7080:
290:
4641:
1231:
6873:
6880:
6831:
6479:
6866:
6838:
6824:
6472:
7115:
7101:
6263:
6859:
6845:
6493:
7108:
6500:
7087:
5346:
6486:
5705:
7094:
4692:
5500:
6465:
3810:
1382:
4686:
4609:
4589:
4655:
5698:
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4648:
4622:
4615:
4582:
5691:
5677:
5663:
5332:
4679:
4674:
4635:
4602:
3803:
3100:
3109:
6315:
5684:
5670:
5656:
5493:
4667:
4662:
4628:
161:
3685:
275:
6343:
6336:
6329:
6322:
5486:
5479:
5472:
5339:
7125:
7049:
6451:
3607:
7056:
6030:
5465:
5311:
3118:
6350:
6852:
6458:
6444:
6037:
6016:
6002:
7035:
5360:
5318:
3136:
5521:
7028:
5304:
3127:
386:
5542:
5535:
5528:
7042:
3472:
6357:
5514:
5507:
48:
6044:
5367:
6023:
5995:
462:
5353:
6009:
5325:
5297:
152:
7063:
3547:
901:
of a regular octahedron between two adjacent triangular faces is 109.47°. This can be obtained from the dihedral angle of an equilateral square pyramid: its dihedral angle between two adjacent triangular faces is the dihedral angle of an equilateral square pyramid between two adjacent triangular
666:
3488:
octahedra, excluding mirror images. More specifically there are 2, 11, 42, 74, 76, 38, 14 for octahedra with 6 to 12 vertices respectively. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one
444:
of sixteen: the appearance is symmetrical by rotating around the axis of symmetry that passing through apices and base's center vertically, and it has mirror symmetry relative to any bisector of the base; it is also symmetrical by reflecting it across a horizontal plane. Therefore, this square
1428:. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. Five octahedra define any given icosahedron in this fashion, and together they define a
893:
3483:
An octahedron can be any polyhedron with eight faces. In a previous example, the regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. There are 257 topologically distinct
1558:. The octahedron is unique among the Platonic solids in having an even number of faces meeting at each vertex. Consequently, it is the only member of that group to possess, among its mirror planes, some that do not pass through any of its faces. The octahedron's
902:
faces, and its dihedral angle between two adjacent triangular faces on the edge in which two equilateral square pyramids are attached is twice the dihedral angle of an equilateral square pyramid between its triangular face and its square base.
533:
2738:, first from an octahedron vertex to an octahedron edge center, then turning 90° to an octahedron face center, then turning 90° to the octahedron center. The orthoscheme has four dissimilar right triangle faces. The exterior face is a
761:
3570:: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral.
404:
A square bipyramid is said to be right if the square pyramids are symmetrically regular and both of their apices are on the line passing through the base's center; otherwise, it is oblique. The resulting bipyramid has
1220:
1031:
3561:
The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it:
360:
by using the
Platonic solids setting into another one and separating them with six spheres resembling the six planets. The ordered solids started from the innermost to the outermost: regular octahedron,
2368:
2051:
1822:
1640:
surrounding the octahedron's center. Three left-handed orthoschemes and three right-handed orthoschemes meet in each of the octahedron's eight faces, the six orthoschemes collectively forming a
538:
3489:
into the other simply by changing the lengths of edges or the angles between edges or faces.) Some of the polyhedrons do have eight faces aside from being square bipyramids in the following:
2461:
1867:
259:
7780:
7740:
7700:
2205:
2993:
2934:
2901:
2846:
2735:
2702:
2643:
2558:
2503:
2118:
2006:
1969:
1904:
1726:
1765:
1409:
the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the
442:
4514:
4496:
4448:
4430:
3509:: The four faces from the tetrahedron are truncated to become regular hexagons, and there are four more equilateral triangle faces where each tetrahedron vertex was truncated.
8393:
4572:
4301:
4253:
5287:
5277:
5267:
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4909:
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4320:
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4224:
3750:
Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see
3335:
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927:
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4842:
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4458:
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4263:
4244:
4209:
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4079:
4069:
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4011:
4002:
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3963:
3306:
3249:
3192:
3173:
3163:
1636:
1626:
1616:
1608:
1598:
3577:, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all three quadrilaterals are planar squares.
1309:
6757:
6728:
3330:
3320:
3291:
3262:
3234:
6747:
6718:
6699:
6689:
6670:
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6515:
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5904:
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5865:
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5836:
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5788:
5778:
5759:
5749:
5730:
5720:
5282:
5272:
5253:
5243:
5224:
5214:
5191:
5181:
5162:
5152:
5133:
5123:
5104:
5094:
5069:
5059:
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5011:
5001:
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4943:
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4856:
4837:
4827:
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4779:
4769:
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4538:
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4472:
4416:
4406:
4383:
4373:
4335:
4325:
4287:
4277:
4239:
4229:
4204:
4194:
4171:
4161:
4142:
4132:
4113:
4103:
4084:
4074:
4055:
4045:
4026:
4016:
3997:
3987:
3968:
3958:
3301:
3272:
3244:
3197:
3187:
3168:
3158:
2328:
2255:
2172:
1631:
1621:
1603:
1593:
1345:
1131:
1073:
7803:
756:
726:
696:
2868:
2813:
2765:
2669:
2610:
2525:
2278:
1932:
1689:
1093:
528:
508:
488:
8260:
8842:
661:{\displaystyle {\begin{aligned}A&=2{\sqrt {3}}a^{2}&\approx 3.464a^{2},\\V&={\frac {1}{3}}{\sqrt {2}}a^{3}&\approx 0.471a^{3}.\end{aligned}}}
401:
constructed by attaching two square pyramids base-to-base. These pyramids cover their square bases, so the resulting polyhedron has eight triangular faces.
1136:
5434:
936:
3535:, also called Dürer's solid: Obtained by truncating two opposite corners of a cube or rhombohedron, this has six pentagon faces and two triangle faces.
3503:: One face is a heptagon (usually regular), and the remaining seven faces are triangles (usually isosceles). All triangular faces can't be equilateral.
1405:. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e.
888:{\displaystyle r_{u}={\frac {\sqrt {2}}{2}}a\approx 0.707a,\qquad r_{i}={\frac {\sqrt {6}}{6}}a\approx 0.408a,\qquad r_{m}={\frac {1}{2}}a=0.5a.}
195:
of the same size, and exactly four triangles meet at each vertex. A regular octahedron is convex, meaning that for any two points within it, the
9748:
9743:
6964:
68:
8357:
9768:
9753:
7273:
constant ≈ 1.618, for which
Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.
8835:
1644:: a triangular pyramid with the octahedron face as its equilateral base, and its cube-cornered apex at the center of the octahedron.
9256:
10440:
9763:
9758:
9516:
5427:
1532:
9708:
9703:
9551:
9521:
7336:
8142:
1354:, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected
8182:
8086:
8039:
7963:
7566:
7512:
7447:
7417:
7387:
9773:
8828:
5633:
1835:
2336:
2019:
1790:
9875:
9728:
9723:
9713:
9664:
9526:
8449:
7379:
8119:
9556:
9531:
9506:
9491:
8048:
5420:
3532:
17:
3836:
2430:
9738:
9546:
9536:
9511:
9496:
8314:
7926:
7895:
7539:
7481:
6425:
5640:
3843:
3393:
1311:, a graph partitioned into three independent sets each consisting of two opposite vertices. More generally, it is a
9733:
9698:
9441:
9353:
8275:
5584:
3768:
3631:
1458:
1258:—its edges remain connected whenever two of more three vertices of a graph are removed. Its graph called the
406:
227:
9683:
9610:
9605:
9436:
9426:
7824:
6957:
3529:
glued over one of their square sides so that no triangle shares an edge with another triangle (Johnson solid 26).
490:
of a regular octahedron can be ascertained by summing all of its eight equilateral triangles, whereas its volume
452:
If the edges of a square bipyramid are all equal in length, then that square bipyramid is a regular octahedron.
233:
7782:, Coxeter's notation for the circumradius, midradius, and inradius, respectively, also noting that Coxeter uses
7745:
7705:
7665:
6098:. If this diagonal is oriented vertically with a height of 1, then the first five slices above occur at heights
9383:
9378:
9368:
8300:
8074:
Art & Science of
Geometric Origami: Create Spectacular Paper Polyhedra, Waves, Spirals, Fractals, and More!
2180:
1739:
10472:
9718:
9585:
9541:
9501:
9476:
8747:
8095:
7918:
7887:
7224:
2967:
2908:
2875:
2820:
2709:
2676:
2617:
2532:
2477:
1447:
like polyhedron in which lateral faces are replaced by alternating equilateral triangles. It is also called
9832:
9654:
9644:
9639:
9615:
9595:
9580:
9481:
9421:
9348:
9343:
9333:
9249:
6245:
1524:
1243:
9431:
9416:
9406:
9041:
8982:
4739:
1547:
forms which are mirror images of each other. The characteristic orthoscheme of a regular polyhedron is a
2398:
2094:
2064:
1982:
1945:
1880:
1702:
10462:
9693:
9590:
9575:
9373:
9071:
9031:
8637:
7194:
6950:
4729:
415:
215:
3583:, a non-convex polyhedron that cannot be partitioned into tetrahedra without introducing new vertices.
9659:
9649:
9600:
9486:
9066:
9061:
6223:), progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With
5579:
5574:
1641:
10467:
9898:
9688:
9466:
9451:
9338:
8631:
8419:
8304:
5611:
3512:
3066:
3008:
of the octahedron, named by the triangular face colors going around each vertex: 1212, 1112, 1111.
1406:
1351:
1255:
352:
334:
3662:
3580:
2473:
If the octahedron has edge length 𝒍 = 2, its characteristic tetrahedron's six edges have lengths
1554:
The faces of the octahedron's characteristic tetrahedron lie in the octahedron's mirror planes of
319:
and the same number of faces meet at each vertex. This ancient set of polyhedrons was named after
9868:
9471:
9411:
9242:
9172:
9167:
9046:
8952:
8759:
8687:
8607:
8442:
8404:
K.J.M. MacLean, A Geometric
Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
8104:, p. 130, §7.6 The symmetry group of the general regular polytope; "simplicial subdivision".
6922:
3913:
3496:: Two faces are parallel regular hexagons; six squares link corresponding pairs of hexagon edges.
2941:
2772:
2569:
1452:
1362:
1254:—its edges of a graph are connected to every vertex without crossing other edges—and
1034:
908:
7079:
9844:
9807:
9634:
9036:
8977:
8967:
8912:
8693:
7214:
4734:
3656:
3652:
1275:
329:
dialogue, related these solids to nature. One of them, the regular octahedron, represented the
8058:
8025:
7950:. Springer Proceedings in Mathematics & Statistics. Vol. 159. Springer. p. 250.
7943:
7556:
7529:
7498:
7407:
289:
10412:
10405:
10398:
9797:
9784:
9056:
8972:
8927:
8875:
8072:
7373:
7016:
5961:
5951:
3930:
3864:
3506:
3020:
2296:
2223:
2140:
1462:
1317:
1247:
1098:
1040:
930:
313:
207:
9937:
9915:
9903:
8107:
7785:
4640:
2742:
which is one-sixth of an octahedron face. The three faces interior to the octahedron are: a
1455:, a polyhedron in which two different polygonal faces are alternating and meet at a vertex.
10069:
10016:
9817:
9456:
9363:
9358:
9016:
8942:
8890:
8309:(Third ed.). Dover. Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction.
8010:
7948:
Symmetries in Graphs, Maps, and
Polytopes: 5th SIGMAP Workshop, West Malvern, UK, July 2014
7847:
7646:
7409:
Number, Shape, & Symmetry: An
Introduction to Number Theory, Geometry, and Group Theory
7219:
7132:
7008:
6998:
6389:
6364:
6272:
6267:
6232:
6216:
6076:
5976:
5644:
5621:
4714:
3888:
3858:
3822:
3792:
1481:
1366:
734:
704:
674:
669:
366:
211:
192:
120:
8351:
7910:
7863:
8:
10477:
10457:
10424:
10323:
10073:
9461:
9182:
9051:
9026:
9011:
8947:
8895:
8386:
7229:
7199:
7003:
6912:
6902:
6805:
6394:
6384:
6081:
6066:
6056:
4749:
4719:
3902:
3847:
3791:
by adding 4 tetrahedra on alternated faces. Adding tetrahedra to all 8 faces creates the
3590:
1563:
1555:
1477:
1425:
1390:
1365:
of its vertices have the same size. The other three polyhedra with this property are the
1355:
1239:
929:
can be placed with its center at the origin and its vertices on the coordinate axes; the
370:
362:
57:
7437:
5390:
The octahedron is topologically related as a part of sequence of regular polyhedra with
3775:
in the 1950s. It is commonly regarded as the strongest building structure for resisting
10293:
10243:
10193:
10150:
10120:
10080:
10043:
9861:
9398:
9325:
9314:
9294:
9281:
9273:
9197:
9162:
9021:
8916:
8865:
8804:
8681:
8675:
8435:
8242:
8224:
7980:
7851:
7607:
7599:
7504:
7317:
7209:
6937:
6236:
4709:
3772:
3586:
3557:
as its equator. The axis of symmetry passes through the plane of the antiparallelogram.
3550:
3500:
2853:
2798:
2750:
2654:
2595:
2510:
2263:
1917:
1674:
1536:
1492:
1358:
1078:
513:
493:
473:
324:
188:
31:
8413:
8403:
7877:
5391:
3343:
1548:
1230:
10432:
9675:
9394:
9309:
9299:
9277:
9177:
8987:
8962:
8906:
8794:
8718:
8670:
8643:
8613:
8367:
8310:
8246:
8212:
8082:
8078:
8035:
7959:
7922:
7891:
7855:
7815:
7638:
7611:
7562:
7535:
7508:
7477:
7443:
7413:
7383:
7239:
7189:
6800:
6478:
6429:
6379:
6224:
6094:
The above shapes may also be realized as slices orthogonal to the long diagonal of a
5981:
5569:
3941:
3554:
3526:
2743:
2739:
1470:
1414:
330:
6872:
5620:. This can be shown by a 2-color face model. With this coloring, the octahedron has
134:
A regular octahedron is the three-dimensional case of the more general concept of a
10436:
10001:
9990:
9979:
9968:
9959:
9950:
9889:
9885:
9822:
9812:
9567:
9116:
8799:
8779:
8601:
8331:
8234:
8146:
8031:
7996:
7951:
7881:
7859:
7833:
7634:
7591:
7351:
7313:
7309:
7297:
7234:
7204:
7170:
6993:
6879:
6830:
5399:
3698:
3689:
3005:
1540:
1503:
1488:
1444:
1398:
1270:
699:
346:
316:
63:
8186:
7355:
6865:
6262:
10026:
10011:
9827:
9792:
9446:
8753:
8665:
8660:
8625:
8580:
8570:
8560:
8555:
8238:
8208:
8006:
7843:
7642:
7467:
7114:
7100:
6837:
6823:
6795:
6785:
6471:
3522:
3516:
3493:
3476:
3145:
2650:
of the octahedron). The 3-edge path along orthogonal edges of the orthoscheme is
1370:
446:
341:
295:
281:
219:
170:
8370:
7107:
6858:
6844:
6492:
5345:
10376:
9626:
9289:
8937:
8860:
8809:
8712:
8575:
8565:
8168:
7269:𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the
7177:
7086:
6499:
6485:
3876:
3371:
3012:
1263:
898:
410:
309:
135:
116:
8408:
8001:
7984:
7955:
6215:
exists in a sequence of symmetries of quasiregular polyhedra and tilings with
1312:
10451:
10393:
10281:
10274:
10267:
10231:
10224:
10217:
10181:
10174:
9304:
9265:
9142:
8998:
8932:
8730:
8724:
8619:
8550:
8540:
7093:
6374:
5971:
5704:
5697:
4691:
4654:
4647:
4614:
3881:
3809:
1570:
1559:
1410:
1381:
6464:
6239:
of symmetry, with generator points at the right angle corner of the domain.
