regular dodecahedron
929:
885:
839:
787:icosahedral symmetry
755:regular dodecahedron
700:, , (332), order 12
687:, , (3*2), order 24
537:tetrahedral symmetry
525:cubic crystal system
435:regular dodecahedron
395:Regular dodecahedron
389:Regular dodecahedron
359:rhombic dodecahedron
352:tetrahedral symmetry
329:icosahedral symmetry
313:regular dodecahedron
4426:pentagonal polytope
4325:Uniform 10-polytope
3885:Fundamental convex
3673:triakis icosahedron
3648:tetrakis hexahedron
3633:triakis tetrahedron
3569:rhombicuboctahedron
2997:Klitzing, Richard.
2598:Triakis tetrahedron
2585:hexagonal antiprism
2556:Hexagonal bipyramid
2470:symmetry, order 40.
2457:Uniform polyhedra:
2255:triakis tetrahedron
2241:triakis tetrahedron
1845:dyakis dodecahedron
1127:= 0.618... for the
764:rotational symmetry
423:
363:octahedral symmetry
28:
27:Common dodecahedra
4295:Uniform 9-polytope
4245:Uniform 8-polytope
4195:Uniform 7-polytope
4152:Uniform 6-polytope
4122:Uniform 5-polytope
4082:Uniform polychoron
4045:Uniform polyhedron
3893:in dimensions 2–10
3643:triakis octahedron
3528:Archimedean solids
3303:regular polyhedron
3297:uniform polyhedron
3259:Hectotriadiohedron
2946:Counting polyhedra
2864:2013-09-18 at the
2758:Roman dodecahedron
2725:regular polychoron
2676:symmetry, order 16
2655:symmetry, order 12
2611:symmetry, order 24
2594:symmetry, order 24
2573:symmetry, order 24
2530:symmetry, order 16
2504:symmetry, order 10
2483:symmetry, order 20
2361:
1907:
1828:
1620:irregular pentagon
1218:Heights 1/2 and 1/
960:
914:
870:
801:
576:isosceles pentagon
491:{5, 5/2}, and the
489:great dodecahedron
457:Great dodecahedron
421:
26:
21:Roman dodecahedron
4459:Individual graphs
4447:
4446:
4434:Polytope families
3891:uniform polytopes
3853:
3852:
3772:
3771:
3609:snub dodecahedron
3584:icosidodecahedron
3439:
3438:
3340:Archimedean solid
3327:convex polyhedron
3235:Icosidodecahedron
3016:Origami Polyhedra
2966:. pp. 87–98.
2763:Snub dodecahedron
2543:symmetry, order 4
2495:Pentagonal cupola
2491:(regular faced):
2441:Other dodecahedra
2382:Archimedean solid
2318:
2317:
2231:Geometric freedom
1922:
1921:
1911:
1910:
1832:
1831:
1692:
1691:
1611:
1582:
1581:
1532:endo-dodecahedron
1312:endo-dodecahedron
1291:Geometric freedom
1288:
1287:
1233:
1232:
1137:for other cases.
