766:~1.618 to the smaller section, and the original edge is in the golden ratio to the larger section). The cut must be made in alternate directions on alternate edges incident to each vertex, in order to have a coherent result. The edges incident to a vertex in the 24-cell are the 8 radii of its cubical vertex figure. The only way to choose alternate radii of a cube is to choose the four radii of a tetrahedron (inscribed in the cube) to be cut at the smaller section of their length from the vertex, and the opposite four radii (of the other tetrahedron that can be inscribed in the cube) to be cut at the larger section of their length from the vertex. There are of course two ways to do this; both produce a cluster of five regular tetrahedra in place of the removed vertex, rather than a cube.
39:
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joined to 4 yellow tetrahedra. Thus, the tetrahedral cells occur in clusters of five (four yellow cells face-bonded around a red central one, each red/yellow pair lying in a different hyperplane). The red central tetrahedron of the five shares each of its six edges with a different icosahedral cell, and with the pair of yellow tetrahedral cells which shares that edge on the icosahedral cell.
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517:. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.
2234:
Same projection as above, now with the central tetrahedral cell in the gap filled in. This tetrahedral cell is joined to 4 other tetrahedral cells, two of which fills the two gaps visible in this image. The other two each lies between a green tetrahedral cell, a magenta cell, and the central cell, to
2237:
Note that in these images, cells facing away from the 4D viewpoint have been culled; hence there are only a total of 1 + 8 + 6 + 24 = 39 cells accounted for here. The other cells lie on the other side of the snub 24-cell. Part of the edge outline of one of them, an icosahedral cell, can be discerned
1006:
720:
or "snub octahedron" can be produced by partitioning the 12 edges of an octahedron in the golden ratio. This can be done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the
2197:
Perspective projection centered on an icosahedral cell, with 4D viewpoint placed at a distance of 5 times the vertex-center radius. The nearest icosahedral cell is rendered in solid color, and the other cells are in edge-outline. Cells facing away from the 4D viewpoint are culled, to reduce visual
1706:
The tetrahedral cells may be divided into two groups, of 96 yellow cells and 24 red cells respectively (as colored in the net illustration). Each yellow tetrahedral cell is joined via its triangular faces to 3 blue icosahedral cells and one red tetrahedral cell, while each red tetrahedral cell is
1702:
Each icosahedral cell is joined to 8 other icosahedral cells at 8 triangular faces in the positions corresponding to an inscribing octahedron. The remaining triangular faces are joined to tetrahedral cells, which occur in pairs that share an edge on the icosahedral cell.
1523:
2271:
Finally, the other 12 tetrahedral cells joined to the 6 yellow tetrahedral cells are shown here. These cells, together with the 8 icosahedral cells shown earlier, comprise all the cells that share at least 1 vertex with the central cell.
2259:
Now the 12 tetrahedral cells joined to the central icosahedral cell and the 6 yellow tetrahedral cells are shown. Each of these cells is surrounded by the central icosahedron and two of the other icosahedral cells shown earlier.
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Truncations remove vertices by cutting through the edges incident to the vertex; forms of truncation differ by where on the edge the cut is made. The common truncations of the 24-cell include the
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759:). In these truncations a cube is produced in place of the removed vertex, because the vertex figure of the 24-cell is a cube and the cuts are equidistant from the vertex.
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From this particular viewpoint, one of the gaps containing tetrahedral cells can be seen. Each of these gaps are filled by 5 tetrahedral cells, not shown here.
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773:") from the octahedron, by the same method. That is how the snub-24 cell's icosahedra are produced from the 24-cell's octahedra during truncation.
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diminishing, when the vertices of a second vertex inscribed 24-cell would be diminished as well. Accordingly, this one is known as the
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The icosahedral cells fit together face-to-face leaving voids between them filled by clusters of five tetrahedral cells.
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and 6 isosceles triangles. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.
2218:
The same projection as above, now with the other 4 icosahedral cells surrounding the central cell shown in magenta. The
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can also be constructed although it is not uniform, being composed of irregular tetrahedra on the alternated vertices.
2250:
In this image, only the nearest icosahedral cell and the 6 yellow tetrahedral cells from the previous image are shown.
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The snub truncation of the 24-cell cuts each edge into two golden sections (such that the larger section is in the
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Conversely, the 600-cell may be constructed from the snub 24-cell by augmenting it with 24 icosahedral pyramids.
611:
The vertices of a snub 24-cell centered at the origin of 4-space, with edges of length 2, are obtained by taking
332:
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direction of its vector. This is equivalent to the snub truncation construction of the 24-cell described below.
4105:
755:(which cuts each edge one-third of its length from the vertex, producing a polytope bounded by 24 cubes and 24
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The 96 vertices of the snub 24-cell, together with the 24 vertices of a 24-cell, form the 120 vertices of the
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3510:
1518:{\displaystyle {\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p}
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2207:
The same projection, now with 4 of the 8 icosahedral cells surrounding the central cell shown in green.
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in the golden ratio in a consistent manner dimensionally analogous to the way the 12 vertices of an
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4182:"Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system"
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769:
This construction has an analogy in 3 dimensions: the construction of the icosahedron (the "
612:
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747:(which cuts each edge at its midpoint, producing a polytope bounded by 24 cubes and 24
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2020:
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846:: by removing 24 vertices from the 600-cell corresponding to those of an inscribed
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of the snub 24-cell, with edges of length φ ≈ 0.618, are the even permutations of
17:
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3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 31
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4505:
4092:(1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions".
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group defines two groups of icosahedra in an 8:16 counts, and finally the D
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The snub 24-cell is the largest facet of the 4-dimensional honeycomb, the
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These 96 vertices can be found by partitioning each of the 96 edges of a
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800:
cells, and the 96 deleted vertex voids create 96 new tetrahedron cells.
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has 144 identical irregular cells. Each cell has faces of two kinds: 3
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520:
Topologically, under its highest symmetry, , as an alternation of a
4259:
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4155:
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736:
The snub 24-cell is derived from the 24-cell by a special form of
446:
26:
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793:
528:
symmetry), 24 regular tetrahedra, and 96 triangular pyramids.