5690:
5676:
5662:
5499:
5331:
4685:
4678:
4673:
4634:
4608:
4601:
4588:
3830:
The octahedron is one of a family of uniform polyhedra related to the cube.
10333:
9802:
8545:
8398:
7838:
7819:
7473:
7270:
6314:
6086:
5683:
5669:
5655:
5492:
5376:
4754:
4666:
4661:
4627:
4621:
4595:
4581:
3802:
3702:
3108:
3099:
1421:
1251:
357:
299:
196:
3684:
3054:. These symmetries can be emphasized by different colorings of the faces.
728:(one that tangent to each of the octahedron's faces), and the radius of a
123:, four of which meet at each vertex. Regular octahedra occur in nature as
10342:
10303:
10253:
10203:
10160:
10130:
10062:
10048:
8585:
8519:
8509:
8499:
8494:
7531:
Mathematics and
Plausible Reasoning: Induction and analogy in mathematics
7463:
6071:
6051:
5966:
5946:
5564:
5554:
4744:
3923:
3817:
3788:
3764:
3642:
3441:
3069:
1577:, have the same symmetry group but different characteristic tetrahedra.
1528:
203:
127:
structures. Many types of irregular octahedra also exist, including both
6328:
6321:
5485:
5478:
5471:
5338:
1373:, and an irregular polyhedron with 12 vertices and 20 triangular faces.
274:
10328:
10312:
10262:
10212:
10169:
10139:
9207:
9095:
8852:
8770:
8524:
8504:
8489:
8479:
8458:
7603:
7469:
The Golden Ratio: The Story of Phi, the World's Most
Astonishing Number
7321:
7048:
6979:
6897:
6892:
6887:
6790:
6450:
6414:, the octahedron is related to the hexagonal dihedral symmetry family.
6342:
6335:
5384:
3776:
3709:
3645:
3539:
1402:
1394:
160:
128:
108:
7055:
6029:
5310:
3606:
10384:
10298:
10248:
10198:
10155:
10125:
10094:
9202:
9192:
9137:
9121:
8957:
8784:
8375:
7124:
7034:
6944:
6917:
6851:
6457:
6443:
6411:
6095:
6036:
6015:
6001:
5627:
Compare this truncation sequence between a tetrahedron and its dual:
5380:
5359:
5317:
3918:
3574:
3567:
3117:
3083:
3077:
3043:
3032:
1531:, all of the same shape characteristic of the polytope. A polytope's
1440:
1215:{\displaystyle \left|x-a\right|+\left|y-b\right|+\left|z-c\right|=r.}
729:
398:
7944:"Faithful Embeddings of Planar Graphs on Orientable Closed Surfaces"
7595:
7027:
5520:
5464:
5303:
3135:
1385:
The octahedron represents the central intersection of two tetrahedra
698:(one that touches the octahedron at all vertices), the radius of an
10358:
10113:
10109:
10036:
9088:
8820:
8484:
7890:, vol. 221 (2nd ed.), Springer-Verlag, pp. 235–244,
7041:
6907:
6780:
6775:
6770:
6349:
5549:
5541:
5534:
5527:
3712:
3638:
3627:
3610:
3126:
3024:
1484:
1420:
One can also divide the edges of an octahedron in the ratio of the
385:
88:
8229:
8215:(2014), "The cross ratio as a shape parameter for Dürer's solid",
6356:
5513:
3471:
1026:{\displaystyle (\pm 1,0,0),\qquad (0,\pm 1,0),\qquad (0,0,\pm 1).}
10367:
10337:
10104:
10099:
10090:
10031:
9234:
9212:
9187:
8789:
8350:
3673:
3619:
3542:: degenerate in Euclidean space, but can be realized spherically.
1612:
which subdivides it into 48 of these characteristic orthoschemes
1584:
can be found by a canonical dissection of the regular octahedron
1539:
property because the polytope is generated by reflections in the
124:
8394:
Editable printable net of an octahedron with interactive 3D view
5506:
2561:
around its exterior right-triangle face (the edges opposite the
10307:
10257:
10207:
10164:
10134:
10085:
10021:
7978:
7265:) uses the greek letter 𝝓 (phi) to represent one of the three
6043:
5366:
3751:
3666:
1544:
1499:
510:
is twice the volume of a square pyramid; if the edge length is
7979:
Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.;
7062:
6022:
5994:
5352:
3767:
of alternating tetrahedra and half-octahedra derived from the
1495:. It has four of the triangular faces, and 3 central squares.
455:
6008:
5324:
5296:
461:
320:
151:
8427:
10057:
8880:
6927:
6061:
4724:
3871:
3670:
3623:
3546:
1574:
1466:
1461:
can be alternated to form a vertex, edge, and face-uniform
374:
223:
174:
3648:
are arranged paralleling the eight faces of an octahedron.
8027:
Connections: The
Geometric Bridge Between Art and Science
7625:
Berman, Martin (1971). "Regular-faced convex polyhedra".
7257:
7255:
1523:
Like all regular convex polytopes, the octahedron can be
7913:(1995). "Chapter 4: Steinitz' Theorem for 3-Polytopes".
7582:
McLean, K. Robin (1990). "Dungeons, dragons, and dice".
3701:, this solid is known as a "d8", one of the more common
8116:, pp. 70–71, Characteristic tetrahedra; Fig. 4.7A.
8207:
7252:
2973:
2914:
2881:
2826:
2715:
2682:
2623:
2538:
2483:
2435:
2342:
2099:
2025:
1987:
1950:
1885:
1840:
1796:
1707:
1432:. A regular icosahedron produced this way is called a
7788:
7748:
7708:
7668:
2970:
2944:
2911:
2878:
2856:
2823:
2801:
2775:
2753:
2712:
2679:
2657:
2620:
2598:
2572:
2535:
2513:
2480:
2433:
2401:
2339:
2299:
2266:
2226:
2183:
2143:
2097:
2067:
2022:
1985:
1948:
1920:
1883:
1838:
1793:
1742:
1705:
1677:
1397:
is an octahedron, and this compound—called the
1320:
1278:
1139:
1101:
1081:
1043:
939:
911:
764:
737:
707:
677:
536:
516:
496:
476:
418:
236:
7921:. Vol. 152. Springer-Verlag. pp. 103–126.
7555:
Alexander, Daniel C.; Koeberlin, Geralyn M. (2014).
7176:
The octahedron can be generated as the case of a 3D
2363:{\displaystyle {\sqrt {\tfrac {2}{3}}}\approx 0.816}
2046:{\displaystyle {\sqrt {\tfrac {1}{3}}}\approx 0.577}
1817:{\displaystyle {\sqrt {\tfrac {4}{3}}}\approx 1.155}
1582:
characteristic tetrahedron of the regular octahedron
8365:
3708:
If each edge of an octahedron is replaced by a one-
55:It has been suggested that this article should be
7797:
7774:
7734:
7694:
7554:
5379:, a polytope formed by certain intersections of a
3596:
2987:
2954:
2928:
2895:
2862:
2840:
2807:
2785:
2759:
2729:
2696:
2663:
2637:
2604:
2582:
2552:
2519:
2497:
2455:
2409:
2362:
2322:
2272:
2249:
2199:
2166:
2112:
2075:
2045:
2000:
1963:
1926:
1898:
1861:
1816:
1759:
1720:
1683:
1543:of its orthoscheme. The orthoscheme occurs in two
1339:
1303:
1214:
1125:
1087:
1067:
1025:
921:
887:
750:
720:
690:
660:
522:
502:
482:
436:
253:
7758:
7718:
7678:
7662:Table I(i), pp. 292–293. See the columns labeled
7472:(First trade paperback ed.). New York City:
7169:Truncation of two opposite vertices results in a
2999:
758:(one that touches the middle of each edge), are:
10449:
8054:
7496:
5609:The regular octahedron can also be considered a
2456:{\displaystyle {\tfrac {{\text{arc sec }}3}{2}}}
1439:The regular octahedron can be considered as the
1242:of a regular octahedron can be represented as a
340:Following its attribution with nature by Plato,
8128:, pp. 292–293, Table I(i); "Octahedron, 𝛽
7561:(6th ed.). Cengage Learning. p. 403.
6420:Uniform hexagonal dihedral spherical polyhedra
3679:
3031:(order 12), the symmetry group of a triangular
7328:
6250:32 orbifold symmetries of quasiregular tilings
3715:, the resistance between opposite vertices is
3630:are commonly octahedral, as the space-filling
1862:{\displaystyle {\tfrac {\pi }{2}}-{\text{𝜿}}}
9869:
9250:
8836:
8443:
7870:
7405:
6958:
5428:
5375:It is also one of the simplest examples of a
3787:A regular octahedron can be augmented into a
1518:
350:sketched each of the Platonic solids. In his
9749:metagyrate diminished rhombicosidodecahedron
9744:paragyrate diminished rhombicosidodecahedron
5412:32 symmetry mutation of regular tilings: {3,
191:. All the faces of a regular octahedron are
7534:. Princeton University Press. p. 138.
7406:Herrmann, Diane L.; Sally, Paul J. (2013).
7300:(1978). "An Infinite Class of Deltahedra".
3466:
3042:(order 16), the symmetry group of a square
456:Metric properties and Cartesian coordinates
254:{\displaystyle \mathrm {O} _{\mathrm {h} }}
9876:
9862:
9769:gyrate bidiminished rhombicosidodecahedron
9754:bigyrate diminished rhombicosidodecahedron
9257:
9243:
8843:
8829:
8450:
8436:
7946:. In Širáň, Jozef; Jajcay, Robert (eds.).
7775:{\displaystyle {}_{2}\!\mathrm {R} /\ell }
7735:{\displaystyle {}_{1}\!\mathrm {R} /\ell }
7695:{\displaystyle {}_{0}\!\mathrm {R} /\ell }
7497:O'Keeffe, Michael; Hyde, Bruce G. (2020).
7442:. Cambridge University Press. p. 55.
7431:
7429:
7367:
7365:
7334:
7164:
6965:
6951:
5435:
5421:
3758:
1650:Characteristics of the regular octahedron
1269:The octahedral graph can be considered as
111:with eight faces. One special case is the
8387:"3D convex uniform polyhedra x3o4o – oct"
8228:
8183:"Polyhedra with 8 Faces and 6-8 Vertices"
8000:
7837:
7500:Crystal Structures: Patterns and Symmetry
1037:, the octahedron with center coordinates
199:connecting them lies entirely within it.
8023:
8017:
7876:
7558:Elementary Geometry for College Students
7435:
7371:
3731:ohm, and that between adjacent vertices
3683:
3605:
3545:
3470:
2200:{\displaystyle {\sqrt {2}}\approx 1.414}
1662:
1659:
1380:
1229:
459:
384:
380:
10441:List of regular polytopes and compounds
9764:metabidiminished rhombicosidodecahedron
9759:parabidiminished rhombicosidodecahedron
9517:elongated pentagonal orthocupolarotunda
8299:
8125:
8113:
8101:
8070:
8064:
7909:
7903:
7814:
7808:
7659:
7426:
7362:
7292:
7290:
7262:
5634:Family of uniform tetrahedral polyhedra
3475:A regular faced convex polyhedron, the
2988:{\displaystyle {\sqrt {\tfrac {2}{3}}}}
2929:{\displaystyle {\sqrt {\tfrac {4}{3}}}}
2896:{\displaystyle {\sqrt {\tfrac {2}{3}}}}
2841:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
2730:{\displaystyle {\sqrt {\tfrac {2}{3}}}}
2697:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
2638:{\displaystyle {\sqrt {\tfrac {2}{3}}}}
2553:{\displaystyle {\sqrt {\tfrac {1}{3}}}}
2498:{\displaystyle {\sqrt {\tfrac {4}{3}}}}
1549:quadrirectangular irregular tetrahedron
1465:. This and the regular tessellation of
312:, a set of polyhedrons whose faces are
14:
10450:
9709:metabiaugmented truncated dodecahedron
9704:parabiaugmented truncated dodecahedron
9552:gyroelongated pentagonal cupolarotunda
9522:elongated pentagonal gyrocupolarotunda
7941:
7935:
7624:
7618:
7581:
7575:
3601:
264:
27:Polyhedron with eight triangular faces
9238:
8824:
8431:
8366:
8361:. Vol. 19 (11th ed.). 1911.
8329:
8258:
7985:"On well-covered triangulations. III"
7820:"Convex polyhedra with regular faces"
7653:
7527:
7462:
7412:. Taylor & Francis. p. 252.
7337:"Junction of Non-composite Polyhedra"
7296:
6405:
3051:
3019:, of order 48, the three dimensional
1417:relate to the other Platonic solids.
308:The regular octahedron is one of the
141:
9774:tridiminished rhombicosidodecahedron
8850:
8384:
7972:
7521:
7456:
7401:
7399:
7287:
3782:
3050:(order 24), the symmetry group of a
1527:into an integral number of disjoint
1451:. Therefore, it has the property of
41:
9729:metabigyrate rhombicosidodecahedron
9724:parabigyrate rhombicosidodecahedron
9714:triaugmented truncated dodecahedron
9665:augmented tridiminished icosahedron
9527:elongated pentagonal orthobirotunda
8330:Huson, Daniel H. (September 1998),
8217:Journal of Mathematics and the Arts
7880:(2003), "13.1 Steinitz's theorem",
7380:Mathematical Association of America
7344:St. Petersburg Mathematical Journal
7180:with all exponent values set to 1.