1134:Geometric freedom
1007:
1006:
958:
950:
949:
935:
912:
904:
891:
868:
860:
856:
845:
739:
738:
711:Pseudoicosahedron
567:
497:great icosahedron
473:
472:
379:other dodecahedra
238:
237:
203:Rhombo-triangular
4481:
4438:Regular polytope
3999:
3988:
3977:
3936:
3879:
3872:
3865:
3856:
3731:
3727:Dihedral uniform
3702:Dihedral regular
3625:
3541:
3490:
3466:
3459:
3452:
3443:
3275:elemental things
3253:Enneacontahedron
3223:Icositetrahedron
3073:
3066:
3059:
3050:
3002:
2968:
2967:
2955:
2949:
2943:
2937:
2936:
2920:
2914:
2903:
2897:
2891:
2885:
2879:
2873:
2855:
2849:
2843:
2837:
2836:
2818:
2792:
2786:
2783:
2510:– 12 triangles,
2346:
2314:
2307:
2300:
2293:
2286:
2279:
2272:
2265:
2246:
2225:
2132:
2110:
2108:
2107:
2099:
2096:
2087:
2085:
2084:
2076:
2073:
2064:
2062:
2061:
2053:
2050:
2029:
2027:
2026:
2018:
2015:
2006:
2004:
2003:
1995:
1992:
1983:
1981:
1980:
1972:
1969:
1884:
1875:
1866:
1854:
1853:
1834:
1806:
1797:
1775:
1766:
1757:
1745:
1744:
1605:
1603:
1589:
1576:
1563:
1554:
1541:
1528:
1519:
1502:
1476:
1474:
1473:
1470:
1467:
1465:
1464:
1450:
1448:
1447:
1444:
1441:
1439:
1438:
1424:
1422:
1421:
1418:
1415:
1413:
1412:
1397:
1395:
1394:
1391:
1388:
1386:
1385:
1326:
1279:
1277:
1276:
1271:
1268:
1254:
1247:
1235:
1212:
1203:
1177:
1168:
1159:
1147:
1146:
1126:
1124:
1123:
1118:
1115:
1097:
1095:
1094:
1091:
1088:
993:
984:
972:
971:
969:
967:
966:
961:
959:
956:
951:
942:
941:
936:
933:
923:
921:
920:
915:
913:
910:
905:
897:
892:
889:
879:
877:
876:
871:
869:
866:
861:
852:
851:
846:
843:
735:
641:
640:
639:
635:
634:
630:
629:
625:
624:
620:
619:
613:
612:
611:
607:
606:
602:
601:
597:
596:
592:
591:
583:Coxeter diagrams
561:
559:
547:
521:symmetry classes
465:
454:
443:
431:
424:
300:
293:
283:
276:
266:
259:
234:
227:
220:
213:
188:Rhombo-hexagonal
166:
159:
152:
145:
88:
81:
74:
67:
29:
4489:
4488:
4484:
4483:
4482:
4480:
4479:
4478:
4469:Platonic solids
4449:
4448:
4417:
4410:
4403:
4286:
4279:
4272:
4236:
4229:
4222:
4186:
4179:
4013:Regular polygon
4006:
3997:
3990:
3986:
3979:
3975:
3966:
3957:
3950:
3946:
3934:
3928:
3924:
3912:
3894:
3883:
3854:
3849:
3838:
3777:Dihedral others
3768:
3747:
3722:
3697:
3626:
3623:
3622:
3613:
3542:
3531:
3530:
3521:
3484:
3482:Platonic solids
3476:
3470:
3440:
3435:
3386:
3375:Star polyhedron
3357:
3322:
3270:
3247:Hexecontahedron
3229:Triacontahedron
3211:
3202:Enneadecahedron
3192:Heptadecahedron
3182:Pentadecahedron
3177:Tetradecahedron
3150:
3089:
3082:
3077:
2996:
2976:
2971:
2957:
2956:
2952:
2944:
2940:
2922:
2921:
2917:
2904:
2900:
2892:
2888:
2880:
2876:
2866:Wayback Machine
2856:
2852:
2844:
2840:
2794:
2793:
2789:
2784:
2780:
2776:
2737:coccolithophore
2717:
2709:Albert Einstein
2698:
2696:Practical usage
2687:
2674:
2653:
2632:
2609:
2592:
2571:
2564:hexagonal prism
2558:– 12 isosceles
2550:face-transitive
2541:
2528:
2515:
2508:Snub disphenoid
2502:
2481:
2468:
2461:Decagonal prism
2443:
2425:. It is also a
2353:
2336:
2323:
2233:
2223:
2217:
2211:
2187:
2158:
2119:
2106:
2100:
2097:
2092:
2091:
2089:
2083:
2077:
2074:
2069:
2068:
2066:
2060:
2054:
2051:
2046:
2045:
2043:
2025:
2019:
2016:
2011:
2010:
2008:
2002:
1996:
1993:
1988:
1987:
1985:
1979:
1973:
1970:
1965:
1964:
1962:
1927:
1897:
1896:
1895:
1894:
1887:
1886:
1885:
1877:
1876:
1868:
1867:
1816:
1815:
1814:
1813:
1809:
1808:
1807:
1799:
1798:
1785:
1784:
1783:
1782:
1778:
1777:
1776:
1768:
1767:
1759:
1758:
1687:face transitive
1627:Conway notation
1604:
1593:
1587:
1577:
1564:
1555:
1542:
1529:
1520:
1508:, with regular
1503:
1471:
1468:
1462:
1460:
1459:
1458:
1456:
1445:
1442:
1436:
1434:
1433:
1432:
1430:
1419:
1416:
1410:
1408:
1406:
1405:
1403:
1392:
1389:
1383:
1381:
1380:
1379:
1377:
1339:
1293:
1283:
1272:
1269:
1266:
1265:
1263:
1227:
1226:
1225:
1224:
1215:
1214:
1213:
1205:
1204:
1191:
1190:
1189:
1188:
1180:
1179:
1178:
1170:
1169:
1161:
1160:
1119:
1116:
1113:
1112:
1110:
1105:Another one is
1092:
1089:
1086:
1085:
1083:
1064:
1052:
1048:
1036:
1032:
1020:
1012:
1003:
1002:
1001:
1000:
996:
995:
994:
986:
985:
927:
926:
883:
882:
837:
836:
806:
752:
730:
721:face transitive
706:Dual polyhedron
685:
637:
632:
627:
622:
617:
615:
614:
609:
604:
599:
594:
589:
587:
560:
545:
517:crystallography
513:
483:. They are the
466:
455:
444:
432:
414:is the regular
412:dual polyhedron
405:Schläfli symbol
401:Platonic solids
397:
391:
198:Trapezo-rhombic
181:
174:
113:
106:
96:
57:Great stellated
47:Small stellated
35:
24:
17:
12:
11:
5:
4487:
4485:
4477:
4476:
4471:
4466:
4461:
4451:
4450:
4445:
4444:
4429:
4428:
4419:
4415:
4408:
4401:
4397:
4388:
4371:
4362:
4351:
4350:
4348:
4346:
4341:
4332:
4327:
4321:
4320:
4318:
4316:
4311:
4302:
4297:
4291:
4290:
4288:
4284:
4277:
4270:
4266:
4261:
4252:
4247:
4241:
4240:
4238:
4234:
4227:
4220:
4216:
4211:
4202:
4197:
4191:
4190:
4188:
4184:
4177:
4173:
4168:
4159:
4154:
4148:
4147:
4145:
4143:
4138:
4129:
4124:
4118:
4117:
4108:
4103:
4098:
4089:
4084:
4078:
4077:
4068:
4066:
4061:
4052:
4047:
4041:
4040:
4035:
4030:
4025:
4020:
4015:
4009:
4008:
4004:
4000:
3995:
3984:
3973:
3964:
3955:
3948:
3942:
3932:
3926:
3920:
3914:
3908:
3902:
3896:
3895:
3884:
3882:
3881:
3874:
3867:
3859:
3851:
3850:
3843:
3840:
3839:
3837:
3836:
3831:
3826:
3821:
3816:
3811:
3806:
3801:
3796:
3791:
3786:
3780:
3778:
3774:
3773:
3770:
3769:
3767:
3766:
3761:
3755:
3753:
3749:
3748:
3746:
3745:
3740:
3734:
3728:
3724:
3723:
3721:
3720:
3713:
3705:
3703:
3699:
3698:
3696:
3695:
3690:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3635:
3629:
3627:
3620:Catalan solids
3618:
3615:
3614:
3612:
3611:
3606:
3601:
3596:
3591:
3586:
3581:
3576:
3571:
3566:
3561:
3559:truncated cube
3556:
3551:
3545:
3543:
3526:
3523:
3522:
3520:
3519:
3514:
3509:
3504:
3499:
3493:
3491:
3478:
3477:
3471:
3469:
3468:
3461:
3454:
3446:
3437:
3436:
3434:
3433:
3431:parallelepiped
3428:
3423:
3418:
3413:
3408:
3403:
3397:
3395:
3388:
3387:
3385:
3384:
3378:
3372:
3365:
3363:
3359:
3358:
3356:
3355:
3349:
3343:
3337:
3334:Platonic solid
3330:
3328:
3324:
3323:
3321:
3320:
3319:
3318:
3312:
3306:
3294:
3289:
3284:
3278:
3276:
3272:
3271:
3269:
3268:
3262:
3256:
3250:
3244:
3238:
3232:
3226:
3219:
3217:
3213:
3212:
3210:
3209:
3204:
3199:
3197:Octadecahedron
3194:
3189:
3187:Hexadecahedron
3184:
3179:
3174:
3169:
3164:
3158:
3156:
3152:
3151:
3149:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3103:
3097:
3095:
3091:
3090:
3087:
3084:
3083:
3078:
3076:
3075:
3068:
3061:
3053:
3047:
3046:
3041:
3035:
3030:
3025:
3019:
3013:
3008:
3003:
2994:
2985:
2975:
2974:External links
2972:
2970:
2969:
2950:
2938:
2915:
2898:
2886:
2874:
2857:Dutch, Steve.