4137:
Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012).
4050:, 5. Detailed analysis of cell structure of the snub 24-cell.
986:
as quaternion orbit weights of D4 under the Weyl group W(D4):
854:
of the remaining vertices. This is equivalent to removing 24
4180:
Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011).
3996:
2905:
is also called a semi-snub 24-cell because it is not a true
2307:
at 24 of its vertices, in fact those of a vertex inscribed
2161:
36:
29:
838:
The snub 24-cell may also be constructed by a particular
4139:"Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)"
4059:
4071:
4047:
4035:
3999:, p. 401, 26. Gosset's semi-snub polyoctahedron.
1693:
with blue icosahedra, and red and yellow tetrahedra.
1615:
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1159:
1098:
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1743:
group has 3 groups of icosahedra with 8:8:8 counts.
1329:{\displaystyle W(H_{4})=\lbrace \oplus ^{*}\rbrace }
2235:the left and right of the yellow tetrahedral cell.
1582:{\displaystyle -1/\varphi \leftrightarrow \varphi }
991:
O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
590:
Sadi (Jonathan Bowers) for snub disicositetrachoron
524:, it contains 24 pyritohedra (an icosahedron with T
2028:Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1954:Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1870:Two sets of tetrahedra: 96 (yellow) and 24 (cyan)
1735:defines 24 interchangeable icosahedra, while the B
1661:
1601:
1581:
1544:
1517:
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1328:
1223:{\displaystyle ^{*}:r\rightarrow r''=p{\bar {r}}q}
1222:
1145:
1084:
1064:
1035:
975:
950:
930:
4233:"4D uniform polytopes (polychora) s3s4o3o - sadi"
4118:; Burgiel, Heidi; Goodman-Strauss, Chaim (2008).
2026:Three sets of 8 icosahedra (red, green, and blue)
784:operation. Half the vertices are deleted, the 24
4038:, pp. 986–988, 6. Dual of the snub 24-cell.
2222:gives a good view on the layout of these cells.
2909:(alternation of an omnitruncated 24-cell). The
1952:Two sets icosahedra: 8, 16 each (red and blue)
1662:{\displaystyle S=\sum _{i=1}^{4}\oplus p^{i}T}
4280:
8:
1323:
1262:
4014:, pp. 151–153, §8.4. The snub {3,4,3}.
2172:Centered on hyperplane of one icosahedron.
4287:
4273:
4265:
3997:Conway, Burgiel & Goodman-Strauss 2008
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2927:symmetry family of uniform 4-polytopes.
2303:can be obtained as a diminishing of the
2238:here, overlying the yellow tetrahedron.
2182:
2043:
1745:
802:
4852:List of regular polytopes and compounds
4023:
4011:
3962:
1609:, we can construct the snub 24-cell as
4060:Koca, Ozdes Koca & Al-Barwani 2012
3984:
938:of order 120. The following describe
931:{\displaystyle O(\Lambda )=W(H_{4})=I}
1786:(Colored as faces in vertex figures)
7:
4254:Print #11: Snub icositetrachoron net
3969:
3799:
3785:
3771:
3750:
1146:{\displaystyle :r\rightarrow r'=prq}
540:made of two or more cells which are
4186:Linear Algebra and Its Applications
4072:Koca, Al-Ajmi & Ozdes Koca 2011
4048:Koca, Al-Ajmi & Ozdes Koca 2011
4036:Koca, Al-Ajmi & Ozdes Koca 2011
2923:The snub 24-cell is a part of the F
776:The snub 24-cell is related to the
894:
548:in his 1900 paper. He called it a
25:
869:Another construction method uses
4110:(3rd ed.). New York: Dover.
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4143:Int. J. Geom. Methods Mod. Phys
2311:. There is also a further such
1868:One set of 24 icosahedra (blue)
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2220:animated version of this image
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1292:
1286:
1280:
1265:
1256:
1243:
1211:
1188:
1173:
1160:
1120:
1111:
1099:
1056:
1030:
1018:
919:
906:
897:
891:
560:cells. (The other two are the
1:
1336:is the symmetry group of the
2147:
2128:
2099:
2088:
2069:
2050:
1987:
1903:
1819:
1545:{\displaystyle p^{\dagger }}
1715:The snub 24-cell has three
4885:
4841:
4268:
2932:
2324:
1065:{\displaystyle {\bar {p}}}
583:Semi-snub polyoctahedron (
4208:10.1016/j.laa.2010.10.005
4165:10.1142/S0219887812500685
2933:24-cell family polytopes
2317:bi-24-diminished 600-cell
2185:
1777:Tridiminished icosahedron
456:Tridiminished icosahedron
430:
420:
410:
331:
320:Tridiminished icosahedron
311:
301:
293:
285:
76:
60:
48:
4120:The Symmetries of Things
4094:Messenger of Mathematics
3928:
3921:
3914:
3907:
3900:
3893:
3886:
3879:
3872:
3865:
3859:
3850:
3843:
3836:
3829:
3822:
3815:
3808:
3792:
3778:
3764:
3757:
3743:
3736:
3729:
3720:
3713:
3706:
3699:
3692:
3685:
3678:
3671:
3664:
3657:
3651:
3642:
3635:
3628:
3621:
3614:
3607:
3600:
3593:
3586:
3579:
3573:
3564:
3557:
3550:
3543:
3536:
3529:
3522:
3515:
3508:
3501:
3494:
2186:Perspective projections
2046:orthographic projections
2040:Orthographic projections
804:Orthogonal projection, F
506:composed of 120 regular
500:snub disicositetrachoron
4026:, pp. 50–52, §3.7.