5604:
1487:of the regular octahedron, sharing
1361:polyhedra, meaning that all of the
96:
24:
9557:gyroelongated pentagonal birotunda
9532:elongated pentagonal gyrobirotunda
9507:elongated pentagonal orthobicupola
9492:elongated triangular orthobicupola
9264:
8467:Listed by number of faces and type
7760:
7720:
7680:
3533:Truncated triangular trapezohedron
2904:, and a right triangle with edges
2113:{\displaystyle {\tfrac {\pi }{4}}}
2001:{\displaystyle {\tfrac {\pi }{3}}}
1964:{\displaystyle {\tfrac {\pi }{4}}}
1899:{\displaystyle {\tfrac {\pi }{2}}}
1721:{\displaystyle {\tfrac {\pi }{2}}}
1376:
428:
280:Sketch of a regular octahedron by
245:
239:
25:
10489:
9739:diminished rhombicosidodecahedron
9547:gyroelongated pentagonal bicupola
9537:gyroelongated triangular bicupola
9512:elongated pentagonal gyrobicupola
9497:elongated triangular gyrobicupola
8343:
8333:Two Dimensional Symmetry Mutation
7627:Journal of the Franklin Institute
7548:
7490:
7396:
1760:{\displaystyle \pi -2{\text{𝟁}}}
1234:The graph of a regular octahedron
437:{\displaystyle D_{4\mathrm {h} }}
228:three-dimensional symmetry groups
9734:trigyrate rhombicosidodecahedron
9699:augmented truncated dodecahedron
9442:gyroelongated pentagonal rotunda
9354:gyroelongated pentagonal pyramid
8416:– The Encyclopedia of Polyhedra
7123:
7113:
7106:
7099:
7092:
7085:
7078:
7061:
7054:
7047:
7040:
7033:
7026:
6878:
6871:
6864:
6857:
6850:
6843:
6836:
6829:
6822:
6760:
6755:
6750:
6745:
6740:
6731:
6726:
6721:
6716:
6711:
6702:
6697:
6692:
6687:
6682:
6673:
6668:
6663:
6658:
6653:
6644:
6639:
6634:
6629:
6624:
6615:
6610:
6605:
6600:
6595:
6586:
6581:
6576:
6571:
6566:
6557:
6552:
6547:
6542:
6537:
6528:
6523:
6518:
6513:
6508:
6498:
6491:
6484:
6477:
6470:
6463:
6456:
6449:
6442:
6355:
6348:
6341:
6334:
6327:
6320:
6313:
6261:
6042:
6035:
6028:
6021:
6014:
6007:
6000:
5993:
5936:
5931:
5926:
5921:
5916:
5907:
5902:
5897:
5892:
5887:
5878:
5873:
5868:
5863:
5858:
5849:
5844:
5839:
5834:
5829:
5820:
5815:
5810:
5805:
5800:
5791:
5786:
5781:
5776:
5771:
5762:
5757:
5752:
5747:
5742:
5733:
5728:
5723:
5718:
5713:
5703:
5696:
5689:
5682:
5675:
5668:
5661:
5654:
5540:
5533:
5526:
5519:
5512:
5505:
5498:
5491:
5484:
5477:
5470:
5463:
5365:
5358:
5351:
5344:
5337:
5330:
5323:
5316:
5309:
5302:
5295:
5285:
5280:
5275:
5270:
5265:
5256:
5251:
5246:
5241:
5236:
5227:
5222:
5217:
5212:
5207:
5194:
5189:
5184:
5179:
5174:
5165:
5160:
5155:
5150:
5145:
5136:
5131:
5126:
5121:
5116:
5107:
5102:
5097:
5092:
5087:
5072:
5067:
5062:
5057:
5052:
5043:
5038:
5033:
5028:
5023:
5014:
5009:
5004:
4999:
4994:
4985:
4980:
4975:
4970:
4965:
4956:
4951:
4946:
4941:
4936:
4927:
4922:
4917:
4912:
4907:
4898:
4893:
4888:
4883:
4878:
4869:
4864:
4859:
4854:
4849:
4840:
4835:
4830:
4825:
4820:
4811:
4806:
4801:
4796:
4791:
4782:
4777:
4772:
4767:
4762:
4690:
4684:
4677:
4672:
4665:
4660:
4653:
4646:
4639:
4633:
4626:
4620:
4613:
4607:
4600:
4594:
4587:
4580:
4570:
4565:
4560:
4551:
4546:
4541:
4536:
4531:
4522:
4517:
4512:
4504:
4499:
4494:
4485:
4480:
4475:
4470:
4465:
4456:
4451:
4446:
4438:
4433:
4428:
4419:
4414:
4409:
4404:
4399:
4386:
4381:
4376:
4371:
4366:
4357:
4352:
4347:
4338:
4333:
4328:
4323:
4318:
4309:
4304:
4299:
4290:
4285:
4280:
4275:
4270:
4261:
4256:
4251:
4242:
4237:
4232:
4227:
4222:
4207:
4202:
4197:
4192:
4187:
4174:
4169:
4164:
4159:
4154:
4145:
4140:
4135:
4130:
4125:
4116:
4111:
4106:
4101:
4096:
4087:
4082:
4077:
4072:
4067:
4058:
4053:
4048:
4043:
4038:
4029:
4024:
4019:
4014:
4009:
4000:
3995:
3990:
3985:
3980:
3971:
3966:
3961:
3956:
3951:
3808:
3801:
3769:Tetrahedral-octahedral honeycomb
3655:six ligands in an octahedral or
3632:tetrahedral-octahedral honeycomb
3515:: The eight faces are congruent
3333:
3328:
3323:
3318:
3313:
3304:
3299:
3294:
3289:
3284:
3275:
3270:
3265:
3260:
3255:
3247:
3242:
3237:
3232:
3227:
3218:
3213:
3208:
3200:
3195:
3190:
3185:
3180:
3171:
3166:
3161:
3156:
3151:
3134:
3125:
3116:
3107:
3098:
1634:
1629:
1624:
1619:
1614:
1606:
1601:
1596:
1591:
1586:
288:
273:
159:
150:
46:
9684:augmented truncated tetrahedron
9611:metabiaugmented hexagonal prism
9606:parabiaugmented hexagonal prism
9437:gyroelongated pentagonal cupola
9427:gyroelongated triangular cupola
8323:
8293:
8261:"Resistance-Distance Sum Rules"
8252:
8201:
8175:
8161:
8135:
7825:Canadian Journal of Mathematics
3597:Octahedra in the physical world
995:
967:
905:An octahedron with edge length
846:
805:
460:
393:Many octahedra of interest are
169:The regular octahedron and its
9384:gyroelongated square bipyramid
9379:elongated pentagonal bipyramid
9369:elongated triangular bipyramid
7805:as the edge length (see p. 2).
7314:10.1080/0025570X.1978.11976675
3692:can approximate an octahedron.
3000:Uniform colorings and symmetry
2816:, a right triangle with edges
1120:
1102:
1062:
1044:
1017:
996:
989:
968:
961:
940:
466:3D model of regular octahedron
202:It is one of the eight convex
13:
1:
9719:gyrate rhombicosidodecahedron
9586:triaugmented triangular prism
9542:gyroelongated square bicupola
9502:elongated square gyrobicupola
9477:pentagonal orthocupolarotunda
8678:(two infinite groups and 75)
8457:
8420:Conway Notation for Polyhedra
8399:Paper model of the octahedron
7919:Graduate Texts in Mathematics
7888:Graduate Texts in Mathematics
7356:10.1090/S1061-0022-10-01105-2
7280:
7225:Octahedral molecular geometry
3589:, a non-convex self-crossing
1401:—is its first and only
407:three-dimensional point group
206:because all of the faces are
9833:triangular hebesphenorotunda
9655:metabidiminished icosahedron
9645:metabiaugmented dodecahedron
9640:parabiaugmented dodecahedron
9616:triaugmented hexagonal prism
9596:biaugmented pentagonal prism
9581:biaugmented triangular prism
9482:pentagonal gyrocupolarotunda
9422:elongated pentagonal rotunda
9349:gyroelongated square pyramid
9344:elongated pentagonal pyramid
9334:elongated triangular pyramid
9223:Degenerate polyhedra are in
8696:(two infinite groups and 50)
8274:(2): 633–649. Archived from
8239:10.1080/17513472.2014.974483
7989:Discrete Applied Mathematics
7639:10.1016/0016-0032(71)90071-8
6231:32 all of these tilings are
3837:Uniform octahedral polyhedra
3680:Octahedra in art and culture
3439:
3391:
3369:
3341:
3143:
3091:
2467:
2465:
2426:
2421:
2419:
2378:
2376:
2374:
2372:
2332:
2288:
2286:
2284:
2282:
2259:
2215:
2213:
2211:
2209:
2176:
2090:
2085:
2060:
2055:
2015:
1978:
1973:
1941:
1936:
1913:
1876:
1871:
1831:
1826:
1786:
1735:
1730:
1698:
1693:
1670:
1656:
1424:to define the vertices of a
298:Platonic solid model of the
7:
9432:gyroelongated square cupola
9417:elongated pentagonal cupola
9407:elongated triangular cupola
9042:pentagonal icositetrahedron
8983:truncated icosidodecahedron
7436:Cromwell, Peter R. (1997).
7183:
6185:is any number in the range
6158:is any number in the range
5988:Duals to uniform polyhedra
4699:Duals to uniform polyhedra
2955:{\displaystyle {\sqrt {2}}}
2786:{\displaystyle {\sqrt {2}}}
2583:{\displaystyle {\sqrt {2}}}
2410:{\displaystyle {\text{𝜿}}}
2394:
2292:
2219:
2136:
2076:{\displaystyle {\text{𝜿}}}
2012:
1910:
1783:
1667:
1654:
922:{\displaystyle {\sqrt {2}}}
356:, Kepler also proposed the
216:equilateral square pyramids
187:is an octahedron that is a
10:
10494:
10430:
9857:
9694:biaugmented truncated cube
9591:augmented pentagonal prism
9576:augmented triangular prism
9374:elongated square bipyramid
9072:pentagonal hexecontahedron
9032:deltoidal icositetrahedron
8259:Klein, Douglas J. (2002).
8143:"Enumeration of Polyhedra"
8055:O'Keeffe & Hyde (2020)
7335:Timofeenko, A. V. (2010).
7195:Centered octahedral number
6816:
6419:
5407:
4698:
3835:
1642:trirectangular tetrahedron
1533:characteristic orthoscheme
1519:Characteristic orthoscheme
1498:A regular octahedron is a
397:. A square bipyramid is a
230:, the octahedral symmetry
61:into a new article titled
29:
9841:
9782:
9673:
9660:tridiminished icosahedron
9650:triaugmented dodecahedron
9624:
9601:augmented hexagonal prism
9565:
9487:pentagonal orthobirotunda
9392:
9323:
9272:
9221:
9155:
9130:
9112:
9105:
9080:
9067:disdyakis triacontahedron
9062:deltoidal hexecontahedron
8996:
8904:
8859:
8769:
8748:Kepler–Poinsot polyhedron
8740:
8705:
8653:
8594:
8533:
8472:
8465:
8414:Virtual Reality Polyhedra
8002:10.1016/j.dam.2009.08.002
7956:10.1007/978-3-319-30451-9
7372:Erickson, Martin (2011).
6424:
6279:
6271:
6260:
6244:
5987:
5639:
5632:
5457:
5451:
5445:
3855:
3842:
3659:octahedral configuration.
1649:
1569:. The octahedron and its
1304:{\displaystyle K_{2,2,2}}
1271:complete tripartite graph
1095:is the set of all points
226:, and they have the same
9689:augmented truncated cube
9467:pentagonal orthobicupola
9452:triangular orthobicupola
9339:elongated square pyramid
7584:The Mathematical Gazette
7245:
3513:Tetragonal trapezohedron
3467:Other types of octahedra
1473:in 3-dimensional space.
1459:Octahedra and tetrahedra
1363:maximal independent sets
1350:The octahedral graph is
1250:, provided the graph is
1225:
353:Mysterium Cosmographicum
9472:pentagonal gyrobicupola
9412:elongated square cupola
9173:gyroelongated bipyramid
9047:rhombic triacontahedron
8953:truncated cuboctahedron
8760:Uniform star polyhedron
8688:quasiregular polyhedron
8358:Encyclopædia Britannica
7165:Other related polyhedra
7075:Spherical tiling image
5398:}, continuing into the
3759:Tetrahedral octet truss
3688:Two identically formed
2323:{\displaystyle _{2}R/l}
2250:{\displaystyle _{1}R/l}
2167:{\displaystyle _{0}R/l}
1340:{\displaystyle T_{6,3}}
1126:{\displaystyle (x,y,z)}
1068:{\displaystyle (a,b,c)}
1035:three dimensional space
131:and non-convex shapes.
9845:List of Johnson solids
9808:augmented sphenocorona
9635:augmented dodecahedron
9168:truncated trapezohedra
9037:disdyakis dodecahedron
9003:(duals of Archimedean)
8978:rhombicosidodecahedron
8968:truncated dodecahedron
8694:semiregular polyhedron
8024:Kappraff, Jay (1991).
7839:10.4153/cjm-1966-021-8
7799:
7798:{\displaystyle 2\ell }
7776:
7736:
7696:
5616:– and can be called a
5458:Noncompact hyperbolic
3693:
3663:Widmanstätten patterns
3614:
3558:
3480:
2989:
2956:
2930:
2897:
2864:
2842:
2809:
2787:
2761:
2731:
2698:
2665:
2639:
2606:
2584:
2554:
2521:
2499:
2457:
2411:
2364:
2324:
2274:
2251:
2201:
2168:
2114:
2077:
2047:
2002:
1965:
1928:
1900:
1863:
1818:
1761:
1722:
1685:
1386:
1341:
1305:
1235:
1216:
1127:
1089:
1069:
1027:
923:
889:
752:
722:
692:
662:
524:
504:
484:
467:
438:
390:
255:
214:made by attaching two
9798:snub square antiprism
9057:pentakis dodecahedron
8973:truncated icosahedron
8928:truncated tetrahedron
8741:non-convex polyhedron
8409:The Uniform Polyhedra
8268:Croatica Chemica Acta
8071:Maekawa, Jun (2022).