2850:
2838:
2809:(3): 847–877.
2787:
2777:
2775:
2772:
2771:
2770:
2765:
2760:
2755:
2750:
2734:
2728:
2716:
2713:
2697:
2694:
2693:
2692:
2691:
2690:
2685:
2677:
2672:
2656:
2651:
2640:– 6 rhombi, 6
2635:
2630:
2614:
2613:
2612:
2607:
2595:
2590:
2574:
2569:
2546:
2545:
2544:
2539:
2531:
2526:
2518:
2513:
2505:
2500:
2489:Johnson solids
2486:
2485:
2484:
2479:
2471:
2466:
2442:
2439:
2404:first of which
2352:
2349:
2348:
2347:
2334:
2322:
2319:
2316:
2315:
2308:
2301:
2294:
2287:
2280:
2273:
2266:
2258:
2257:
2232:
2229:
2228:
2227:
2221:
2215:
2209:
2185:
2180:
2156:
2151:
2134:
2113:
2112:
2104:
2081:
2058:
2023:
2000:
1977:
1926:
1923:
1920:
1919:
1909:
1908:
1898:
1889:
1888:
1879:
1878:
1870:
1869:
1861:
1860:
1859:
1858:
1857:
1839:
1838:
1830:
1829:
1817:
1811:
1810:
1801:
1800:
1792:
1791:
1790:
1789:
1788:
1786:
1780:
1779:
1770:
1769:
1761:
1760:
1752:
1751:
1750:
1749:
1748:
1714:(T). Like the
1690:
1689:
1680:
1676:
1675:
1669:
1667:Symmetry group
1663:
1662:
1659:
1653:
1652:
1649:
1643:
1642:
1639:
1633:
1632:
1629:
1623:
1622:
1617:
1613:
1612:
1596:
1595:
1586:
1583:
1580:
1579:
1570:
1557:
1548:
1535:
1522:
1513:
1504:Regular star,
1496:
1492:
1491:
1488:
1485:
1482:
1478:
1477:
1454:
1451:
1428:
1425:
1401:
1398:
1374:
1368:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1342:
1341:
1336:this animation
1331:
1330:
1297:limiting cases
1292:
1289:
1286:
1285:
1280:
1256:
1255:
1248:
1240:
1239:
1231:
1230:
1228:
1217:
1216:
1207:
1206:
1198:
1197:
1196:
1195:
1194:
1192:
1182:
1181:
1172:
1171:
1163:
1162:
1154:
1153:
1152:
1151:
1150:
1131:. See section
1062:
1050:
1046:
1034:
1030:
1018:
1011:
1008:
1005:
1004:
998:
997:
988:
987:
979:
978:
977:
976:
975:
954:
948:
945:
939:
908:
903:
900:
895:
864:
859:
855:
849:
830:dihedral angle
814:crystal habits
810:crystal pyrite
805:
804:Crystal pyrite
802:
798:crystal models
775:Platonic solid
750:
737:
736:
724:
723:
718:
714:
713:
708:
702:
701:
695:
693:Rotation group
689:
688:
683:
679:
677:Symmetry group
673:
672:
669:
663:
662:
659:
653:
652:
649:
643:
642:
585:
579:
578:
573:
569:
568:
552:
551:
544:
541:
512:
509:
487:{5/2, 5}, the
471:
470:
459:
448:
437:
393:Main article:
390:
387:
317:Platonic solid
236:
235:
228:
221:
214:
206:
205:
200:
195:
190:
184:
183:
179:
176:
172:
168:
167:
160:
153:
146:
138:
137:
132:
127:
122:
116:
115:
111:
108:
104:
101:
98:
94:
90:
89:
82:
75:
68:
60:
59:
54:
49:
44:
38:
37:
33:
15:
13:
10:
9:
6:
4:
3:
2:
4486:
4475:
4472:
4470:
4467:
4465:
4464:Planar graphs
4462:
4460:
4457:
4456:
4454:
4443:
4439:
4435:
4430:
4427:
4423:
4420:
4418:
4411:
4404:
4398:
4396:
4392:
4389:
4387:
4383:
4379:
4375:
4372:
4370:
4366:
4363:
4361:
4357:
4353:
4352:
4349:
4347:
4345:
4342:
4340:
4336:
4333:
4331:
4328:
4326:
4323:
4322:
4319:
4317:
4315:
4312:
4310:
4306:
4303:
4301:
4298:
4296:
4293:
4292:
4289:
4287:
4280:
4273:
4267:
4265:
4262:
4260:
4256:
4253:
4251:
4248:
4246:
4243:
4242:
4239:
4237:
4230:
4223:
4217:
4215:
4212:
4210:
4206:
4203:
4201:
4198:
4196:
4193:
4192:
4189:
4187:
4180:
4174:
4172:
4169:
4167:
4163:
4160:
4158:
4155:
4153:
4150:
4149:
4146:
4144:
4142:
4139:
4137:
4133:
4130:
4128:
4125:
4123:
4120:
4119:
4116:
4112:
4109:
4107:
4104:
4102:
4101:Demitesseract
4099:
4097:
4093:
4090:
4088:
4085:
4083:
4080:
4079:
4076:
4072:
4069:
4067:
4065:
4062:
4060:
4056:
4053:
4051:
4048:
4046:
4043:
4042:
4039:
4036:
4034:
4031:
4029:
4026:
4024:
4021:
4019:
4016:
4014:
4011:
4010:
4007:
4001:
3998:
3994:
3987:
3983:
3976:
3972:
3967:
3963:
3958:
3954:
3949:
3947:
3945:
3941:
3931:
3927:
3925:
3923:
3919:
3915:
3913:
3911:
3907:
3903:
3901:
3898:
3897:
3892:
3888:
3880:
3875:
3873:
3868:
3866:
3861:
3860:
3857:
3847:
3841:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3781:
3779:
3775:
3765:
3762:
3760:
3757:
3756:
3754:
3750:
3744:
3741:
3739:
3736:
3735:
3732:
3729:
3725:
3719:
3718:
3714:
3712:
3711:
3707:
3706:
3704:
3700:
3694:
3691:
3689:
3686:
3684:
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3630:
3628:
3621:
3616:
3610:
3607:
3605:
3602:
3600:
3597:
3595:
3592:
3590:
3587:
3585:
3582:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3554:cuboctahedron
3552:
3550:
3547:
3546:
3544:
3539:
3535:
3529:
3524:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3494:
3492:
3488:
3483:
3479:
3475:
3467:
3462:
3460:
3455:
3453:
3448:
3447:
3444:
3432:
3429:
3427:
3424:
3422:
3419:
3417:
3414:
3412:
3409:
3407:
3404:
3402:
3399:
3398:
3396:
3393:
3389:
3382:
3379:
3376:
3373:
3370:
3367:
3366:
3364:
3360:
3353:
3352:Johnson solid
3350:
3347:
3346:Catalan solid
3344:
3341:
3338:
3335:
3332:
3331:
3329:
3325:
3316:
3313:
3310:
3307:
3304:
3301:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3279:
3277:
3273:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3241:Hexoctahedron
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3220:
3218:
3214:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3172:Tridecahedron
3170:
3168:
3165:
3163:
3162:Hendecahedron
3160:
3159:
3157:
3153:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3098:
3096:
3092:
3085:
3081:
3074:
3069:
3067:
3062:
3060:
3055:
3054:
3051:
3045:
3042:
3039:
3036:
3034:
3031:
3029:
3026:
3023:
3020:
3017:
3014:
3012:
3009:
3007:
3004:
3000:
2995:
2993:
2989:
2986:
2984:
2981:
2978:
2977:
2973:
2965:
2961:
2954:
2951:
2947:
2942:
2939:
2934:
2930:
2926:
2919:
2916:
2912:
2908:
2902:
2899:
2895:
2894:The Tetartoid
2890:
2887:
2883:
2882:Crystal Habit
2878:
2875:
2871:
2867:
2863:
2860:
2854:
2851:
2847:
2846:Crystal Habit
2842:
2839:
2834:
2830:
2826:
2822:
2817:
2812:
2808:
2804:
2803:
2798:
2791:
2788:
2782:
2779:
2773:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2748:
2745:
2744:phytoplankton
2742:
2738:
2733:
2730:
2729:
2726:
2722:
2719:
2718:
2714:
2712:
2710:
2706:
2702:
2695:
2688:
2681:
2678:
2675:
2668:
2664:
2660:
2657:
2654:
2647:
2643:
2639:
2636:
2633:
2626:
2622:
2619:Hendecagonal
2618:
2617:
2615:
2610:
2603:
2599:
2596:
2593:
2586:
2582:
2578:
2575:
2572:
2565:
2561:
2557:
2554:
2553:
2551:
2547:
2542:
2535:
2532:
2529:
2522:
2519:
2516:
2509:
2506:
2503:
2496:
2493:
2492:
2490:
2487:
2482:
2475:
2472:
2469:
2462:
2459:
2458:
2456:
2455:
2454:
2451:
2448:
2440:
2438:
2436:
2435:spacefillings
2432:
2428:
2424:
2420:
2416:
2411:
2409:
2405:
2401:
2396:
2394:
2390:
2385:
2383:
2379:
2378:cuboctahedron
2376:
2372:
2368:
2367:
2357:
2350:
2345:
2341:
2340:
2339:
2332:
2328:
2320:
2313:
2309:
2306:
2302:
2299:
2295:
2292:
2288:
2285:
2281:
2278:
2274:
2271:
2267:
2264:
2260:
2259:
2256:
2252:
2247:
2244:
2242:
2238:
2230:
2220:
2214:
2210:
2207:
2203:
2199:
2195:
2191:
2184:
2181:
2178:
2174:
2170:
2166:
2162:
2155:
2152:
2149:
2145:
2142:
2138:
2135:
2131:
2127:
2123:
2118:
2117:
2116:
2103:
2095:
2080:
2072:
2057:
2049:
2041:
2037:
2033:
2022:
2014:
1999:
1991:
1976:
1968:
1960:
1956:
1952:
1948:
1944:
1940:
1936:
1935:
1934:
1932:
1924:
1918:
1916:
1915:crystal model
1905:Crystal model
1903:
1899:
1892:
1883:
1874:
1865:
1856:
1855:
1852:
1850:
1846:
1841:
1840:
1836:
1835:
1826:
1822:
1818:
1805:
1796:
1787:
1774:
1765:
1756:
1747:
1746:
1743:
1741:
1736:
1731:
1729:
1723:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1688:
1684:
1681:
1678:
1677:
1673:
1670:
1668:
1665:
1664:
1660:
1658:
1655:
1654:
1651:30 (6+12+12)
1650:
1648:
1645:
1644:
1640:
1638:
1635:
1634:
1630:
1628:
1625:
1624:
1621:
1618:
1615:
1614:
1609:
1602:
1597:
1590:
1584:
1575:
1571:
1568:
1562:
1558:
1553:
1549:
1546:
1540:
1536:
1533:
1527:
1523:
1518:
1514:
1511:
1507:
1501:
1497:
1494:
1493:
1489:
1486:
1483:
1480:
1479:
1455:
1429:
1402:
1375:
1373:
1369:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1343:
1337:
1332:
1327:
1324:
1322:
1318:
1313:
1308:
1306:
1302:
1298:
1290:
1281:
1275:
1261:
1258:
1257:
1253:
1249:
1246:
1242:
1241:
1236:
1229:
1223:
1222:
1211:
1202:
1193:
1186:
1176:
1167:
1158:
1149:
1148:
1145:
1143:
1138:
1136:
1135:
1130:
1122:
1108:
1103:
1101:
1081:
1076:
1074:
1070:
1066:
1060:
1056:
1044:
1040:
1028:
1024:
1015:
1009:
992:
983:
974:
973:
970:
952:
946:
943:
937:
924:
906:
901:
898:
893:
880:
862:
857:
853:
847:
834:
831:
827:
823:
819:
815:
811:
803:
799:
794:
790:
788:
784:
780:
779:quasicrystals
776:
772:
767:
765:
760:
756:
748:
744:
734:
729:
722:
719:
716:
715:
712:
709:
707:
704:
703:
699:
696:
694:
691:
690:
686:
680:
678:
675:
674:
670:
668:
665:
664:
660:
658:
655:
654:
650:
648:
645:
644:
586:
584:
581:
580:
577:
574:
571:
570:
565:
558:
553:
550:Pyritohedron
548:
542:
540:
538:
534:
530:
526:
522:
518:
510:
508:
506:
502:
498:
494:
490:
486:
482:
478:
469:
464:
460:
458:
453:
449:
447:
442:
438:
436:
430:
426:
419:
417:
413:
408:
406:
402:
396:
388:
386:
382:
380:
376:
375:space-filling
372:
368:
364:
360:
355:
353:
349:
345:
341:
337:
332:
331:, order 120.