2179:Perspective projections
654:unit-radius coordinates
538:semiregular 4-polytopes
4260:Snub icositetrachoron
3946:Snub 24-cell honeycomb
2985:runcitruncated 24-cell
2973:cantitruncated 24-cell
2918:snub 24-cell honeycomb
1984:Omnisnub demitesseract
1900:Snub rectified 16-cell
1762:Coxeter-Dynkin diagram
1663:
1642:
1603:
1583:
1546:
1519:
1367:
1366:{\displaystyle p\in T}
1330:
1224:
1147:
1086:
1066:
1037:
977:
952:
932:
850:, and then taking the
484:
2992:omnitruncated 24-cell
2169:Orthogonal projection
1719:colorings based on a
1664:
1622:
1604:
1584:
1547:
1520:
1368:
1331:
1225:
1148:
1087:
1067:
1038:
1036:{\displaystyle (p,q)}
978:
953:
933:
577:Snub icositetrachoron
450:
1721:Wythoff construction
1613:
1593:
1556:
1529:
1377:
1351:
1237:
1157:
1096:
1076:
1072:is the conjugate of
1047:
1015:
962:
942:
885:
875:Icosahedral symmetry
856:icosahedral pyramids
786:truncated octahedron
532:Semiregular polytope
4869:Uniform 4-polytopes
4836:pentagonal polytope
4735:Uniform 10-polytope
4295:Fundamental convex
4256:, George Olshevsky.
4250:, George Olshevsky.
4231:Klitzing, Richard.
2967:bitruncated 24-cell
2962:cantellated 24-cell
2048:
858:from the 600-cell.
809:
757:truncated octahedra
536:It is one of three
4705:Uniform 9-polytope
4655:Uniform 8-polytope
4605:Uniform 7-polytope
4562:Uniform 6-polytope
4532:Uniform 5-polytope
4492:Uniform polychoron
4455:Uniform polyhedron
4303:in dimensions 2–10
2979:runcinated 24-cell
2329:uniform polychora
2149:Dihedral symmetry
2044:
1758:Constructive name
1659:
1599:
1579:
1552:as an exchange of
1542:
1515:
1363:
1326:
1220:
1143:
1082:
1062:
1033:
976:{\displaystyle T'}
973:
948:
928:
803:
580:Snub demitesseract
566:rectified 600-cell
552:for being made of
504:uniform 4-polytope
485:
50:Uniform 4-polytope
18:Snub demitesseract
4857:
4856:
4844:Polytope families
4301:uniform polytopes
4262:- Data and images
4129:978-1-56881-220-5
4107:Regular Polytopes
3951:Dual snub 24-cell
3937:
3936:
3089:
3080:
3071:
3060:
3049:
3038:
3027:
3013:
2994:
2987:
2975:
2957:rectified 24-cell
2947:truncated 24-cell
2911:full snub 24-cell
2899:
2898:
2295:Related polytopes
2285:Dual snub 24-cell
2276:
2275:
2176:
2175:
2157:
2156:
2091:Dihedral symmetry
2032:
2031:
1727:from which it is
1717:vertex-transitive
1697:
1696:
1602:{\displaystyle p}
1497:
1459:
1421:
1389:
1310:
1283:
1214:
1085:{\displaystyle p}
1059:
1011:With quaternions
1000:O(0001) : V3
997:O(0010) : V2
994:O(1000) : V1
951:{\displaystyle T}
836:
835:
778:truncated 24-cell
753:truncated 24-cell
745:rectified 24-cell
613:even permutations
593:Tetricosahedric (
572:Alternative names
522:truncated 24-cell
445:
444:
412:Dual snub 24-cell
16:(Redirected from
4876:
4848:Regular polytope
4409:
4398:
4387:
4346:
4289:
4282:
4275:
4266:
4236:
4227:
4201:
4176:
4158:
4133:
4111:
4097:
4075:
4069:
4063:
4057:
4051:
4045:
4039:
4033:
4027:
4021:
4015:
4009:
4000:
3994:
3988:
3982:
3973:
3967:
3933:
3926:
3919:
3912:
3905:
3898:
3891:
3884:
3877:
3870:
3855:
3848:
3841:
3834:
3827:
3820:
3804:
3797:
3790:
3783:
3776:
3769:
3762:
3755:
3748:
3741:
3725:
3718:
3711:
3704:
3697:
3690:
3683:
3676:
3669:
3662:
3647:
3640:
3633:
3626:
3619:
3612:
3605:
3598:
3591:
3584:
3569:
3562:
3555:
3548:
3541:
3534:
3527:
3520:
3513:
3506:
3490:
3489:
3488:
3484:
3483:
3479:
3478:
3474:
3473:
3469:
3468:
3464:
3463:
3459:
3458:
3451:
3450:
3449:
3445:
3444:
3440:
3439:
3435:
3434:
3430:
3429:
3425:
3424:
3420:
3419:
3412:
3411:
3410:
3406:
3405:
3401:
3400:
3396:
3395:
3391:
3390:
3386:
3385:
3381:
3380:
3373:
3372:
3371:
3367:
3366:
3362:
3361:
3357:
3356:
3352:
3351:
3347:
3346:
3342:
3341:
3334:
3333:
3332:
3328:
3327:
3323:
3322:
3318:
3317:
3313:
3312:
3308:
3307:
3303:
3302:
3295:
3294:
3293:
3289:
3288:
3284:
3283:
3279:
3278:
3274:
3273:
3269:
3268:
3264:
3263:
3256:
3255:
3254:
3250:
3249:
3245:
3244:
3240:
3239:
3235:
3234:
3230:
3229:
3225:
3224:
3217:
3216:
3215:
3211:
3210:
3206:
3205:
3201:
3200:
3196:
3195:
3191:
3190:
3186:
3185:
3178:
3177:
3176:
3172:
3171:
3167:
3166:
3162:
3161:
3157:
3156:
3152:
3151:
3147:
3146:
3139:
3138:
3137:
3133:
3132:
3128:
3127:
3123:
3122:
3118:
3117:
3113:
3112:
3108:
3107:
3087:
3078:
3069:
3058:
3047:
3036:
3025:
3011:
2990:
2983:
2971:
2930:
2825:
2818:
2811:
2804:
2797:
2790:
2783:
2776:
2767:
2766:
2765:
2761:
2760:
2756:
2755:
2751:
2750:
2744:
2743:
2742:
2738:
2737:
2733:
2732:
2728:
2727:
2723:
2722:
2715:
2714:
2713:
2709:
2708:
2704:
2703:
2699:
2698:
2692:
2691:
2690:
2686:
2685:
2681:
2680:
2676:
2675:
2671:
2670:
2663:
2662:
2661:
2657:
2656:
2652:
2651:
2647:
2646:
2640:
2639:
2638:
2634:
2633:
2629:
2628:
2624:
2623:
2619:
2618:
2611:
2610:
2609:
2605:
2604:
2600:
2599:
2595:
2594:
2588:
2587:
2586:
2582:
2581:
2577:
2576:
2572:
2571:
2567:
2566:
2559:
2558:
2557:
2553:
2552:
2548:
2547:
2543:
2542:
2538:
2537:
2531:
2530:
2529:
2525:
2524:
2520:
2519:
2515:
2514:
2510:
2509:
2502:
2501:
2500:
2496:
2495:
2491:
2490:
2486:
2485:
2481:
2480:
2474:
2473:
2472:
2468:
2467:
2463:
2462:
2458:
2457:
2453:
2452:
2445:
2444:
2443:
2439:
2438:
2434:
2433:
2429:
2428:
2424:
2423:
2417:
2416:
2415:
2411:
2410:
2406:
2405:
2401:
2400:
2396:
2395:
2388:
2387:
2386:
2382:
2381:
2377:
2376:
2372:
2371:
2367:
2366:
2360:
2359:
2358:
2354:
2353:
2349:
2348:
2344:
2343:
2339:
2338:
2322:
2269:
2257:
2248:
2232:
2216:
2205:
2195:
2183:
2166:
2159:
2144:
2137:
2085:
2078:
2049:
2023:
2014:
2013:
2012:
2008:
2007:
2003:
2002:
1998:
1997:
1993:
1992:
1973:
1971:
1970:
1967:
1964:
1949:
1940:
1939:
1938:
1934:
1933:
1929:
1928:
1924:
1923:
1919:
1918:
1914:
1913:
1909:
1908:
1889:
1887:
1886:
1883:
1880:
1865:
1856:
1855:
1854:
1850:
1849:
1845:
1844:
1840:
1839:
1835:
1834:
1830:
1829:
1825:
1824:
1805:
1803:
1802:
1799:
1796:
1746:
1683:
1676:
1668:
1666:
1665:
1660:
1655:
1654:
1641:
1636:
1608:
1606:
1605:
1600:
1588:
1586:
1585:
1580:
1569:
1551:
1549:
1548:
1543:
1541:
1540:
1524:
1522:
1521:
1516:
1505:
1504:
1499:
1498:
1490:
1483:
1482:
1467:
1466:
1461:
1460:
1452:
1445:
1444:
1429:
1428:
1423:
1422:
1414:
1407:
1406:
1391:
1390:
1382:
1372:
1370:
1369:
1364:
1344:of order 14400.
1335:
1333:
1332:
1327:
1322:
1321:
1312:
1311:
1303:
1285:
1284:
1276:
1255:
1254:
1229:
1227:
1226:
1221:
1216:
1215:
1207:
1198:
1181:
1180:
1152:
1150:
1149:
1144:
1130:
1091:
1089:
1088:
1083:
1071:
1069:
1068:
1063:
1061:
1060:
1052:
1042:
1040:
1039:
1034:
1008:
982:
980:
979:
974:
972:
957:
955:
954:
949:
937:
935:
934:
929:
918:
917:
832:
825:
810:
788:cells become 24
707:
705:
704:
701:
698:
691:
689:
688:
685:
682:
675:
673:
672:
669:
666:
644:
642:
641:
638:
635:
634:
633:
562:rectified 5-cell
544:, discovered by
470:
465:
397:
395:
394:
391:
388:
373:
371:
370:
367:
364:
349:
347:
346:
343:
340:
316:
278:
266:
254:
231:
230:
229:
225:
224:
220:
219:
215:
214:
208:
207:
206:
202:
201:
197:
196:
192:
191:
187:
186:
180:
179:
178:
174:
173:
169:
168:
164:
163:
159:
158:
152:
151:
150:
146:
145:
141:
140:
136:
135:
131:
130:
126:
125:
121:
120:
114:
113:
112:
108:
107:
103:
102:
98:
97:
93:
92:
88:
87:
83:
82:
41:
27:
21:
4884:
4883:
4879:
4878:
4877:
4875:
4874:
4873:
4859:
4858:
4827:
4820:
4813:
4696:
4689:
4682:
4646:
4639:
4632:
4596:
4589:
4423:Regular polygon
4416:
4407:
4400:
4396:
4389:
4385:
4376:
4367:
4360:
4356:
4344:
4338:
4334:
4322:
4304:
4293:
4244:
4239:
4230:
4179:
4136:
4130:
4114:
4102:Coxeter, H.S.M.