7915:Lectures on Polytopes
7800:
7777:
7737:
7697:
7375:Beautiful Mathematics
7267:characteristic angles
7017:Apeirogonal antiprism
6233:Wythoff constructions
6217:vertex configurations
3687:
3609:
3581:Schönhardt polyhedron
3549:
3507:Truncated tetrahedron
3474:
3052:rectified tetrahedron
3021:hyperoctahedral group
2990:
2957:
2931:
2898:
2865:
2843:
2810:
2788:
2762:
2732:
2699:
2666:
2640:
2607:
2585:
2563:characteristic angles
2555:
2522:
2500:
2458:
2412:
2365:
2325:
2275:
2252:
2202:
2169:
2115:
2078:
2048:
2003:
1966:
1929:
1901:
1864:
1819:
1762:
1723:
1686:
1463:tessellation of space
1384:
1342:
1306:
1233:
1217:
1128:
1090:
1070:
1028:
933:of the vertices are:
931:Cartesian coordinates
924:
890:
753:
751:{\displaystyle r_{m}}
723:
721:{\displaystyle r_{i}}
693:
691:{\displaystyle r_{u}}
663:
525:
505:
485:
465:
439:
388:
381:As a square bipyramid
256:
208:equilateral triangles
193:equilateral triangles
121:equilateral triangles
10473:Prismatoid polyhedra
9818:hebesphenomegacorona
9457:square orthobicupola
9364:pentagonal bipyramid
9359:triangular bipyramid
9017:rhombic dodecahedron
8943:truncated octahedron
8169:"Counting polyhedra"
7786:
7746:
7706:
7666:
7302:Mathematics Magazine
7220:Truncated octahedron
7009:Heptagonal antiprism
6999:Pentagonal antiprism
6987:Triangular antiprism
6211:The octahedron as a
5622:tetrahedral symmetry
3823:stellated octahedron
3793:stellated octahedron
3618:Natural crystals of
2968:
2942:
2909:
2876:
2854:
2821:
2799:
2773:
2751:
2710:
2677:
2655:
2648:characteristic radii
2646:(edges that are the
2618:
2596:
2570:
2533:
2511:
2478:
2431:
2399:
2337:
2297:
2264:
2224:
2181:
2141:
2095:
2065:
2020:
1983:
1946:
1918:
1881:
1836:
1791:
1740:
1703:
1675:
1482:tetrahedral symmetry
1389:The interior of the
1367:pentagonal dipyramid
1318:
1276:
1137:
1099:
1079:
1041:
937:
909:
762:
735:
705:
675:
670:circumscribed sphere
534:
514:
494:
474:
416:
367:regular dodecahedron
234:
212:composite polyhedron
10425:pentagonal polytope
10324:Uniform 10-polytope
9884:Fundamental convex
9847:, a sortable table)
9462:square gyrobicupola
9052:triakis icosahedron
9027:tetrakis hexahedron
9012:triakis tetrahedron
8948:rhombicuboctahedron
8385:Klitzing, Richard.
8189:on 17 November 2014
7981:Plummer, Michael D.
7942:Negami, S. (2016).
7230:Octahedral symmetry
7200:Spinning octahedron
7120:Plane tiling image
7004:Hexagonal antiprism
6972:
3602:Octahedra in nature
3591:flexible polyhedron
3073:(Tetratetrahedron)
1478:tetrahemihexahedron
1426:regular icosahedron
371:regular tetrahedron
363:regular icosahedron
265:As a Platonic solid
30:For the album, see
10294:Uniform 9-polytope
10244:Uniform 8-polytope
10194:Uniform 7-polytope
10151:Uniform 6-polytope
10121:Uniform 5-polytope
10081:Uniform polychoron
10044:Uniform polyhedron
9892:in dimensions 2–10
9676:Archimedean solids
9315:pentagonal rotunda
9295:pentagonal pyramid
9022:triakis octahedron
8907:Archimedean solids
8682:regular polyhedron
8676:uniform polyhedron
8638:Hectotriadiohedron
8368:Weisstein, Eric W.
8352:"Octahedron"
8149:on 10 October 2011
7911:Ziegler, Günter M.
7816:Johnson, Norman W.
7795:
7772:
7732:
7692:
7528:Polya, G. (1954).
7505:Dover Publications
7476:. pp. 70–71.
7215:Hexakis octahedron
7210:Triakis octahedron
6935:
6817:Duals to uniforms
6406:Trigonal antiprism
6237:fundamental domain
3773:Buckminster Fuller
3694:
3615:
3587:Bricard octahedron
3559:
3551:Bricard octahedron
3481:
2985:
2982:
2952:
2926:
2923:
2893:
2890:
2860:
2838:
2835:
2805:
2783:
2757:
2727:
2724:
2694:
2691:
2661:
2635:
2632:
2602:
2580:
2565:𝟀, 𝝉, 𝟁), plus
2550:
2547:
2517:
2495:
2492:
2453:
2451:
2407:
2360:
2351:
2320:
2270:
2247:
2197:
2164:
2110:
2108:
2073:
2043:
2034:
1998:
1996:
1961:
1959:
1924:
1896:
1894:
1859:
1849:
1814:
1805:
1757:
1718:
1716:
1681:
1493:vertex arrangement
1471:uniform honeycombs
1469:are the only such
1449:trigonal antiprism
1387:
1337:
1301:
1248:Steinitz's theorem
1236:
1212:
1123:
1085:
1065:
1023:
919:
885:
748:
718:
688:
658:
656:
520:
500:
480:
468:
434:
391:
251:
189:regular polyhedron
185:regular octahedron
142:Regular octahedron
119:composed of eight
113:regular octahedron
18:Regular octahedron
10463:Individual graphs
10446:
10445:
10433:Polytope families
9890:uniform polytopes
9852:
9851:
9785:elementary solids
9310:pentagonal cupola
9300:triangular cupola
9232:
9231:
9151:
9150:
8988:snub dodecahedron
8963:icosidodecahedron
8818:
8817:
8719:Archimedean solid
8706:convex polyhedron
8614:Icosidodecahedron
8306:Regular Polytopes
8088:978-1-4629-2398-4
8041:978-981-281-139-4
7965:978-3-319-30451-9
7568:978-1-285-19569-8
7514:978-0-486-83654-6
7449:978-0-521-55432-9
7419:978-1-4665-5464-1
7389:978-1-61444-509-8
7298:Trigg, Charles W.
7240:Octahedral sphere
7190:Octahedral number
7162:
7161:
7023:Polyhedron image
6980:Digonal antiprism
6933:
6932:
6403:
6402:
6225:orbifold notation
6092:
6091:
5602:
5601:
5373:
5372:
3828:
3827:
3783:Related polyhedra
3699:roleplaying games
3555:antiparallelogram
3527:triangular prisms
3464:
3463:
3011:The octahedron's
3006:uniform colorings
2995:
2983:
2981:
2962:
2950:
2936:
2924:
2922:
2903:
2891:
2889:
2870:
2863:{\displaystyle 1}
2848:
2836:
2834:
2815:
2808:{\displaystyle 1}
2793:
2781:
2767:
2760:{\displaystyle 1}
2744:45-90-45 triangle
2740:90-60-30 triangle
2737:
2725:
2723:
2704:
2692:
2690:
2671:
2664:{\displaystyle 1}
2645:
2633:
2631:
2612:
2605:{\displaystyle 1}
2590:
2578:
2560:
2548:
2546:
2527:
2520:{\displaystyle 1}
2505:
2493:
2491:
2471:
2470:
2463:
2450:
2441:
2424:
2417:
2405:
2370:
2352:
2350:
2330:
2280:
2273:{\displaystyle 1}
2257:
2207:
2189:
2174:
2120:
2107:
2088:
2083:
2071:
2058:
2053:
2035:
2033:
2008:
1995:
1976:
1971:
1958:
1939:
1934:
1927:{\displaystyle 1}
1906:
1893:
1874:
1869:
1857:
1848:
1829:
1824:
1806:
1804:
1767:
1755:
1733:
1728:
1715:
1696:
1691:
1684:{\displaystyle 2}
1415:icosidodecahedron
1256:3-connected graph
1088:{\displaystyle r}
917:
868:
829:
825:
788:
784:
621:
614:
559:
523:{\displaystyle a}
503:{\displaystyle V}
483:{\displaystyle A}
470:The surface area
395:square bipyramids
331:classical element
85:
84:
16:(Redirected from
10485:
10437:Regular polytope
9998:
9987:
9976:
9935:
9878:
9871:
9864:
9855:
9854:
9823:disphenocingulum
9813:sphenomegacorona
9259:
9252:
9245:
9236:
9235:
9110:
9109:
9106:Dihedral uniform
9081:Dihedral regular
9004:
8920:
8869:
8845:
8838:
8831:
8822:
8821:
8654:elemental things
8632:Enneacontahedron
8602:Icositetrahedron
8452:
8445:
8438:
8429:
8428:
8390:
8381:
8380:
8362:
8354:
8337:
8336:
8327:
8321:
8320:
8297:
8291:
8290:
8288:
8286:
8280:
8265:
8256:
8250:
8249:
8232:
8223:(3–4): 111–119,
8205:
8199:
8198:
8196:
8194:
8185:. Archived from
8179:
8173:
8172:
8165:
8159:
8158:
8156:
8154:
8145:. Archived from
8139:
8133:
8123:
8117:
8111:
8105:
8099:
8093:
8092:
8068:
8062:
8052:
8046:
8045:
8032:World Scientific
8030:(2nd ed.).
8021:
8015:
8014:
8004:
7976:
7970:
7969:
7939:
7933:
7932:
7907:
7901:
7900:
7883:Convex Polytopes
7878:Grünbaum, Branko
7874:
7868:
7867:
7841:
7812:
7806:
7804:
7802:
7801:
7796:
7781:
7779:
7778:
7773:
7768:
7763:
7757:
7756:
7751:
7741:
7739:
7738:
7733:
7728:
7723:
7717:
7716:
7711:
7701:
7699:
7698:
7693:
7688:
7683:
7677:
7676:
7671:
7657:
7651:
7650:
7622:
7616:
7615:
7590:(469): 243–256.
7579:
7573:
7572:
7552:
7546:
7545:
7525:
7519:
7518:
7494:
7488:
7487:
7460:
7454:
7453:
7433:
7424:
7423:
7403:
7394:
7393:
7369:
7360:
7359:
7341:
7332:
7326:
7325:
7294:
7274:
7259:
7235:Octahedral graph
7205:Stella octangula
7171:square bifrustum
7127:
7117:
7110:
7103:
7096:
7089:
7082:
7065:
7058:
7051:
7044:
7037:
7030:
6994:Square antiprism
6973:
6967:
6960:
6953:
6934:
6882:
6875:
6868:
6861:
6854:
6847:
6840:
6833:
6826:
6765:
6764:
6763:
6759:
6758:
6754:
6753:
6749:
6748:
6744:
6743:
6736:
6735:
6734:
6730:
6729:
6725:
6724:
6720:
6719:
6715:
6714:
6707:
6706:
6705:
6701:
6700:
6696:
6695:
6691:
6690:
6686:
6685:
6678:
6677:
6676:
6672:
6671:
6667:
6666:
6662:
6661:
6657:
6656:
6649:
6648:
6647:
6643:
6642:
6638:
6637:
6633:
6632:
6628:
6627:
6620:
6619:
6618:
6614:
6613:
6609:
6608:
6604:
6603:
6599:
6598:
6591:
6590:
6589:
6585:
6584:
6580:
6579:
6575:
6574:
6570:
6569:
6562:
6561:
6560:
6556:
6555:
6551:
6550:
6546:
6545:
6541:
6540:
6533:
6532:
6531:
6527:
6526:
6522:
6521:
6517:
6516:
6512:
6511:
6502:
6495:
6488:
6481:
6474:
6467:
6460:
6453:
6446:
6417:
6416:
6359:
6352:
6345:
6338:
6331:
6324:
6317:
6265:
6242:
6241:
6213:tetratetrahedron
6207:
6201:
6199:
6198:
6195:
6192:
6180:
6179:
6177:
6176:
6173:
6170:
6149:
6147:
6146:
6143:
6140:
6133:
6131:
6130:
6127:
6124:
6117:
6115:
6114:
6111:
6108:
6046:
6039:
6032:
6025:
6018:
6011:
6004:
5997:
5941:
5940:
5939:
5935:
5934:
5930:
5929:
5925:
5924:
5920:
5919:
5912:
5911:
5910:
5906:
5905:
5901:
5900:
5896:
5895:
5891:
5890:
5883:
5882:
5881:
5877:
5876:
5872:
5871:
5867:
5866:
5862:
5861:
5854:
5853:
5852:
5848:
5847:
5843:
5842:
5838:
5837:
5833:
5832:
5825:
5824:
5823:
5819:
5818:
5814:
5813:
5809:
5808:
5804:
5803:
5796:
5795:
5794:
5790:
5789:
5785:
5784:
5780:
5779:
5775:
5774:
5767:
5766:
5765:
5761:
5760:
5756:
5755:
5751:
5750:
5746:
5745:
5738:
5737:
5736:
5732:
5731:
5727:
5726:
5722:
5721:
5717:
5716:
5707:
5700:
5693:
5686:
5679:
5672:
5665:
5658:
5630:
5629:
5618:tetratetrahedron
5605:Tetratetrahedron
5544:
5537:
5530:
5523:
5516:
5509:
5502:
5495:
5488:
5481:
5474:
5467:
5437:
5430:
5423:
5405:
5404:
5400:hyperbolic plane
5392:Schläfli symbols
5369:
5362:
5355:
5348:
5341:
5334:
5327:
5320:
5313:
5306:
5299:
5290:
5289:
5288:
5284:
5283:
5279:
5278:
5274:
5273:
5269:
5268:
5261:
5260:
5259:
5255:
5254:
5250:
5249:
5245:
5244:
5240:
5239:
5232:
5231:
5230:
5226:
5225:
5221:
5220:
5216:
5215:
5211:
5210:
5199:
5198:
5197:
5193:
5192:
5188:
5187:
5183:
5182:
5178:
5177:
5170:
5169:
5168:
5164:
5163:
5159:
5158:
5154:
5153:
5149:
5148:
5141:
5140:
5139:
5135:
5134:
5130:
5129:
5125:
5124:
5120:
5119:
5112:
5111:
5110:
5106:
5105:
5101:
5100:
5096:
5095:
5091:
5090:
5077:
5076:
5075:
5071:
5070:
5066:
5065:
5061:
5060:
5056:
5055:
5048:
5047:
5046:
5042:
5041:
5037:
5036:
5032:
5031:
5027:
5026:
5019:
5018:
5017:
5013:
5012:
5008:
5007:
5003:
5002:
4998:
4997:
4990:
4989:
4988:
4984:
4983:
4979:
4978:
4974:
4973:
4969:
4968:
4961:
4960:
4959:
4955:
4954:
4950:
4949:
4945:
4944:
4940:
4939:
4932:
4931:
4930:
4926:
4925:
4921:
4920:
4916:
4915:
4911:
4910:
4903:
4902:
4901:
4897:
4896:
4892:
4891:
4887:
4886:
4882:
4881:
4874:
4873:
4872:
4868:
4867:
4863:
4862:
4858:
4857:
4853:
4852:
4845:
4844:
4843:
4839:
4838:
4834:
4833:
4829:
4828:
4824:
4823:
4816:
4815:
4814:
4810:
4809:
4805:
4804:
4800:
4799:
4795:
4794:
4787:
4786:
4785:
4781:
4780:
4776:
4775:
4771:
4770:
4766:
4765:
4694:
4688:
4681:
4676:
4669:
4664:
4657:
4650:
4643:
4637:
4630:
4624:
4617:
4611:
4604:
4598:
4591:
4584:
4575:
4574:
4573:
4569:
4568:
4564:
4563:
4556:
4555:
4554:
4550:
4549:
4545:
4544:
4540:
4539:
4535:
4534:
4527:
4526:
4525:
4521:
4520:
4516:
4515:
4509:
4508:
4507:
4503:
4502:
4498:
4497:
4490:
4489:
4488:
4484:
4483:
4479:
4478:
4474:
4473:
4469:
4468:
4461:
4460:
4459:
4455:
4454:
4450:
4449:
4443:
4442:
4441:
4437:
4436:
4432:
4431:
4424:
4423:
4422:
4418:
4417:
4413:
4412:
4408:
4407:
4403:
4402:
4391:
4390:
4389:
4385:
4384:
4380:
4379:
4375:
4374:
4370:
4369:
4362:
4361:
4360:
4356:
4355:
4351:
4350:
4343:
4342:
4341:
4337:
4336:
4332:
4331:
4327:
4326:
4322:
4321:
4314:
4313:
4312:
4308:
4307:
4303:
4302:
4295:
4294:
4293:
4289:
4288:
4284:
4283:
4279:
4278:
4274:
4273:
4266:
4265:
4264:
4260:
4259:
4255:
4254:
4247:
4246:
4245:
4241:
4240:
4236:
4235:
4231:
4230:
4226:
4225:
4212:
4211:
4210:
4206:
4205:
4201:
4200:
4196:
4195:
4191:
4190:
4179:
4178:
4177:
4173:
4172:
4168:
4167:
4163:
4162:
4158:
4157:
4150:
4149:
4148:
4144:
4143:
4139:
4138:
4134:
4133:
4129:
4128:
4121:
4120:
4119:
4115:
4114:
4110:
4109:
4105:
4104:
4100:
4099:
4092:
4091:
4090:
4086:
4085:
4081:
4080:
4076:
4075:
4071:
4070:
4063:
4062:
4061:
4057:
4056:
4052:
4051:
4047:
4046:
4042:
4041:
4034:
4033:
4032:
4028:
4027:
4023:
4022:
4018:
4017:
4013:
4012:
4005:
4004:
4003:
3999:
3998:
3994:
3993:
3989:
3988:
3984:
3983:
3976:
3975:
3974:
3970:
3969:
3965:
3964:
3960:
3959:
3955:
3954:
3833:
3832:
3812:
3805:
3798:
3797:
3771:was invented by
3746:
3744:
3743:
3740:
3737:
3730:
3728:
3727:
3724:
3721:
3651:Many metal ions
3366:{ } + { } + { }
3338:
3337:
3336:
3332:
3331:
3327:
3326:
3322:
3321:
3317:
3316:
3309:
3308:
3307:
3303:
3302:
3298:
3297:
3293:
3292:
3288:
3287:
3280:
3279:
3278:
3274:
3273:
3269:
3268:
3264:
3263:
3259:
3258:
3252:
3251:
3250:
3246:
3245:
3241:
3240:
3236:
3235:
3231:
3230:
3223:
3222:
3221:
3217:
3216:
3212:
3211:
3205:
3204:
3203:
3199:
3198:
3194:
3193:
3189:
3188:
3184:
3183:
3176:
3175:
3174:
3170:
3169:
3165:
3164:
3160:
3159:
3155:
3154:
3138:
3129:
3120:
3111:
3102:
3095:(Face coloring)
3057:
3056:
3023:. This group's
2994:
2992:
2991:
2986:
2984:
2974:
2972:
2964:
2961:
2959:
2958:
2953:
2951:
2946:
2938:
2935:
2933:
2932:
2927:
2925:
2915:
2913:
2905:
2902:
2900:
2899:
2894:
2892:
2882:
2880:
2872:
2869:
2867:
2866:
2861:
2850:
2847:
2845:
2844:
2839:
2837:
2827:
2825:
2817:
2814:
2812:
2811:
2806:
2795:
2792:
2790:
2789:
2784:
2782:
2777:
2769:
2766:
2764:
2763:
2758:
2747:
2736:
2734:
2733:
2728:
2726:
2716:
2714:
2706:
2703:
2701:
2700:
2695:
2693:
2683:
2681:
2673:
2670:
2668:
2667:
2662:
2651:
2644:
2642:
2641:
2636:
2634:
2624:
2622:
2614:
2611:
2609:
2608:
2603:
2592:
2589:
2587:
2586:
2581:
2579:
2574:
2566:
2559:
2557:
2556:
2551:
2549:
2539:
2537:
2529:
2526:
2524:
2523:
2518:
2507:
2504:
2502:
2501:
2496:
2494:
2484:
2482:
2474:
2462:
2460:
2459:
2454:
2452:
2446:
2442:
2439:
2436:
2427:
2422:
2416:
2414:
2413:
2408:
2406:
2403:
2395:
2369:
2367:
2366:
2361:
2353:
2343:
2341:
2333:
2329:
2327:
2326:
2321:
2316:
2308:
2307:
2293:
2279:
2277:
2276:
2271:
2260:
2256:
2254:
2253:
2248:
2243:
2235:
2234:
2220:
2206:
2204:
2203:
2198:
2190:
2185:
2177:
2173:
2171:
2170:
2165:
2160:
2152:
2151:
2137:
2119:
2117:
2116:
2111:
2109:
2100:
2091:
2086:
2082:
2080:
2079:
2074:
2072:
2069:
2061:
2056:
2052:
2050:
2049:
2044:
2036:
2026:
2024:
2016:
2007:
2005:
2004:
1999:
1997:
1988:
1979:
1974:
1970:
1968:
1967:
1962:
1960:
1951:
1942:
1937:
1933:
1931:
1930:
1925:
1914:
1905:
1903:
1902:
1897:
1895:
1886:
1877:
1872:
1868:
1866:
1865:
1860:
1858:
1855:
1850:
1841:
1832:
1827:
1823:
1821:
1820:
1815:
1807:
1797:
1795:
1787:
1766:
1764:
1763:
1758:
1756:
1753:
1736:
1731:
1727:
1725:
1724:
1719:
1717:
1708:
1699:
1694:
1690:
1688:
1687:
1682:
1671:
1647:
1646:
1639:
1638:
1637:
1633:
1632:
1628:
1627:
1623:
1622:
1618:
1617:
1611:
1610:
1609:
1605:
1604:
1600:
1599:
1595:
1594:
1590:
1589:
1509:
1430:regular compound
1399:stella octangula
1346:
1344:
1343:
1338:
1336:
1335:
1310:
1308:
1307:
1302:
1300:
1299:
1260:octahedral graph
1221:
1219:
1218:
1213:
1202:
1198:
1180:
1176:
1158:
1154:
1132:
1130:
1129:
1124:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1032:
1030:
1029:
1024:
928:
926:
925:
920:
918:
913:
894:
892:
891:
886:
869:
861:
856:
855:
830:
821:
820:
815:
814:
789:
780:
779:
774:
773:
757:
755:
754:
749:
747:
746:
727:
725:
724:
719:
717:
716:
700:inscribed sphere
697:
695:
694:
689:
687:
686:
668:The radius of a
667:
665:
664:
659:
657:
650:
649:
632:
631:
622:
617:
615:
607:
588:
587:
570:
569:
560:
555:
529:
527:
526:
521:
509:
507:
506:
501:
489:
487:
486:
481:
464:
443:
441:
440:
435:
433:
432:
431:
389:Square bipyramid
347:Harmonices Mundi
317:regular polygons
292:
277:
260:
258:
257:
252:
250:
249:
248:
242:
163:
154:
98:
80:
77:
64:Square bipyramid
50:
49:
42:
21:
10493:
10492:
10488:
10487:
10486:
10484:
10483:
10482:
10468:Platonic solids
10448:
10447:
10416:
10409:
10402:
10285:
10278:
10271:
10235:
10228:
10221:
10185:
10178:
10012:Regular polygon
10005:
9996:
9989:
9985:
9978:
9974:
9965:
9956:
9949:
9945:
9933:
9927:
9923:
9911:
9893:
9882:
9853:
9848:
9837:
9828:bilunabirotunda
9793:snub disphenoid
9778:
9669:
9627:Platonic solids
9620:
9561:
9447:gyrobifastigium
9388:
9319:
9268:
9263:
9233:
9228:
9217:
9156:Dihedral others
9147:
9126:
9101:
9076:
9005:
9002:
9001:
8992:
8921:
8910:
8909:
8900:
8863:
8861:Platonic solids
8855:
8849:
8819:
8814:
8765:
8754:Star polyhedron
8736:
8701:
8649:
8626:Hexecontahedron
8608:Triacontahedron
8590:
8581:Enneadecahedron
8571:Heptadecahedron
8561:Pentadecahedron
8556:Tetradecahedron
8529:
8468:
8461:
8456:
8349:
8346:
8341:
8340:
8328:
8324:
8317:
8301:Coxeter, H.S.M.
8298:
8294:
8284:
8282:
8281:on 10 June 2007
8278:
8263:
8257:
8253:
8206:
8202:
8192:
8190:
8181:
8180:
8176:
8167:
8166:
8162:
8152:
8150:
8141:
8140:
8136:
8131:
8124:
8120:
8112:
8108:
8100:
8096:
8089:
8069:
8065:
8053:
8049:
8042:
8034:. p. 475.
8022:
8018:
7977:
7973:
7966:
7940:
7936:
7929:
7908:
7904:
7898:
7875:
7871:
7813:
7809:
7787:
7784:
7783:
7764:
7759:
7752:
7750:
7749:
7747:
7744:
7743:
7724:
7719:
7712:
7710:
7709:
7707:
7704:
7703:
7684:
7679:
7672:
7670:
7669:
7667:
7664:
7663:
7658:
7654:
7623:
7619:
7596:10.2307/3619822
7580:
7576:
7569:
7553:
7549:
7542:
7526:
7522:
7515:
7507:. p. 141.
7495:
7491:
7484:
7461:
7457:
7450:
7434:
7427:
7420:
7404:
7397:
7390:
7370:
7363:
7339:
7333:
7329:
7295:
7288:
7283:
7278:
7277:
7260:
7253:
7248:
7186:
7167:
6992:
6985:
6976:Antiprism name
6971:
6761:
6756:
6751:
6746:
6741:
6739:
6732:
6727:
6722:
6717:
6712:
6710:
6703:
6698:
6693:
6688:
6683:
6681:
6674:
6669:
6664:
6659:
6654:
6652:
6645:
6640:
6635:
6630:
6625:
6623:
6616:
6611:
6606:
6601:
6596:
6594:
6587:
6582:
6577:
6572:
6567:
6565:
6558:
6553:
6548:
6543:
6538:
6536:
6529:
6524:
6519:
6514:
6509:
6507:
6408:
6309:
6266:
6196:
6193:
6190:
6189:
6187:
6186:
6174:
6171:
6168:
6167:
6165:
6159:
6144:
6141:
6138:
6137:
6135:
6128:
6125:
6122:
6121:
6119:
6112:
6109:
6106:
6105:
6103:
5937:
5932:
5927:
5922:
5917:
5915:
5908:
5903:
5898:
5893:
5888:
5886:
5879:
5874:
5869:
5864:
5859:
5857:
5850:
5845:
5840:
5835:
5830:
5828:
5821:
5816:
5811:
5806:
5801:
5799:
5792:
5787:
5782:
5777:
5772:
5770:
5763:
5758:
5753:
5748:
5743:
5741:
5734:
5729:
5724:
5719:
5714:
5712:
5607:
5452:Compact hyper.