330:
326:
322:
318:
314:
310:
306:
305:duodecahedron
302:
299:
292:
288:
285:
282:
275:
271:
268:
265:
258:
254:
251:
250:Ancient Greek
247:
243:
233:
229:
226:
222:
219:
215:
212:
208:
204:
201:
199:
196:
194:
193:Rhombo-square
191:
189:
186:
165:
161:
158:
154:
151:
147:
144:
140:
139:
136:
133:
131:
128:
126:
123:
121:
118:
117:
109:
102:
99:
92:
91:
87:
83:
80:
76:
73:
69:
66:
62:
61:
58:
55:
53:
50:
48:
45:
43:
40:
39:
30:
22:
4421:
4390:
4381:
4373:
4364:
4355:
4335:10-orthoplex
4071:Dodecahedron
3992:
3981:
3970:
3961:
3952:
3943:
3939:
3929:
3921:
3917:
3909:
3905:
3845:
3764:trapezohedra
3715:
3708:
3512:dodecahedron
3265:Apeirohedron
3216:>20 faces
3167:Dodecahedron
3166:
2979:
2963:
2953:
2941:
2932:
2928:
2918:
2901:
2889:
2877:
2853:
2841:
2806:
2800:
2790:
2781:
2701:Armand Spitz
2699:
2662:
2452:
2446:
2444:
2423:golden ratio
2412:
2397:
2393:pyritohedron
2388:
2386:
2375:quasiregular
2364:
2362:
2330:
2324:
2234:
2218:
2212:
2205:
2201:
2197:
2193:
2189:
2182:
2176:
2172:
2168:
2164:
2160:
2153:
2147:
2143:
2140:
2136:
2129:
2125:
2121:
2114:
2101:
2093:
2078:
2070:
2055:
2047:
2039:
2035:
2031:
2020:
2012:
1997:
1989:
1974:
1966:
1958:
1954:
1950:
1946:
1942:
1938:
1928:
1912:
1842:
1732:
1724:
1707:
1703:
1699:
1695:
1693:
1661:20 (4+4+12)
1616:Face polygon
1531:
1371:
1311:
1309:
1294:
1220:
1184:
1139:
1132:
1106:
1104:
1079:
1077:
1068:
1067:
1058:
1057:), 0, ±(1 +
1054:
1042:
1038:
1026:
1022:
1016:
1013:
925:
881:
835:
826:Miller index
809:
807:
768:
747:pyritohedral
743:pyritohedron
742:
740:
671:20 (8 + 12)
661:30 (6 + 24)
572:Face polygon
543:Pyritohedron
514:
501:pentagrammic
474:
409:
398:
383:
356:
346:, while the
336:pyritohedron
333:
304:
297:
294:
287:
280:
277:
270:
263:
260:
253:
246:dodecahedron
245:
239:
120:Pyritohedron
100:T, order 12
36:, order 120
4474:12 (number)
4344:10-demicube
4305:9-orthoplex
4255:8-orthoplex
4205:7-orthoplex
4162:6-orthoplex
4132:5-orthoplex
4087:Pentachoron
4075:Icosahedron
4050:Tetrahedron
3534:semiregular
3517:icosahedron
3497:tetrahedron
3207:Icosahedron
3155:11–20 faces
3141:Enneahedron
3131:Heptahedron
3121:Pentahedron
3116:Tetrahedron
2992:stellations
2913:, Slovenia.