4100:
4090:Gosset, Thorold
4088:
4084:
4079:
4078:
4070:
4066:
4058:
4054:
4046:
4042:
4034:
4030:
4022:
4018:
4010:
4003:
3995:
3991:
3983:
3976:
3968:
3964:
3959:
3942:
3863:
3812:
3733:
3655:
3577:
3497:
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3454:
3447:
3442:
3437:
3432:
3427:
3422:
3417:
3415:
3408:
3403:
3398:
3393:
3388:
3383:
3378:
3376:
3369:
3364:
3359:
3354:
3349:
3344:
3339:
3337:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3298:
3291:
3286:
3281:
3276:
3271:
3266:
3261:
3259:
3252:
3247:
3242:
3237:
3232:
3227:
3222:
3220:
3213:
3208:
3203:
3198:
3193:
3188:
3183:
3181:
3174:
3169:
3164:
3159:
3154:
3149:
3144:
3142:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3103:
3098:
3090:
3081:
3072:
3063:
3061:
3052:
3050:
3041:
3039:
3030:
3028:
3016:
3014:
3001:
2926:
2894:
2887:
2880:
2873:
2866:
2862:
2855:
2851:
2844:
2840:
2833:
2763:
2758:
2753:
2748:
2746:
2745:
2740:
2735:
2730:
2725:
2720:
2718:
2711:
2706:
2701:
2696:
2694:
2693:
2688:
2683:
2678:
2673:
2668:
2666:
2659:
2654:
2649:
2644:
2642:
2641:
2636:
2631:
2626:
2621:
2616:
2614:
2607:
2602:
2597:
2592:
2590:
2589:
2584:
2579:
2574:
2569:
2564:
2562:
2555:
2550:
2545:
2540:
2535:
2533:
2532:
2527:
2522:
2517:
2512:
2507:
2505:
2498:
2493:
2488:
2483:
2478:
2476:
2475:
2470:
2465:
2460:
2455:
2450:
2448:
2441:
2436:
2431:
2426:
2421:
2419:
2418:
2413:
2408:
2403:
2398:
2393:
2391:
2384:
2379:
2374:
2369:
2364:
2362:
2361:
2356:
2351:
2346:
2341:
2336:
2334:
2328:
2297:
2281:
2270:
2258:
2249:
2233:
2217:
2206:
2196:
2181:
2171:
2167:
2125:
2121:
2115:
2111:
2107:
2066:
2060:
2042:
2037:
2027:
2015:
2010:
2005:
2000:
1995:
1990:
1988:
1979:
1978:
1977:
1968:
1965:
1962:
1961:
1959:
1953:
1941:
1936:
1931:
1926:
1921:
1916:
1911:
1906:
1904:
1895:
1894:
1893:
1884:
1881:
1878:
1877:
1875:
1869:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1820:
1811:
1810:
1809:
1800:
1797:
1794:
1793:
1791:
1785:
1774:
1767:Schläfli symbol
1764:
1752:
1742:
1738:
1734:
1713:
1684:
1674:
1646:
1611:
1610:
1591:
1590:
1554:
1553:
1532:
1527:
1526:
1487:
1474:
1449:
1436:
1411:
1398:
1375:
1374:
1349:
1348:
1313:
1246:
1235:
1234:
1191:
1172:
1155:
1154:
1123:
1094:
1093:
1074:
1073:
1045:
1044:
1013:
1012:
987:
965:
960:
959:
940:
939:
909:
883:
882:
867:
807:
771:snub octahedron
734:
702:
699:
696:
695:
693:
686:
683:
680:
679:
677:
670:
667:
664:
663:
661:
645:≈ 1.618 is the
639:
636:
631:
629:
627:
626:
624:
619:(0, ±1, ±φ, ±φ)
609:
604:
574:
550:tetricosahedric
542:Platonic solids
534:
527:
471:
460:
458:
401:
392:
389:
386:
385:
383:
379:
377:
368:
365:
362:
361:
359:
353:
344:
341:
338:
337:
335:
328:Symmetry groups
317:
267:
255:
249:(oblique)
227:
222:
217:
212:
210:
204:
199:
194:
189:
184:
182:
181:
176:
171:
166:
161:
156:
154:
148:
143:
138:
133:
128:
123:
118:
116:
115:
110:
105:
100:
95:
90:
85:
80:
78:
72:
64:
62:
57:Schläfli symbol
23:
22:
15:
12:
11:
5:
4882:
4880:
4872:
4871:
4861:
4860:
4855:
4854:
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4807:
4798:
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4772:
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4508:
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4405:
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4330:
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4318:
4312:
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4294:
4292:
4291:
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4277:
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4264:
4263:
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4251:
4243:
4242:External links
4240:
4238:
4237:
4228:
4192:(4): 977–989.
4177:
4134:
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4112:
4098:
4085:
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2959:
2954:
2949:
2944:
2939:
2935:
2934:
2924:
2897:
2896:
2895:s{3}=s{3,4,3}
2889:
2888:t{3}=t{3,4,3}
2882:
2881:r{3}=r{3,4,3}
2875:
2868:
2864:
2857:
2853:
2846:
2842:
2835:
2827:
2826:
2819:
2812:
2805:
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2326:
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2280:
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2251:
2241:
2240:
2226:
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2208:
2199:
2188:
2187:
2180:
2177:
2174:
2173:
2155:
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2150:
2146:
2145:
2138:
2131:
2127:
2126:
2123:
2119:
2116:
2113:
2109:
2105:
2102:
2101:Coxeter plane
2098:
2097:
2095:
2093:
2087:
2086:
2079:
2072:
2068:
2067:
2064:
2061:
2058:
2055:
2041:
2038:
2036:
2033:
2030:
2029:
2024:
2017:
1986:
1981:
1975:
1956:
1955:
1950:
1943:
1902:
1897:
1891:
1872:
1871:
1866:
1859:
1818:
1813:
1807:
1788:
1787:
1780:
1769:
1759:
1756:
1740:
1736:
1732:
1712:
1709:
1695:
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1673:
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1598:
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1564:
1561:
1539:
1535:
1514:
1511:
1508:
1503:
1496:
1493:
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1481:
1477:
1473:
1470:
1465:
1458:
1455:
1448:
1443:
1439:
1435:
1432:
1427:
1420:
1417:
1410:
1405:
1401:
1397:
1394:
1388:
1385:
1362:
1359:
1356:
1325:
1320:
1316:
1309:
1306:
1300:
1297:
1294:
1291:
1288:
1282:
1279:
1273:
1270:
1267:
1264:
1261:
1258:
1253:
1249:
1245:
1242:
1219:
1213:
1210:
1204:
1201:
1197:
1194:
1190:
1187:
1184:
1179:
1175:
1171:
1168:
1165:
1162:
1142:
1139:
1136:
1133:
1129:
1126:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1081:
1058:
1055:
1032:
1029:
1026:
1023:
1020:
1002:
1001:
998:
995:
992:
971:
968:
947:
927:
924:
921:
916:
912:
908:
905:
902:
899:
896:
893:
890:
866:
863:
834:
833:
826:
818:
817:
814:
808:Coxeter plane
805:
792:cells, the 24
733:
730:
710:
709:
621:
620:
608:
605:
603:
600:
599:
598:
595:Thorold Gosset
591:
588:
581:
578:
573:
570:
546:Thorold Gosset
533:
530:
525:
443:
442:
429:
428:Uniform index
425:
424:
419:
415:
414:
409:
405:
404:
399:
375:
351:
330:
324:
323:
310:
304:
303:
300:
296:
295:
292:
288:
287:
284:
280:
279:
241:
238:
234:
233:
75:
71:Coxeter-Dynkin
67:
66:
59:
53:
52:
47:
43:
42:
34:
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4881:
4870:
4867:
4866:
4864:
4853:
4849:
4845:
4840:
4837:
4833:
4830:
4828:
4821:
4814:
4808:
4806:
4802:
4799:
4797:
4793:
4789:
4785:
4782:
4780:
4776:
4773:
4771:
4767:
4763:
4762:
4759:
4757:
4755:
4752:
4750:
4746:
4743:
4741:
4738:
4736:
4733:
4732:
4729:
4727:
4725:
4722:
4720:
4716:
4713:
4711:
4708:
4706:
4703:
4702:
4699:
4697:
4690:
4683:
4677:
4675:
4672:
4670:
4666:
4663:
4661:
4658:
4656:
4653:
4652:
4649:
4647:
4640:
4633:
4627:
4625:
4622:
4620:
4616:
4613:
4611:
4608:
4606:
4603:
4602:
4599:
4597:
4590:
4584:
4582:
4579:
4577:
4573:
4570:
4568:
4565:
4563:
4560:
4559:
4556:
4554:
4552:
4549:
4547:
4543:
4540:
4538:
4535:
4533:
4530:
4529:
4526:
4522:
4519:
4517:
4514:
4512:
4511:Demitesseract
4509:
4507:
4503:
4500:
4498:
4495:
4493:
4490:
4489:
4486:
4482:
4479:
4477:
4475:
4472:
4470:
4466:
4463:
4461:
4458:
4456:
4453:
4452:
4449:
4446:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4424:
4421:
4420:
4417:
4411:
4408:
4404:
4397:
4393:
4386:
4382:
4377:
4373:
4368:
4364:
4359:
4357:
4355:
4351:
4341:
4337:
4335:
4333:
4329:
4325:
4323:
4321:
4317:
4313:
4311:
4308:
4307:
4302:
4298:
4290:
4285:
4283:
4278:
4276:
4271:
4270:
4267:
4261:
4258:
4255:
4252:
4249:
4246:
4245:
4241:
4234:
4229:
4225:
4221:
4217:
4213:
4209:
4205:
4200:
4195:
4191:
4187:
4183:
4178:
4174:
4170:
4166:
4162:
4157:
4152:
4148:
4144:
4140:
4135:
4131:
4125:
4121:
4117:
4113:
4109:
4108:
4103:
4099:
4095:
4091:
4087:
4086:
4081:
4073:
4068:
4065:
4061:
4056:
4053:
4049:
4044:
4041:
4037:
4032:
4029:
4025:
4020:
4017:
4013:
4008:
4006:
4002:
3998:
3993:
3990:
3986:
3981:
3979:
3975:
3971:
3966:
3963:
3956:
3952:
3949:
3947:
3944:
3943:
3939:
3932:
3925:
3918:
3911:
3904:
3897:
3890:
3883:
3876:
3869:
3858:
3854:
3847:
3840:
3833:
3826:
3819:
3807:
3803:
3796:
3789:
3782:
3775:
3768:
3761:
3754:
3747:
3740:
3728:
3724:
3717:
3710:
3703:
3696:
3689:
3682:
3675:
3668:
3661:
3650:
3646:
3639:
3632:
3625:
3618:
3611:
3604:
3597:
3590:
3583:
3572:
3568:
3561:
3554:
3547:
3540:
3533:
3526:
3519:
3512:
3505:
3499:
3493:
3453:
3414:
3375:
3336:
3297:
3258:
3219:
3180:
3141:
3102:
3100:
3095:
3094:
3084:
3075:
3066:
3055:
3044:
3033:
3022:
3019:
3008:
3005:
3003:
2998:
2997:
2993:
2989:
2986:
2982:
2980:
2977:
2974:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2937:
2936:
2931:
2928:
2921:
2919:
2914:
2912:
2908:
2904:
2893:
2890:
2886:
2883:
2879:
2876:
2872:
2869:
2861:
2858:
2850:
2847:
2839:
2836:
2832:
2829:
2828:
2824:
2820:
2817:
2813:
2810:
2806:
2803:
2799:
2796:
2792:
2789:
2785:
2782:
2778:
2775:
2771:
2770:
2717:
2665:
2613:
2561:
2504:
2447:
2390:
2333:
2332:
2323:
2320:
2318:
2314:
2310:
2306:
2302:
2294:
2292:
2290:
2286:
2278:
2268:
2264:
2256:
2252:
2247:
2243:
2239:
2231:
2227:
2225:
2221:
2215:
2211:
2204:
2200:
2194:
2190:
2184:
2178:
2170:
2165:
2160:
2153:
2151:
2148:
2143:
2139:
2136:
2132:
2129:
2117:
2103:
2100:
2096:
2094:
2092:
2089:
2084:
2080:
2077:
2073:
2070:
2062:
2056:
2054:
2053:Coxeter plane
2051:
2047:
2039:
2034:
2025:
2022:
2018:
1985:
1982:
1958:
1957:
1951:
1948:
1944:
1901:
1898:
1874:
1873:
1867:
1864:
1860:
1817:
1814:
1790:
1789:
1784:
1781:
1778:
1773:
1772:Vertex figure
1770:
1768:
1763:
1760:
1757:
1755:
1751:
1748:
1747:
1744:
1730:
1726:
1725:Coxeter group
1722:
1718:
1710:
1708:
1704:
1700:
1692:
1688:
1682:
1678:
1677:
1671:
1669:
1656:
1651:
1647:
1643:
1638:
1633:
1630:
1627:
1623:
1619:
1616:
1596:
1576:
1570:
1566:
1562:
1559:
1537:
1533:
1512:
1509:
1506:
1501:
1491:
1484:
1479:
1475:
1471:
1468:
1463:
1453:
1446:
1441:
1437:
1433:
1430:
1425:
1415:
1408:
1403:
1399:
1395:
1392:
1383:
1360:
1357:
1354:
1345:
1343:
1339:
1318:
1304:
1298:
1295:
1289:
1277:
1271:
1268:
1259:
1251:
1247:
1240:
1233:
1232:Coxeter group
1217:
1208:
1202:
1199:
1195:
1192:
1185:
1182:
1177:
1169:
1166:
1163:
1140:
1137:
1134:
1131:
1127:
1124:
1117:
1114:
1108:
1105:
1102:
1079:
1053:
1027:
1024:
1021:
1009:
1007:
999:
996:
993:
990:
989:
988:
985:
969:
966:
945:
925:
922:
914:
910:
903:
900:
888:
880:
876:
872:
864:
862:
859:
857:
853:
849:
845:
841:
831:
827:
824:
820:
819:
815:
813:Snub 24-cell
812:
811:
801:
799:
795:
791:
787:
783:
779:
774:
772:
767:
765:
760:
758:
754:
750:
746:
741:
739:
732:Constructions
731:
729:
727:
722:
719:
715:
659:
658:
657:
655:
650:
648:
618:
617:
616:
614:
606:
601:
596:
592:
589:
586:
582:
579:
576:
575:
571:
569:
567:
563:
559:
555:
551:
547:
543:
539:
531:
529:
523:
518:
516:
513:
509:
505:
501:
497:
496:
490:
483:
482:
477:
476:
469:
464:
457:
453:
452:Vertex figure
449:
441:
440:
435:
434:
426:
423:
416:
413:
406:
403:
381:
357:
333:
329:
325:
321:
315:
309:
308:Vertex figure
305:
297:
289:
281:
277:
273:
272:
265:
261:
260:
253:
248:
247:
242:
239:
235:
232:
74:
68:
58:
54:
51:
44:
40:
35:
32:
28:
19:
4831:
4800:
4791:
4783:
4774:
4765:
4745:10-orthoplex
4481:Dodecahedron
4402:
4391:
4380:
4371:
4362:
4353:
4349:
4339:
4331:
4327:
4319:
4315:
4189:
4185:
4146:
4142:
4119:
4116:Conway, John
4106:
4096:. Macmillan.
4093:
4067:
4055:
4043:
4031:
4024:Coxeter 1973
4019:
4012:Coxeter 1973
3992:
3965:
2952:snub 24-cell
2951:
2922:
2915:
2903:snub 24-cell
2902:
2900:
2891:
2874:{3}={3,4,3}
2312:
2301:snub 24-cell
2300:
2298:
2282:
2236:
2223:
1816:Snub 24-cell
1714:
1705:
1701:
1698:
1691:snub 24-cell
1690:
1346:
1010:
1003:
868:
860:
839:
837:
775:
768:
764:golden ratio
761:
749:cuboctahedra
742:
735:
723:
711:
653:
651:
647:golden ratio
622:
610:
549:
535:
519:
502:is a convex
499:
492:
486:
480:
474:
455:
437:
431:
355:
354:, order 576
270:
258:
245:
77:
31:Snub 24-cell
30:
4754:10-demicube
4715:9-orthoplex
4665:8-orthoplex
4615:7-orthoplex
4572:6-orthoplex
4542:5-orthoplex
4497:Pentachoron
4485:Icosahedron
4460:Tetrahedron
3985:Gosset 1900
2035:Projections
1230:, then the
871:quaternions
865:Weyl orbits
852:convex hull
840:diminishing
798:tetrahedron
790:icosahedron
782:alternation
751:), and the
718:icosahedron
607:Coordinates
585:John Conway
558:icosahedron
554:tetrahedron
512:icosahedral
508:tetrahedral
418:Properties
402:, order 96
378:, order 192
4740:10-simplex
4724:9-demicube
4674:8-demicube
4624:7-demicube
4581:6-demicube
4551:5-demicube
4465:Octahedron
4082:References
3064:tr{3,4,3}
3053:2t{3,4,3}
3042:rr{3,4,3}
1942:sr{3,3,4}
1729:alternated
1373:such that
879:Weyl group
796:become 24
738:truncation
623:where φ =
4788:orthoplex
4710:9-simplex
4660:8-simplex
4610:7-simplex
4567:6-simplex
4537:5-simplex
4506:Tesseract
4216:0024-3795
4199:0906.2109
4173:119288632
4156:1106.3433
4104:(1973) .
3957:Citations
3031:r{3,4,3}
3020:s{3,4,3}
3017:t{3,4,3}
2834:h{4,3,3}
2198:clutter.
1858:s{3,4,3}
1765:Extended
1672:Structure
1644:⊕
1624:∑
1577:φ
1574:↔
1571:φ
1560:−
1538:†
1510:±
1495:¯
1472:±
1457:¯
1434:±
1419:¯
1396:±
1387:¯
1358:∈
1319:∗
1308:¯
1290:⊕
1281:¯
1212:¯
1189:→
1178:∗
1121:→
1057:¯
895:Λ
816:600-cell
481:3.3.3.3.3
299:Vertices
271:3.3.3.3.3
63:sr{3,3,4}
4863:Category
4842:Topics:
4805:demicube
4770:polytope
4764:Uniform
4525:600-cell
4521:120-cell
4474:Demicube
4448:Pentagon
4428:Triangle
4224:18278359
3970:Klitzing
3940:See also
3496:Schlegel
3091:{3,4,3}
3082:{3,4,3}
3073:{3,4,3}
3006:{3,4,3}
3000:Schläfli
2867:{4,3,3}
2856:{4,3,3}
2845:{4,3,3}
2305:600-cell
1750:Symmetry
1711:Symmetry
1342:120-cell
1340:and the
1338:600-cell
1196:″
1128:′
984:24-cells
970:′
873:and the
844:600-cell
726:600-cell
602:Geometry
489:geometry
459:8 faces:
286:480 {3}
73:diagrams
61:s{3,4,3}
4779:simplex
4749:10-cube
4516:24-cell
4502:16-cell
4443:Hexagon
4297:regular
3498:diagram
3099:diagram
3097:Coxeter
3088:0,1,2,3
3062:{3,4,3}
3051:{3,4,3}
3040:{3,4,3}
3029:{3,4,3}
3015:{3,4,3}
2942:24-cell
2892:sr{3,3}
2885:tr{3,3}
2878:rr{3,3}
2860:2t{3,3}
2838:2r{3,3}
2309:24-cell
1972:
1960:
1888:
1876:
1804:
1792:
1754:(order)
1689:of the
1589:within
881:orbits
848:24-cell
842:of the
714:24-cell
706:
694:
690:
678:
674:
662:
643:
630:√
625:
510:and 24
495:24-cell
396:
384:
372:
360:
348:
336:
4719:9-cube
4669:8-cube
4619:7-cube
4576:6-cube
4546:5-cube
4433:Square
4310:Family
4222:
4214:
4171:
4126:
3002:symbol
2871:r{3,3}
2849:t{3,3}
2130:Graph
2071:Graph
1896:(192)
1812:(576)
1347:Given
1043:where
780:by an
491:, the
478:and 3
422:convex
291:Edges
283:Faces
237:Cells
4438:p-gon
4220:S2CID
4194:arXiv
4169:S2CID
4151:arXiv
4149:(8).