5441:
5286:
5281:
5276:
5271:
5266:
5264:
5257:
5252:
5247:
5242:
5237:
5235:
5228:
5223:
5218:
5213:
5208:
5206:
5195:
5190:
5185:
5180:
5175:
5173:
5166:
5161:
5156:
5151:
5146:
5144:
5137:
5132:
5127:
5122:
5117:
5115:
5108:
5103:
5098:
5093:
5088:
5086:
5073:
5068:
5063:
5058:
5053:
5051:
5044:
5039:
5034:
5029:
5024:
5022:
5015:
5010:
5005:
5000:
4995:
4993:
4986:
4981:
4976:
4971:
4966:
4964:
4957:
4952:
4947:
4942:
4937:
4935:
4928:
4923:
4918:
4913:
4908:
4906:
4899:
4894:
4889:
4884:
4879:
4877:
4870:
4865:
4860:
4855:
4850:
4848:
4841:
4836:
4831:
4826:
4821:
4819:
4812:
4807:
4802:
4797:
4792:
4790:
4783:
4778:
4773:
4768:
4763:
4761:
4689:
4638:
4625:
4612:
4599:
4571:
4566:
4561:
4559:
4558:
4552:
4547:
4542:
4537:
4532:
4530:
4523:
4518:
4513:
4511:
4505:
4500:
4495:
4493:
4492:
4486:
4481:
4476:
4471:
4466:
4464:
4457:
4452:
4447:
4445:
4439:
4434:
4429:
4427:
4426:
4420:
4415:
4410:
4405:
4400:
4398:
4387:
4382:
4377:
4372:
4367:
4365:
4358:
4353:
4348:
4346:
4344:
4339:
4334:
4329:
4324:
4319:
4317:
4310:
4305:
4300:
4298:
4296:
4291:
4286:
4281:
4276:
4271:
4269:
4262:
4257:
4252:
4250:
4248:
4243:
4238:
4233:
4228:
4223:
4221:
4208:
4203:
4198:
4193:
4188:
4186:
4175:
4170:
4165:
4160:
4155:
4153:
4146:
4141:
4136:
4131:
4126:
4124:
4117:
4112:
4107:
4102:
4097:
4095:
4088:
4083:
4078:
4073:
4068:
4066:
4059:
4054:
4049:
4044:
4039:
4037:
4030:
4025:
4020:
4015:
4010:
4008:
4001:
3996:
3991:
3986:
3981:
3979:
3972:
3967:
3962:
3957:
3952:
3950:
3944:
3937:
3934:
3926:
3909:
3905:
3898:
3891:
3884:
3863:
3857:
3852:
3785:
3761:
3741:
3738:
3735:
3734:
3732:
3725:
3722:
3719:
3718:
3716:
3703:polyhedral dice
3682:
3604:
3599:
3523:Gyrobifastigium
3494:Hexagonal prism
3477:gyrobifastigium
3469:
3453:
3435:
3428:
3421:
3417:
3415:
3408:
3401:
3383:
3365:
3360:
3355:
3344:Schläfli symbol
3334:
3329:
3324:
3319:
3314:
3312:
3305:
3300:
3295:
3290:
3285:
3283:
3276:
3271:
3266:
3261:
3256:
3254:
3253:
3248:
3243:
3238:
3233:
3228:
3226:
3219:
3214:
3209:
3207:
3201:
3196:
3191:
3186:
3181:
3179:
3172:
3167:
3162:
3157:
3152:
3150:
3146:Coxeter diagram
3139:
3130:
3121:
3112:
3103:
3094:
3072:
3049:
3040:
3030:
3018:
3002:
2971:
2969:
2966:
2965:
2945:
2943:
2940:
2939:
2912:
2910:
2907:
2906:
2879:
2877:
2874:
2873:
2855:
2852:
2851:
2824:
2822:
2819:
2818:
2800:
2797:
2796:
2776:
2774:
2771:
2770:
2752:
2749:
2748:
2713:
2711:
2708:
2707:
2680:
2678:
2675:
2674:
2656:
2653:
2652:
2621:
2619:
2616:
2615:
2597:
2594:
2593:
2573:
2571:
2568:
2567:
2536:
2534:
2531:
2530:
2512:
2509:
2508:
2481:
2479:
2476:
2475:
2438:
2437:
2434:
2432:
2429:
2428:
2402:
2400:
2397:
2396:
2340:
2338:
2335:
2334:
2312:
2303:
2300:
2298:
2295:
2294:
2265:
2262:
2261:
2239:
2230:
2227:
2225:
2222:
2221:
2184:
2182:
2179:
2178:
2156:
2147:
2144:
2142:
2139:
2138:
2098:
2096:
2093:
2092:
2068:
2066:
2063:
2062:
2023:
2021:
2018:
2017:
1986:
1984:
1981:
1980:
1949:
1947:
1944:
1943:
1919:
1916:
1915:
1884:
1882:
1879:
1878:
1854:
1839:
1837:
1834:
1833:
1794:
1792:
1789:
1788:
1752:
1741:
1738:
1737:
1706:
1704:
1701:
1700:
1676:
1673:
1672:
1635:
1630:
1625:
1620:
1615:
1613:
1607:
1602:
1597:
1592:
1587:
1585:
1567:
1521:
1512:
1505:
1434:snub octahedron
1379:
1377:Related figures
1371:snub disphenoid
1325:
1321:
1319:
1316:
1315:
1283:
1279:
1277:
1274:
1273:
1228:
1188:
1184:
1166:
1162:
1144:
1140:
1138:
1135:
1134:
1100:
1097:
1096:
1080:
1077:
1076:
1042:
1039:
1038:
938:
935:
934:
912:
910:
907:
906:
860:
851:
847:
819:
810:
806:
778:
769:
765:
763:
760:
759:
742:
738:
736:
733:
732:
712:
708:
706:
703:
702:
682:
678:
676:
673:
672:
655:
654:
645:
641:
633:
627:
623:
616:
606:
599:
593:
592:
583:
579:
571:
565:
561:
554:
544:
537:
535:
532:
531:
515:
512:
511:
495:
492:
491:
475:
472:
471:
458:
447:face-transitive
427:
423:
419:
417:
414:
413:
383:
342:Johannes Kepler
310:Platonic solids
306:
305:
304:
303:
302:
293:
285:
284:
282:Johannes Kepler
278:
267:
244:
243:
238:
237:
235:
232:
231:
220:dual polyhedron
181:
180:
179:
178:
171:dual polyhedron
166:
165:
164:
156:
155:
144:
81:
75:
72:
51:
47:
38:
28:
23:
22:
15:
12:
11:
5:
10491:
10481:
10480:
10475:
10470:
10465:
10460:
10444:
10443:
10428:
10427:
10418:
10414:
10407:
10400:
10396:
10387:
10370:
10361:
10350:
10349:
10347:
10345:
10340:
10331:
10326:
10320:
10319:
10317:
10315:
10310:
10301:
10296:
10290:
10289:
10287:
10283:
10276:
10269:
10265:
10260:
10251:
10246:
10240:
10239:
10237:
10233:
10226:
10219:
10215:
10210:
10201:
10196:
10190:
10189:
10187:
10183:
10176:
10172:
10167:
10158:
10153:
10147:
10146:
10144:
10142:
10137:
10128:
10123:
10117:
10116:
10107:
10102:
10097:
10088:
10083:
10077:
10076:
10067:
10065:
10060:
10051:
10046:
10040:
10039:
10034:
10029:
10024:
10019:
10014:
10008:
10007:
10003:
9999:
9994:
9983:
9972:
9963:
9954:
9947:
9941:
9931:
9925:
9919:
9913:
9907:
9901:
9895:
9894:
9883:
9881:
9880:
9873:
9866:
9858:
9850:
9849:
9842:
9839:
9838:
9836:
9835:
9830:
9825:
9820:
9815:
9810:
9805:
9800:
9795:
9789:
9787:
9780:
9779:
9777:
9776:
9771:
9766:
9761:
9756:
9751:
9746:
9741:
9736:
9731:
9726:
9721:
9716:
9711:
9706:
9701:
9696:
9691:
9686:
9680:
9678:
9671:
9670:
9668:
9667:
9662:
9657:
9652:
9647:
9642:
9637:
9631:
9629:
9622:
9621:
9619:
9618:
9613:
9608:
9603:
9598:
9593:
9588:
9583:
9578:
9572:
9570:
9563:
9562:
9560:
9559:
9554:
9549:
9544:
9539:
9534:
9529:
9524:
9519:
9514:
9509:
9504:
9499:
9494:
9489:
9484:
9479:
9474:
9469:
9464:
9459:
9454:
9449:
9444:
9439:
9434:
9429:
9424:
9419:
9414:
9409:
9403:
9401:
9390:
9389:
9387:
9386:
9381:
9376:
9371:
9366:
9361:
9356:
9351:
9346:
9341:
9336:
9330:
9328:
9321:
9320:
9318:
9317:
9312:
9307:
9302:
9297:
9292:
9290:square pyramid
9286:
9284:
9270:
9269:
9266:Johnson solids
9262:
9261:
9254:
9247:
9239:
9230:
9229:
9222:
9219:
9218:
9216:
9215:
9210:
9205:
9200:
9195:
9190:
9185:
9180:
9175:
9170:
9165:
9159:
9157:
9153:
9152:
9149:
9148:
9146:
9145:
9140:
9134:
9132:
9128:
9127:
9125:
9124:
9119:
9113:
9107:
9103:
9102:
9100:
9099:
9092:
9084:
9082:
9078:
9077:
9075:
9074:
9069:
9064:
9059:
9054:
9049:
9044:
9039:
9034:
9029:
9024:
9019:
9014:
9008:
9006:
8999:Catalan solids
8997:
8994:
8993:
8991:
8990:
8985:
8980:
8975:
8970:
8965:
8960:
8955:
8950:
8945:
8940:
8938:truncated cube
8935:
8930:
8924:
8922:
8905:
8902:
8901:
8899:
8898:
8893:
8888:
8883:
8878:
8872:
8870:
8857:
8856:
8848:
8847:
8840:
8833:
8825:
8816:
8815:
8813:
8812:
8810:parallelepiped
8807:
8802:
8797:
8792:
8787:
8782:
8776:
8774:
8767:
8766:
8764:
8763:
8757:
8751:
8744:
8742:
8738:
8737:
8735:
8734:
8728:
8722:
8716:
8713:Platonic solid
8709:
8707:
8703:
8702:
8700:
8699:
8698:
8697:
8691:
8685:
8673:
8668:
8663:
8657:
8655:
8651:
8650:
8648:
8647:
8641:
8635:
8629:
8623:
8617:
8611:
8605:
8598:
8596:
8592:
8591:
8589:
8588:
8583:
8578:
8576:Octadecahedron
8573:
8568:
8566:Hexadecahedron
8563:
8558:
8553:
8548:
8543:
8537:
8535:
8531:
8530:
8528:
8527:
8522:
8517:
8512:
8507:
8502:
8497:
8492:
8487:
8482:
8476:
8474:
8470:
8469:
8466:
8463:
8462:
8455:
8454:
8447:
8440:
8432:
8426:
8425:
8424:
8423:
8411:
8406:
8401:
8396:
8391:
8382:
8363:
8345:
8344:External links
8342:
8339:
8338:
8322:
8315:
8292:
8251:
8211:; Frantz, M.;
8200:
8174:
8160:
8134:
8129:
8118:
8106:
8094:
8087:
8081:. p. 42.
8063:
8047:
8040:
8016:
7995:(8): 894–912.
7971:
7964:
7934:
7927:
7902:
7896:
7869:
7807:
7794:
7791:
7771:
7767:
7762:
7755:
7731:
7727:
7722:
7715:
7691:
7687:
7682:
7675:
7660:Coxeter (1973)
7652:
7633:(5): 329–352.
7617:
7574:
7567:
7547:
7540:
7520:
7513:
7489:
7482:
7474:Broadway Books
7455:
7448:
7425:
7418:
7395:
7388:
7382:. p. 62.
7361:
7350:(3): 483–512.
7327:
7285:
7284:
7282:
7279:
7276:
7275:
7250:
7249:
7247:
7244:
7243:
7242:
7237:
7232:
7227:
7222:
7217:
7212:
7207:
7202:
7197:
7192:
7185:
7182:
7178:superellipsoid
7166:
7163:
7160:
7159:
7156:
7153:
7150:
7147:
7144:
7141:
7138:
7135:
7133:Vertex config.
7129:
7128:
7121:
7118:
7111:
7104:
7097:
7090:
7083:
7076:
7072:
7071:
7069:
7066:
7059:
7052:
7045:
7038:
7031:
7024:
7020:
7019:
7014:
7011:
7006:
7001:
6996:
6989:
6982:
6977:
6970:
6969:
6962:
6955:
6947:
6931:
6930:
6925:
6920:
6915:
6910:
6905:
6900:
6895:
6890:
6884:
6883:
6876:
6869:
6862:
6855:
6848:
6841:
6834:
6827:
6819:
6818:
6814:
6813:
6808:
6803:
6798:
6793:
6788:
6783:
6778:
6773:
6767:
6766:
6737:
6708:
6679:
6650:
6621:
6592:
6563:
6534:
6504:
6503:
6496:
6489:
6482:
6475:
6468:
6461:
6454:
6447:
6439:
6438:
6435:
6432:
6422:
6421:
6410:As a trigonal
6407:
6404:
6401:
6400:
6397:
6392:
6387:
6382:
6377:
6372:
6367:
6361:
6360:
6353:
6346:
6339:
6332:
6325:
6318:
6311:
6305:
6304:
6301:
6298:
6295:
6292:
6289:
6286:
6282:
6281:
6278:
6275:
6270:
6258:
6257:
6090:
6089:
6084:
6079:
6074:
6069:
6064:
6059:
6054:
6048:
6047:
6040:
6033:
6026:
6019:
6012:
6005:
5998:
5990:
5989:
5985:
5984:
5979:
5974:
5969:
5964:
5959:
5954:
5949:
5943:
5942:
5913:
5884:
5855:
5826:
5797:
5768:
5739:
5709:
5708:
5701:
5694:
5687:
5680:
5673:
5666:
5659:
5651:
5650:
5647:
5637:
5636:
5606:
5603:
5600:
5599:
5596:
5593:
5590:
5587:
5582:
5577:
5572:
5567:
5562:
5557:
5552:
5546:
5545:
5538:
5531:
5524:
5517:
5510:
5503:
5496:
5489:
5482:
5475:
5468:
5460:
5459:
5456:
5453:
5450:
5447:
5443:
5442:
5440:
5439:
5432:
5425:
5417:
5371:
5370:
5363:
5356:
5349:
5342:
5335:
5328:
5321:
5314:
5307:
5300:
5292:
5291:
5262:
5233:
5204:
5202:
5200:
5171:
5142:
5113:
5084:
5082:
5079:
5078:
5049:
5020:
4991:
4962:
4933:
4904:
4875:
4846:
4817:
4788:
4758:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4722:
4717:
4712:
4707:
4701:
4700:
4696:
4695:
4682:
4670:
4658:
4651:
4644:
4631:
4618:
4605:
4592:
4585:
4577:
4576:
4528:
4462:
4396:
4394:
4392:
4363:
4315:
4267:
4219:
4217:
4214:
4213:
4184:
4182:
4180:
4151:
4122:
4093:
4064:
4035:
4006:
3977:
3947:
3946:
3939:
3932:
3928:
3921:
3916:
3911:
3907:
3900:
3893:
3886:
3879:
3874:
3868:
3867:
3861:
3854:
3850:
3840:
3839:
3826:
3825:
3820:
3814:
3813:
3806:
3784:
3781:
3760:
3757:
3756:
3755:
3748:
3706:
3697:Especially in
3690:Rubik's Snakes
3681:
3678:
3677:
3676:
3660:
3649:
3637:The plates of
3635:
3603:
3600:
3598:
3595:
3594:
3593:
3584:
3578:
3571:
3544:
3543:
3536:
3530:
3525:: Two uniform
3520:
3510:
3504:
3497:
3468:
3465:
3462:
3461:
3458:
3455:
3450:
3447:
3444:
3438:
3437:
3433:
3430:
3426:
3423:
3419:
3413:
3410:
3406:
3403:
3399:
3396:
3390:
3389:
3387:
3385:
3380:
3377:
3374:
3372:Wythoff symbol
3368:
3367:
3362:
3357:
3352:
3349:
3346:
3340:
3339:
3310:
3281:
3224:
3177:
3148:
3142:
3141:
3132:
3123:
3114:
3105:
3096:
3090:
3089:
3088:Rhombic fusil
3086:
3080:
3074:
3064:
3061:
3047:
3038:
3028:
3016:
3013:symmetry group
3001:
2998:
2980:
2977:
2949:
2921:
2918:
2888:
2885:
2859:
2833:
2830:
2804:
2780:
2756:
2722:
2719:
2689:
2686:
2660:
2630:
2627:
2601:
2577:
2545:
2542:
2516:
2490:
2487:
2469:
2468:
2466:
2464:
2449:
2445:
2425:
2420:
2418:
2392:
2391:
2389:
2387:
2385:
2383:
2380:
2379:
2377:
2375:
2373:
2371:
2359:
2356:
2349:
2346:
2331:
2319:
2315:
2311:
2306:
2302:
2290:
2289:
2287:
2285:
2283:
2281:
2269:
2258:
2246:
2242:
2238:
2233:
2229:
2217:
2216:
2214:
2212:
2210:
2208:
2196:
2193:
2188:
2175:
2163:
2159:
2155:
2150:
2146:
2134:
2133:
2131:
2129:
2127:
2125:
2122:
2121:
2106:
2103:
2089:
2084:
2059:
2054:
2042:
2039:
2032:
2029:
2014:
2010:
2009:
1994:
1991:
1977:
1972:
1957:
1954:
1940:
1935:
1923:
1912:
1908:
1907:
1892:
1889:
1875:
1870:
1853:
1847:
1844:
1830:
1825:
1813:
1810:
1803:
1800:
1785:
1781:
1780:
1778:
1776:
1774:
1772:
1769:
1768:
1751:
1748:
1745:
1734:
1729:
1714:
1711:
1697:
1692:
1680:
1669:
1665:
1664:
1661:
1658:
1655:
1652:
1651:
1565:
1560:symmetry group
1520:
1517:
1510:
1378:
1375:
1334:
1331:
1328:
1324:
1298:
1295:
1292:
1289:
1286:
1282:
1264:Platonic graph
1227:
1224:
1211:
1208:
1205:
1201:
1197:
1194:
1191:
1187:
1183:
1179:
1175:
1172:
1169:
1165:
1161:
1157:
1153:
1150:
1147:
1143:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1084:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
994:
991:
988:
985:
982:
979:
976:
973:
970:
966:
963:
960:
957:
954:
951:
948:
945:
942:
916:
899:dihedral angle
884:
881:
878:
875:
872:
867:
864:
859:
854:
850:
845:
842:
839:
836:
833:
828:
824:
818:
813:
809:
804:
801:
798:
795:
792:
787:
783:
777:
772:
768:
745:
741:
715:
711:
685:
681:
653:
648:
644:
640:
637:
634:
630:
626:
620:
613:
610:
605:
602:
600:
598:
595:
594:
591:
586:
582:
578:
575:
572:
568:
564:
558:
553:
550:
547:
545:
543:
540:
539:
519:
499:
479:
457:
454:
449:or isohedral.