2741:unicellular
2400:stellations
1366:1 : 1
1363:0 : 1
1360:1 : 1
1357:2 : 1
1354:1 : 1
1351:0 : 1
1348:1 : 1
1301:convex hull
1299:of a cubic
1238:Animations
934:Short sides
477:stellations
416:icosahedron
325:stellations
269:; from
264:dōdekáedron
257:δωδεκάεδρον
182:, order 12
175:, order 16
107:, order 48
97:, order 24
4453:Categories
4330:10-simplex
4314:9-demicube
4264:8-demicube
4214:7-demicube
4171:6-demicube
4141:5-demicube
4055:Octahedron
3829:prismatoid
3759:bipyramids
3743:antiprisms
3717:hosohedron
3507:octahedron
3392:prismatoid
3377:(infinite)
3146:Decahedron
3136:Octahedron
3126:Hexahedron
3101:Monohedron
3094:1–10 faces
2816:1811.04131
2774:References
2644:– dual of
2642:trapezoids
2634:, order 11
2625:hendecagon
2583:, dual of
2562:, dual of
2427:zonohedron
2406:is also a
2371:zonohedron
2327:anticupola
1720:pentagonal
1679:Properties
1484:−0.618...
1481:−1.618...
1321:pentagrams
759:pentagonal
717:Properties
531:, and the
309:polyhedron
248:(from
135:Triangular
110:Johnson (J
4378:orthoplex
4300:9-simplex
4250:8-simplex
4200:7-simplex
4157:6-simplex
4127:5-simplex
4096:Tesseract
3824:birotunda
3814:bifrustum
3579:snub cube
3474:polyhedra
3406:antiprism
3111:Trihedron
3080:Polyhedra
2929:Curr. Sci
2833:119318080
2560:triangles
2517:, order 8
1825:Cobaltite
1728:cobaltite
1696:tetartoid
1592:Tetartoid
1585:Tetartoid
1510:pentagram
1490:1.618...
1487:0.618...
1260:Honeycomb
1041:), ±(1 −
1025:), ±(1 −
1021:0, ±(1 +
957:Long side
953:⋅
911:Long side
907:⋅
867:Long side
863:⋅
816:shown by
808:The name
781:(such as
533:tetartoid
348:tetartoid
125:Tetartoid
4432:Topics:
4395:demicube
4360:polytope
4354:Uniform
4115:600-cell
4111:120-cell
4064:Demicube
4038:Pentagon
4018:Triangle
3804:bicupola
3784:pyramids
3710:dihedron
3106:Dihedron
2935:: 64–72.
2862:Archived
2721:120-cell
2715:See also
2667:hexagons
2431:Bilinski
2333:It has D
1735:topology
1657:Vertices
667:Vertices
407:{5, 3}.
242:geometry
4369:simplex
4339:10-cube
4106:24-cell
4092:16-cell
4033:Hexagon
3887:regular
3846:italics
3834:scutoid
3819:rotunda
3809:frustum
3538:uniform
3487:regular
3472:Convex
3426:pyramid
3411:frustum
2621:pyramid
2338:match.
2109:
2090:
2086:
2067:
2063:
2044:
2028:
2009:
2005:
1986:
1982:
1963:
1475:
1461:√
1457:
1449:
1435:√
1431:
1423:
1409:√
1404:
1396:
1382:√
1378:
1278:
1264:
1125:
1111:
1096:
1084:
785:) with
523:of the
433:Convex
307:is any
130:Rhombic
42:Regular
4309:9-cube
4259:8-cube
4209:7-cube
4166:6-cube
4136:5-cube
4023:Square
3900:Family
3799:cupola
3752:duals:
3738:prisms
3416:cupola
3292:vertex
2872:, U.S.
2831:
2447:convex
2402:, the
1891:Chiral
1706:, and
1698:(also
1683:convex
1512:faces
1495:Image
1345:Ratio
1053:±(1 −
1037:±(1 +
844:Height
818:pyrite
771:pyrite
365:. The
342:, has
340:pyrite
281:dṓdeka
274:δώδεκα
4028:p-gon
3421:wedge
3401:prism
3261:(132)
2829:S2CID
2811:arXiv
2747:algae
2581:kites
2579:– 12
2369:is a
2042:); (−
2030:); (−
1961:); (−
1949:); (−
1647:Edges
1637:Faces
1606:(See
1187:= 1/2
1073:wedge
890:Width
657:Edges
647:Faces
562:(See
535:with
298:hédra
252:
52:Great
4386:cube
4059:Cube
3889:and
3502:cube
3383:(57)
3354:(92)
3348:(13)
3342:(13)
3311:(16)
3287:edge
3282:face
3255:(90)
3249:(60)
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