3079:0,1,3
3059:0,1,2
2938:Name
2831:{3,3}
2289:kites
2016:s{3}
1980:(96)
1783:Cells
1723:on a
794:cubes
515:cells
493:snub
475:3.3.3
408:Dual
259:3.3.3
246:3.3.3
65:s{3}
46:Type
4796:cube
4469:Cube
4299:and
4212:ISSN
4124:ISBN
3813:(b)
3734:(a)
2907:snub
2901:The
2299:The
2283:The
2279:Dual
1525:and
1153:and
1092:and
958:and
708:, 0)
652:The
564:and
556:and
294:432
240:144
4345:(p)
4204:doi
4190:434
4161:doi
3070:0,3
3048:1,2
3037:0,2
3012:0,1
2865:2,3
2313:bi-
2122:/ A
2112:/ A
2108:/ B
1731:: F
1687:net
877:of
740:.
692:, ±
676:, ±
615:of
568:.)
498:or
487:In
436:31
302:96
268:24
256:24
243:96
209:or
153:or
4865::
4850:•
4846:•
4826:21
4822:•
4819:k1
4815:•
4812:k2
4790:•
4747:•
4717:•
4695:21
4691:•
4688:41
4684:•
4681:42
4667:•
4645:21
4641:•
4638:31
4634:•
4631:32
4617:•
4595:21
4591:•
4588:22
4574:•
4544:•
4523:•
4504:•
4483:•
4467:•
4399:/
4388:/
4378:/
4369:/
4347:/
4218:.
4210:.
4202:.
4188:.
4184:.
4167:.
4159:.
4147:09
4145:.
4141:.
4122:.
4004:^
3977:^
2920:.
2319:.
1779:)
1685:A
728:.
660:(±
649:.
628:1+
472:5
454::
439:32
433:30
382:,
358:,
334:,
322:)
4834:-
4832:n
4824:k
4817:2
4810:1
4803:-
4801:n
4794:-
4792:n
4786:-
4784:n
4777:-
4775:n
4768:-
4766:n
4693:4
4686:2
4679:1
4643:3
4636:2
4629:1
4593:2
4586:1
4415:n
4413:H
4406:2
4403:G
4395:4
4392:F
4384:8
4381:E
4375:7
4372:E
4366:6
4363:E
4354:n
4350:D
4343:2
4340:I
4332:n
4328:B
4320:n
4316:A
4288:e
4281:t
4274:v
4235:.
4226:.
4206::
4196::
4175:.
4163::
4153::
4132:.
4074:.
4062:.
3987:.
3972:.
3862:2
3860:B
3811:3
3809:B
3732:3
3730:B
3654:4
3652:B
3576:4
3574:F
3085:t
3076:t
3067:t
3056:t
3045:t
3034:t
3026:1
3023:t
3009:t
2925:4
2863:h
2854:2
2852:h
2843:3
2841:h
2327:4
2325:D
2124:3
2120:2
2118:B
2114:2
2110:3
2106:4
2104:D
2065:4
2063:B
2059:4
2057:F
1976:4
1974:D
1969:2
1966:/
1963:1
1892:4
1890:B
1885:2
1882:/
1879:1
1808:4
1806:F
1801:2
1798:/
1795:1
1775:(
1741:4
1737:4
1733:4
1657:T
1652:i
1648:p
1639:4
1634:1
1631:=
1628:i
1620:=
1617:S
1597:p
1567:/
1563:1
1534:p
1513:p
1507:=
1502:4
1492:p
1485:,
1480:2
1476:p
1469:=
1464:3
1454:p
1447:,
1442:3
1438:p
1431:=
1426:2
1416:p
1409:,
1404:4
1400:p
1393:=
1384:p
1361:T
1355:p
1324:}
1315:]
1305:p
1299:,
1296:p
1293:[
1287:]
1278:p
1272:,
1269:p
1266:[
1263:{
1260:=
1257:)
1252:4
1248:H
1244:(
1241:W
1218:q
1209:r
1203:p
1200:=
1193:r
1186:r
1183::
1174:]
1170:q
1167:,
1164:p
1161:[
1141:q
1138:r
1135:p
1132:=
1125:r
1118:r
1115::
1112:]
1109:q
1106:,
1103:p
1100:[
1080:p
1054:p
1031:)
1028:q
1025:,
1022:p
1019:(
967:T
946:T
926:I
923:=
920:)
915:4
911:H
907:(
904:W
901:=
898:)
892:(
889:O
806:4
703:2
700:/
697:φ
687:2
684:/
681:1
671:2
668:/
665:φ
640:2
637:/
632:5
597:)
587:)
526:h
400:4
398:D
393:2
390:/
387:1
376:4
374:B
369:2
366:/
363:1
352:4
350:F
345:2
342:/
339:1
318:(
20:)
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