430:
426:
422:
411:dihedral group
382:
379:
294:
287:
286:
279:
272:
271:
270:
269:
268:
266:
263:
247:
241:
168:
167:
158:
157:
149:
148:
147:
146:
145:
143:
140:
136:cross-polytope
117:Platonic solid
83:
82:
54:
52:
45:
26:
9:
6:
4:
3:
2:
10490:
10479:
10476:
10474:
10471:
10469:
10466:
10464:
10461:
10459:
10456:
10455:
10453:
10442:
10438:
10434:
10429:
10426:
10422:
10419:
10417:
10410:
10403:
10397:
10395:
10391:
10388:
10386:
10382:
10378:
10374:
10371:
10369:
10365:
10362:
10360:
10356:
10352:
10351:
10348:
10346:
10344:
10341:
10339:
10335:
10332:
10330:
10327:
10325:
10322:
10321:
10318:
10316:
10314:
10311:
10309:
10305:
10302:
10300:
10297:
10295:
10292:
10291:
10288:
10286:
10279:
10272:
10266:
10264:
10261:
10259:
10255:
10252:
10250:
10247:
10245:
10242:
10241:
10238:
10236:
10229:
10222:
10216:
10214:
10211:
10209:
10205:
10202:
10200:
10197:
10195:
10192:
10191:
10188:
10186:
10179:
10173:
10171:
10168:
10166:
10162:
10159:
10157:
10154:
10152:
10149:
10148:
10145:
10143:
10141:
10138:
10136:
10132:
10129:
10127:
10124:
10122:
10119:
10118:
10115:
10111:
10108:
10106:
10103:
10101:
10100:Demitesseract
10098:
10096:
10092:
10089:
10087:
10084:
10082:
10079:
10078:
10075:
10071:
10068:
10066:
10064:
10061:
10059:
10055:
10052:
10050:
10047:
10045:
10042:
10041:
10038:
10035:
10033:
10030:
10028:
10025:
10023:
10020:
10018:
10015:
10013:
10010:
10009:
10006:
10000:
9997:
9993:
9986:
9982:
9975:
9971:
9966:
9962:
9957:
9953:
9948:
9946:
9944:
9940:
9930:
9926:
9924:
9922:
9918:
9914:
9912:
9910:
9906:
9902:
9900:
9897:
9896:
9891:
9887:
9879:
9874:
9872:
9867:
9865:
9860:
9859:
9856:
9846:
9840:
9834:
9831:
9829:
9826:
9824:
9821:
9819:
9816:
9814:
9811:
9809:
9806:
9804:
9801:
9799:
9796:
9794:
9791:
9790:
9788:
9786:
9781:
9775:
9772:
9770:
9767:
9765:
9762:
9760:
9757:
9755:
9752:
9750:
9747:
9745:
9742:
9740:
9737:
9735:
9732:
9730:
9727:
9725:
9722:
9720:
9717:
9715:
9712:
9710:
9707:
9705:
9702:
9700:
9697:
9695:
9692:
9690:
9687:
9685:
9682:
9681:
9679:
9677:
9672:
9666:
9663:
9661:
9658:
9656:
9653:
9651:
9648:
9646:
9643:
9641:
9638:
9636:
9633:
9632:
9630:
9628:
9623:
9617:
9614:
9612:
9609:
9607:
9604:
9602:
9599:
9597:
9594:
9592:
9589:
9587:
9584:
9582:
9579:
9577:
9574:
9573:
9571:
9569:
9564:
9558:
9555:
9553:
9550:
9548:
9545:
9543:
9540:
9538:
9535:
9533:
9530:
9528:
9525:
9523:
9520:
9518:
9515:
9513:
9510:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9488:
9485:
9483:
9480:
9478:
9475:
9473:
9470:
9468:
9465:
9463:
9460:
9458:
9455:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9413:
9410:
9408:
9405:
9404:
9402:
9400:
9396:
9391:
9385:
9382:
9380:
9377:
9375:
9372:
9370:
9367:
9365:
9362:
9360:
9357:
9355:
9352:
9350:
9347:
9345:
9342:
9340:
9337:
9335:
9332:
9331:
9329:
9327:
9322:
9316:
9313:
9311:
9308:
9306:
9305:square cupola
9303:
9301:
9298:
9296:
9293:
9291:
9288:
9287:
9285:
9283:
9279:
9275:
9271:
9267:
9260:
9255:
9253:
9248:
9246:
9241:
9240:
9237:
9226:
9220:
9214:
9211:
9209:
9206:
9204:
9201:
9199:
9196:
9194:
9191:
9189:
9186:
9184:
9181:
9179:
9176:
9174:
9171:
9169:
9166:
9164:
9161:
9160:
9158:
9154:
9144:
9141:
9139:
9136:
9135:
9133:
9129:
9123:
9120:
9118:
9115:
9114:
9111:
9108:
9104:
9098:
9097:
9093:
9091:
9090:
9086:
9085:
9083:
9079:
9073:
9070:
9068:
9065:
9063:
9060:
9058:
9055:
9053:
9050:
9048:
9045:
9043:
9040:
9038:
9035:
9033:
9030:
9028:
9025:
9023:
9020:
9018:
9015:
9013:
9010:
9009:
9007:
9000:
8995:
8989:
8986:
8984:
8981:
8979:
8976:
8974:
8971:
8969:
8966:
8964:
8961:
8959:
8956:
8954:
8951:
8949:
8946:
8944:
8941:
8939:
8936:
8934:
8933:cuboctahedron
8931:
8929:
8926:
8925:
8923:
8918:
8914:
8908:
8903:
8897:
8894:
8892:
8889:
8887:
8884:
8882:
8879:
8877:
8874:
8873:
8871:
8867:
8862:
8858:
8854:
8846:
8841:
8839:
8834:
8832:
8827:
8826:
8823:
8811:
8808:
8806:
8803:
8801:
8798:
8796:
8793:
8791:
8788:
8786:
8783:
8781:
8778:
8777:
8775:
8772:
8768:
8761:
8758:
8755:
8752:
8749:
8746:
8745:
8743:
8739:
8732:
8731:Johnson solid
8729:
8726:
8725:Catalan solid
8723:
8720:
8717:
8714:
8711:
8710:
8708:
8704:
8695:
8692:
8689:
8686:
8683:
8680:
8679:
8677:
8674:
8672:
8669:
8667:
8664:
8662:
8659:
8658:
8656:
8652:
8645:
8642:
8639:
8636:
8633:
8630:
8627:
8624:
8621:
8620:Hexoctahedron
8618:
8615:
8612:
8609:
8606:
8603:
8600:
8599:
8597:
8593:
8587:
8584:
8582:
8579:
8577:
8574:
8572:
8569:
8567:
8564:
8562:
8559:
8557:
8554:
8552:
8551:Tridecahedron
8549:
8547:
8544:
8542:
8541:Hendecahedron
8539:
8538:
8536:
8532:
8526:
8523:
8521:
8518:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8496:
8493:
8491:
8488:
8486:
8483:
8481:
8478:
8477:
8475:
8471:
8464:
8460:
8453:
8448:
8446:
8441:
8439:
8434:
8433:
8430:
8421:
8418:
8417:
8415:
8412:
8410:
8407:
8405:
8402:
8400:
8397:
8395:
8392:
8388:
8383:
8378:
8377:
8372:
8369:
8364:
8360:
8359:
8353:
8348:
8347:
8335:
8334:
8326:
8318:
8316:0-486-61480-8
8312:
8308:
8307:
8302:
8296:
8277:
8273:
8269:
8262:
8255:
8248:
8244:
8240:
8236:
8231:
8226:
8222:
8218:
8214:
8210:
8204:
8188:
8184:
8178:
8170:
8164:
8148:
8144:
8138:
8127:
8122:
8115:
8110:
8103:
8098:
8090:
8084:
8080:
8076:
8075:
8067:
8060:
8056:
8051:
8043:
8037:
8033:
8029:
8028:
8020:
8012:
8008:
8003:
7998:
7994:
7990:
7986:
7982:
7975:
7967:
7961:
7957:
7953:
7949:
7945:
7938:
7930:
7928:0-387-94365-X
7924:
7920:
7916:
7912:
7906:
7899:
7897:0-387-40409-0
7893:
7889:
7885:
7884:
7879:
7873:
7865:
7861:
7857:
7853:
7849:
7845:
7840:
7835:
7831:
7827:
7826:
7821:
7817:
7811:
7792:
7789:
7769:
7765:
7753:
7729:
7725:
7713:
7689:
7685:
7673:
7661:
7656:
7648:
7644:
7640:
7636:
7632:
7628:
7621:
7613:
7609:
7605:
7601:
7597:
7593:
7589:
7585:
7578:
7570:
7564:
7560:
7559:
7551:
7543:
7541:0-691-02509-6
7537:
7533:
7532:
7524:
7516:
7510:
7506:
7502:
7501:
7493:
7485:
7483:0-7679-0816-3
7479:
7475:
7471:
7470:
7465:
7459:
7451:
7445:
7441:
7440:
7432:
7430:
7421:
7415:
7411:
7410:
7402:
7400:
7391:
7385:
7381:
7377:
7376:
7368:
7366:
7357:
7353:
7349:
7345:
7338:
7331:
7323:
7319:
7315:
7311:
7307:
7303:
7299:
7293:
7291:
7286:
7272:
7268:
7264:
7258:
7256:
7251:
7241:
7238:
7236:
7233:
7231:
7228:
7226:
7223:
7221:
7218:
7216:
7213:
7211:
7208:
7206:
7203:
7201:
7198:
7196:
7193:
7191:
7188:
7187:
7181:
7179:
7174:
7172:
7157:
7154:
7151:
7148:
7145:
7142:
7139:
7136:
7134:
7131:
7130:
7126:
7122:
7119:
7116:
7112:
7109:
7105:
7102:
7098:
7095:
7091:
7088:
7084:
7081:
7077:
7074:
7073:
7070:
7067:
7064:
7060:
7057:
7053:
7050:
7046:
7043:
7039:
7036:
7032:
7029:
7025:
7022:
7021:
7018:
7015:
7012:
7010:
7007:
7005:
7002:
7000:
6997:
6995:
6990:
6988:
6983:
6981:
6978:
6975:
6974:
6968:
6963:
6961:
6956:
6954:
6949:
6948:
6946:
6942:
6939:
6929:
6926:
6924:
6921:
6919:
6916:
6914:
6911:
6909:
6906:
6904:
6901:
6899:
6896:
6894:
6891:
6889:
6886:
6885:
6881:
6877:
6874:
6870:
6867:
6863:
6860:
6856:
6853:
6849:
6846:
6842:
6839:
6835:
6832:
6828:
6825:
6821:
6820:
6815:
6812:
6809:
6807:
6804:
6802:
6799:
6797:
6794:
6792:
6789:
6787:
6784:
6782:
6779:
6777:
6774:
6772:
6769:
6768:
6738:
6709:
6680:
6651:
6622:
6593:
6564:
6535:
6506:
6505:
6501:
6497:
6494:
6490:
6487:
6483:
6480:
6476:
6473:
6469:
6466:
6462:
6459:
6455:
6452:
6448:
6445:
6441:
6440:
6436:
6433:
6430:
6427:
6423:
6418:
6415:
6413:
6398:
6396:
6393:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6362:
6358:
6354:
6351:
6347:
6344:
6340:
6337:
6333:
6330:
6326:
6323:
6319:
6316:
6312:
6307:
6306:
6302:
6299:
6296:
6293:
6290:
6287:
6284:
6283:
6276:
6274:
6269:
6264:
6259:
6255:
6251:
6249:
6243:
6240:
6238:
6234:
6230:
6227:symmetry of *
6226:
6222:
6218:
6214:
6209:
6205:
6184:
6163:
6157:
6153:
6101:
6097:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6065:
6063:
6060:
6058:
6055:
6053:
6050:
6049:
6045:
6041:
6038:
6034:
6031:
6027:
6024:
6020:
6017:
6013:
6010:
6006:
6003:
5999:
5996:
5992:
5991:
5986:
5983:
5980:
5978:
5975:
5973:
5970:
5968:
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5948:
5945:
5944:
5914:
5885:
5856:
5827:
5798:
5769:
5740:
5711:
5710:
5706:
5702:
5699:
5695:
5692:
5688:
5685:
5681:
5678:
5674:
5671:
5667:
5664:
5660:
5657:
5653:
5652:
5648:
5645:
5642:
5638:
5635:
5631:
5628:
5625:
5623:
5619:
5615:
5613:
5597:
5594:
5591:
5588:
5586:
5583:
5581:
5578:
5576:
5573:
5571:
5568:
5566:
5563:
5561:
5558:
5556:
5553:
5551:
5548:
5547:
5543:
5539:
5536:
5532:
5529:
5525:
5522:
5518:
5515:
5511:
5508:
5504:
5501:
5497:
5494:
5490:
5487:
5483:
5480:
5476:
5473:
5469:
5466:
5462:
5461:
5454:
5448:
5444:
5438:
5433:
5431:
5426:
5424:
5419:
5418:
5415:
5411:
5406:
5403:
5401:
5397:
5393:
5388:
5386:
5382:
5378:
5368:
5364:
5361:
5357:
5354:
5350:
5347:
5343:
5340:
5336:
5333:
5329:
5326:
5322:
5319:
5315:
5312:
5308:
5305:
5301:
5298:
5294:
5293:
5263:
5234:
5205:
5203:
5201:
5172:
5143:
5114:
5085:
5083:
5081:
5080:
5050:
5021:
4992:
4963:
4934:
4905:
4876:
4847:
4818:
4789:
4760:
4759:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4702:
4697:
4693:
4687:
4683:
4680:
4675:
4671:
4668:
4663:
4659:
4656:
4652:
4649:
4645:
4642:
4636:
4632:
4629:
4623:
4619:
4616:
4610:
4606:
4603:
4597:
4593:
4590:
4586:
4583:
4579:
4578:
4529:
4463:
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3431:
3424:
3411:
3404:
3397:
3395:
3392:
3388:
3386:
3384:| 2 3 2
3382:2 | 6 2
3381:
3379:2 | 4 3
3378:
3376:4 | 3 2
3375:
3373:
3370:
3363:
3358:
3353:
3350:
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3014:
3009:
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2886:
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2802:
2778:
2754:
2745:
2741:
2720:
2717:
2687:
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2658:
2649:
2628:
2625:
2599:
2575:
2564:
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2514:
2488:
2485:
2447:
2443:
2440:arc sec
2393:
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2388:
2386:
2384:
2382:
2381:
2357:
2354:
2347:
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2317:
2313:
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2267:
2244:
2240:
2236:
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2228:
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2194:
2191:
2186:
2161:
2157:
2153:
2148:
2145:
2135:
2132:
2130:
2128:
2126:
2124:
2123:
2104:
2101:
2040:
2037:
2030:
2027:
2011:
1992:
1989:
1955:
1952:
1921:
1909:
1890:
1887:
1851:
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1842:
1811:
1808:
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1779:
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1712:
1709:
1678:
1666:
1653:
1648:
1645:
1643:
1583:
1578:
1576:
1572:
1571:dual polytope
1568:
1561:
1557:
1552:
1550:
1546:
1542:
1538:
1534:
1530:
1526:
1516:
1514:
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1450:
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1442:
1437:
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1431:
1427:
1423:
1418:
1416:
1412:
1411:cuboctahedron
1408:
1404:
1400:
1396:
1392:
1383:
1374:
1372:
1368:
1364:
1360:
1357:
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1293:
1290:
1287:
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1280:
1272:
1267:
1265:
1261:
1257:
1253:
1249:
1246:according to
1245:
1241:
1232:
1223:
1209:
1206:
1203:
1199:
1195:
1192:
1189:
1185:
1181:
1177:
1173:
1170:
1167:
1163:
1159:
1155:
1151:
1148:
1145:
1141:
1117:
1114:
1111:
1108:
1105:
1082:
1059:
1056:
1053:
1050:
1047:
1036:
1020:
1014:
1011:
1008:
1005:
1002:
999:
992:
986:
983:
980:
977:
974:
971:
964:
958:
955:
952:
949:
946:
943:
932:
914:
903:
900:
895:
882:
879:
876:
873:
870:
865:
862:
857:
852:
848:
843:
840:
837:
834:
831:
826:
822:
816:
811:
807:
802:
799:
796:
793:
790:
785:
781:
775:
770:
766:
743:
739:
731:
713:
709:
701:
683:
679:
671:
651:
646:
642:
638:
635:
628:
624:
618:
611:
608:
603:
601:
596:
589:
584:
580:
576:
573:
566:
562:
556:
551:
548:
546:
541:
517:
497:
477:
463:
453:
450:
448:
445:bipyramid is
424:
420:
412:
408:
402:
400:
396:
387:
378:
376:
372:
368:
364:
359:
355:
354:
349:
348:
343:
338:
336:
332:
328:
327:
322:
318:
315:
311:
301:
297:
291:
283:
276:
262:
229:
225:
221:
217:
213:
209:
205:
200:
198:
194:
190:
186:
176:
172:
162:
153:
139:
137:
132:
130:
126:
122:
118:
114:
110:
106:
102:
94:
90:
79:
70:
66:
65:
60:
59:
53:
44:
43:
40:
36:
34:
19:
10420:
10389:
10380:
10372:
10363:
10354:
10334:10-orthoplex
10070:Dodecahedron
10053:
9991:
9980:
9969:
9960:
9951:
9942:
9938:
9928:
9920:
9916:
9908:
9904:
9803:sphenocorona
9224:
9143:trapezohedra
9094:
9087:
8891:dodecahedron
8885:
8644:Apeirohedron
8595:>20 faces
8546:Dodecahedron
8514:
8374:
8371:"Octahedron"
8356:
8332:
8325:
8305:
8295:
8285:30 September
8283:. Retrieved
8276:the original
8271:
8267:
8254:
8220:
8216:
8213:Crannell, A.
8209:Futamura, F.
8203:
8191:. Retrieved
8187:the original
8177:
8163:
8151:. Retrieved
8147:the original
8137:
8126:Coxeter 1973
8121:
8114:Coxeter 1973
8109:
8102:Coxeter 1973
8097:
8073:
8066:
8050:
8026:
8019:
7992:
7988:
7974:
7947:
7937:
7914:
7905:
7882:
7872:
7829:
7823:
7810:
7655:
7630:
7626:
7620:
7587:
7583:
7577:
7557:
7550:
7530:
7523:
7499:
7492:
7468:
7464:Livio, Mario
7458:
7438:
7408:
7374:
7347:
7343:
7330:
7308:(1): 55–57.
7305:
7301:
7271:golden ratio
7266:
7263:Coxeter 1973
7175:
7168:
6991:(Tetragonal)
6986:
6940:
6810:
6409:
6399:(3.∞)
6369:
6308:Quasiregular
6268:Construction
6253:
6247:
6228:
6220:
6212:
6210:
6203:
6182:
6161:
6155:
6151:
6099:
6093:
5956:
5626:
5617:
5610:
5608:
5559:
5413:
5409:
5395:
5389:
5377:hypersimplex
5374:
4704:
3895:
3829:
3786:
3762:
3560:
3485:
3482:
3036:
3010:
3004:There are 3
3003:
2647:
2562:
2472:
1581:
1579:
1553:
1529:orthoschemes
1522:
1506:
1497:
1476:The uniform
1475:
1457:
1453:quasiregular
1448:
1438:
1433:
1429:
1419:
1393:of two dual
1388:
1359:well-covered
1349:
1268:
1259:
1237:
904:
896:
469:
451:
403:
394:
392:
358:Solar System
351:
345:
339:
325:
323:who, in his
307:
300:Solar System
201:
197:line segment
184:
182:
133:
112:
104:
100:
92:
86:
73:
62:
56:
39:
32:
10343:10-demicube
10304:9-orthoplex
10254:8-orthoplex
10204:7-orthoplex
10161:6-orthoplex
10131:5-orthoplex
10086:Pentachoron
10074:Icosahedron
10049:Tetrahedron
8913:semiregular
8896:icosahedron
8876:tetrahedron
8586:Icosahedron
8534:11–20 faces
8520:Enneahedron
8510:Heptahedron
8500:Pentahedron
8495:Tetrahedron
7832:: 169–200.
6303:*∞32
6280:Hyperbolic
5614:tetrahedron
3818:tetrahedron
3789:tetrahedron
3765:space frame
3643:octahedrite
3613:octahedron.
3573:Tetragonal
3566:Triangular
3499:Heptagonal
3436:, , (*222)
3429:, , (*422)
3409:, , (*332)
3402:, , (*432)
3076:Triangular
3070:tetrahedron
3063:Octahedron
2746:with edges
1562:is denoted
1537:fundamental
1504:Manhattan (
1422:golden mean
1352:4-connected
1313:Turán graph
1075:and radius
105:octahedrons
76:August 2024
10478:Bipyramids
10458:Deltahedra
10452:Categories
10329:10-simplex
10313:9-demicube
10263:8-demicube
10213:7-demicube
10170:6-demicube
10140:5-demicube
10054:Octahedron
9843:(See also
9566:Augmented
9208:prismatoid
9138:bipyramids
9122:antiprisms
9096:hosohedron
8886:octahedron
8771:prismatoid
8756:(infinite)
8525:Decahedron
8515:Octahedron
8505:Hexahedron
8480:Monohedron
8473:1–10 faces
8422:– Try: dP4
8057:, p.
7864:0132.14603
7281:References
6984:(Trigonal)
6945:antiprisms
6936:Family of
6277:Euclidean
6087:V3.3.3.3.3
5446:Spherical
5385:hyperplane
3779:stresses.
3777:cantilever
3653:coordinate
3646:meteorites
3575:bipyramids
3568:antiprisms
3540:hosohedron
3538:Octagonal
3422:, , (322)
3361:{ } + {4}
1407:rectifying
1403:stellation
1395:tetrahedra
1356:simplicial
1133:such that
210:. It is a
204:deltahedra
109:polyhedron
93:octahedron
33:Octahedron
10377:orthoplex
10299:9-simplex
10249:8-simplex
10199:7-simplex
10156:6-simplex
10126:5-simplex
10095:Tesseract
9674:Modified
9625:Modified
9393:Modified
9324:Modified
9203:birotunda
9193:bifrustum
8958:snub cube
8853:polyhedra
8785:antiprism
8490:Trihedron
8459:Polyhedra
8376:MathWorld
8247:120958490
8230:1405.6481
8193:14 August
7856:122006114
7793:ℓ
7770:ℓ
7730:ℓ
7690:ℓ
7612:195047512
7466:(2003) .
7439:Polyhedra
6431:, (*622)
6412:antiprism
6273:Spherical
6235:within a
6096:tesseract
5646:, (*332)
5612:rectified
5381:hypercube
3657:distorted
3641:alloy in
3416:, , (2*3)
3084:bipyramid
3078:antiprism
3067:Rectified
3044:bipyramid
3033:antiprism
3027:include D
3025:subgroups
2423:35°15′52″
2404:𝜿
2355:≈
2192:≈
2102:π
2070:𝜿
2057:35°15′52″
2038:≈
1990:π
1953:π
1888:π
1856:𝜿
1852:−
1843:π
1809:≈
1754:𝟁
1747:−
1744:π
1710:π
1663:dihedral
1525:dissected
1441:antiprism
1193:−
1171:−
1149:−
1012:±
978:±
944:±
835:≈
794:≈
730:midsphere
636:≈
574:≈
399:bipyramid
314:congruent
101:octahedra
10431:Topics:
10394:demicube
10359:polytope
10353:Uniform
10114:600-cell
10110:120-cell
10063:Demicube
10037:Pentagon
10017:Triangle
9399:rotundae
9326:pyramids
9282:rotundae
9274:Pyramids
9183:bicupola
9163:pyramids
9089:dihedron
8485:Dihedron
8303:(1973).
7983:(2010).
7818:(1966).
7184:See also
7158:∞.3.3.3
7152:7.3.3.3
7149:6.3.3.3
7146:5.3.3.3
7143:4.3.3.3
7140:3.3.3.3
7137:2.3.3.3
6928:V3.3.3.3
6923:V3.3.3.6
6437:, (2*3)
6434:, (622)
6426:Symmetry
6310:figures
6300:*832...
6154:, where
6077:V3.4.3.4
6062:V3.3.3.3
5649:, (332)
5641:Symmetry
5455:Paraco.
5449:Euclid.
3844:Symmetry
3713:resistor
3674:crystals
3639:kamacite
3628:fluorite
3611:Fluorite
3553:with an
3394:Symmetry
3364:ftr{2,2}
3356:sr{2,3}
1828:54°44′8″
1556:symmetry
1513:) metric
1485:faceting
1391:compound
1240:skeleton
296:Kepler's
89:geometry
10368:simplex
10338:10-cube
10105:24-cell
10091:16-cell
10032:Hexagon
9886:regular
9395:cupolae
9278:cupolae
9225:italics
9213:scutoid
9198:rotunda
9188:frustum
8917:uniform
8866:regular
8851:Convex
8805:pyramid
8790:frustum
8011:2602814
7848:0185507
7647:0290245
7604:3619822
7322:2689647
6943:-gonal
6938:uniform
6918:V4.4.12
6806:sr{6,2}
6801:tr{6,2}
6796:rr{6,2}
6200:
6188:
6178:
6166:
6160:0 <
6148:
6136:
6132:
6120:
6116:
6104:
5982:sr{3,3}
5977:tr{3,3}
5972:rr{3,3}
5383:with a
3938:t{3,3}
3919:sr{4,3}
3914:tr{4,3}
3903:rr{4,3}
3745:
3733:
3729:
3717:
3620:diamond
3501:pyramid
3359:ft{2,4}
3351:r{3,3}
3140:(1111)
3131:(1111)
3122:(1112)
3113:(1212)
3104:(1111)
3082:Square
3046:; and T
1732:109°28′
1502:in the
344:in his
326:Timaeus
222:is the
125:crystal
107:) is a
69:discuss
35:(album)
10308:9-cube
10258:8-cube
10208:7-cube
10165:6-cube
10135:5-cube
10022:Square
9899:Family
9783:Other
9568:prisms
9178:cupola
9131:duals:
9117:prisms
8795:cupola
8671:vertex
8313:
8245:
8085:
8079:Tuttle
8038:
8009:
7962:
7925:
7894:
7862:
7854:
7846:
7742:, and
7645:
7610:
7602:
7565:
7538:
7511:
7480:
7446:
7416:
7386:
7320:
6913:V4.4.6
6903:V4.4.6
6811:s{2,6}
6786:t{2,6}
6781:r{6,2}
6776:t{6,2}
6365:Vertex
6206:< 1
6181:, and
6150:, and
6082:V4.6.6
6072:V3.3.3
6067:V3.6.6
6057:V3.6.6
6052:V3.3.3
5962:t{3,3}
5957:r{3,3}
5952:t{3,3}
4735:V4.6.8
4715:V(3.4)
3942:s{3,4}
3927:{3,3}
3924:h{4,3}
3910:{3,4}
3889:t{3,4}
3882:r{4,3}
3877:t{4,3}
3859:(*332)
3853:(432)
3848:(*432)
3752:hexany
3667:nickel
3486:convex
3354:s{2,6}
3348:{3,4}
1573:, the
1545:chiral
1541:facets
1500:3-ball
1369:, the
1252:planar
373:, and
218:. Its
173:, the
129:convex
10027:p-gon
8800:wedge
8780:prism
8640:(132)
8279:(PDF)
8264:(PDF)
8243:S2CID
8225:arXiv
8153:2 May
7852:S2CID
7608:S2CID
7600:JSTOR
7340:(PDF)
7318:JSTOR
7246:Notes
6791:{2,6}
6771:{6,2}
6395:(3.8)
6390:(3.7)
6385:(3.6)
6380:(3.5)
6375:(3.4)
6370:(3.3)
6297:*732
6294:*632
6291:*532
6288:*432
6285:*332
6252:: (3.
5967:{3,3}
5947:{3,3}
3945:s{3}
3935:{4,3}
3896:{3,4}
3892:t{3}
3885:r{3}
3872:{4,3}
3865:(3*2)
3517:kites
3442:Order
3093:Image
3060:Name
2358:0.816
2195:1.414
2041:0.577
1812:1.155
1657:edge
1535:is a
1480:is a
1467:cubes
1445:prism
1244:graph
1226:Graph
838:0.408
797:0.707
639:0.471
577:3.464
321:Plato
91:, an
58:split
10385:cube
10058:Cube
9888:and
9397:and
9280:and
8881:cube
8762:(57)
8733:(92)
8727:(13)
8721:(13)
8690:(16)
8666:edge
8661:face
8634:(90)
8628:(60)
8622:(48)
8616:(32)
8610:(30)
8604:(24)
8311:ISBN
8287:2006
8195:2016
8155:2006
8083:ISBN
8036:ISBN
7960:ISBN
7923:ISBN
7892:ISBN
7563:ISBN
7536:ISBN
7509:ISBN
7478:ISBN
7444:ISBN
7414:ISBN
7384:ISBN
7155:...
7068:...
7013:...
4750:V3.6
4740:V3.4
4730:V3.4
4720:V4.6
4710:V3.8
3899:{3}
3846:: ,
3747:ohm.
3671:iron
3624:alum
3015:is O
1660:arc
1580:The
1575:cube
1491:and
1489:edge
1443:, a
1413:and
1262:, a
1238:The
897:The
375:cube
335:wind
224:cube
175:cube
115:, a
9934:(p)
8915:or
8750:(4)
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