2911:
2600:
2656:
1718:
1769:
2735:
2791:
1042:
2860:
1593:
1649:
1364:
3127:
457:
306:
3685:
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3488:
3401:
3346:
3259:
3209:
3038:
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2457:
2400:
2311:
2256:
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2112:
2023:
1973:
1896:
1846:
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3443:
3301:
3169:
2693:
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2266:
1933:
1551:
450:
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1990:
1983:
1940:
1863:
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1688:
1681:
1610:
1603:
1558:
402:
340:
1281:
1308:
769:
372:
1254:
1152:
487:
439:
354:
755:
707:
281:
20:
659:
642:
738:
690:
97:
generalized skew polyhedra to allow for skew faces as well. Grünbaum discovered an additional 23 skew apeirohedra in 3-dimensional
Euclidean space and 3 in 2-dimensional space which are skew by virtue of their faces. 12 of Grünbaum's polyhedra were formed using the blending operation on 2-dimensional
82:
derived a third, the mutetrahedron, and proved that the these three were complete. Under
Coxeter and Petrie's definition, requiring convex faces and allowing a skew vertex figure, the three were not only the only skew apeirohedra in 3-dimensional Euclidean space, but they were the only skew polyhedra
3747:
Polytopes produced as a non-trivial blend have a degree of freedom corresponding to the relative scaling of their components. For this reason some authors count these as infinite families rather than a single polytope. This article counts two polytopes as equal when there is an
520:
Alternatively they can be constructed using the apeir operation on regular polygons. While the
Petrial is used the classical construction, it does not generalize well to higher ranks. In contrast, the apeir operation is used to construct higher rank skew honeycombs.
1182:
A polytope is considered pure if it cannot be expressed as a non-trivial blend of two polytopes. A blend is considered trivial if it contains the result as one of the components. Alternatively a pure polytope is one whose symmetry group contains no non-trivial
939:
Blended polyhedra in 3-dimensional space can be made by blending 2-dimensional polyhedra with 1-dimensional polytopes. The only 2-dimensional polyhedra are the 6 honeycombs (3 Euclidean tilings and 3
71:
are typically required to tile the 2-dimensional plane, Petrie considered cases where the faces were still convex but were not required to lie flat in the plane, they could have a skew polygon
3670:
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2440:
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2142:
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2713:
2578:
2373:
2296:
2231:
2085:
2013:
1953:
1886:
1826:
1696:
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465:
380:
2895:
2775:
2640:
1753:
1633:
324:
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3615:
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3568:
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3533:
3518:
3513:
3478:
3473:
3458:
3431:
3426:
3416:
3411:
3391:
3376:
3371:
3336:
3331:
3316:
3289:
3284:
3274:
3269:
3239:
3234:
3199:
3184:
3152:
3142:
3137:
3115:
3110:
3095:
3068:
3063:
3053:
3048:
3018:
3013:
2978:
2963:
2936:
2926:
2921:
2885:
2801:
2765:
2760:
2708:
2671:
2666:
2630:
2625:
2573:
2536:
2531:
2445:
2425:
2388:
2383:
2368:
2341:
2336:
2326:
2321:
2301:
2286:
2281:
2246:
2241:
2226:
2199:
2194:
2184:
2179:
2157:
2137:
2100:
2095:
2080:
2053:
2048:
2038:
2033:
2003:
1998:
1963:
1948:
1921:
1911:
1906:
1876:
1871:
1836:
1821:
1794:
1784:
1779:
1743:
1659:
1623:
1618:
1566:
1529:
1524:
475:
422:
412:
314:
254:
2848:
2811:
2723:
2681:
2588:
2546:
1706:
1669:
1581:
1539:
1056:
390:
264:
3105:
3058:
3023:
2973:
2931:
2890:
2843:
2806:
2635:
2583:
2541:
1748:
1701:
1664:
385:
319:
259:
3665:
3610:
3563:
3523:
3468:
3421:
3381:
3326:
3279:
3244:
3194:
3147:
2770:
2718:
2676:
2435:
2378:
2331:
2291:
2236:
2189:
2147:
2090:
2043:
2008:
1958:
1916:
1881:
1831:
1789:
1628:
1576:
1534:
470:
417:
4249:(Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues, Proceedings of the London Mathematics Society, Ser. 2, Vol 43, 1937.)
3157:
1052:
105:
In 1985 Dress found an additional pure regular skew apeirohedron in 3-space, and proved that with this additional skew apeirohedron the list was complete.
1018:
Each pair between these produces a valid distinct regular skew apeirohedron in 3-dimensional
Euclidean space, for a total of 12 blended skew apeirohedra.
1388:
These represent 14 compact and 17 paracompact regular skew polyhedra in hyperbolic space, constructed from the symmetry of a subset of linear and cyclic
1055:
2450:
2393:
1077:
1076:
1075:
1074:
1073:
2853:
1072:
1054:
1051:
2900:
1445:
1280:
1253:
1049:
2593:
1307:
1079:
2645:
1711:
1455:
vertex figure with the honeycomb, but only the zig-zag edge faces of the vertex figure are realized, while the other faces make holes.
2162:
2105:
1758:
1045:
2728:
1060:
1059:
102:
could not be formed by a non-trivial blend. Grünbaum conjectured that this new list was complete for the parameters considered.
2780:
510:
4202:
4157:
1071:
1070:
4197:, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
137:, constructed by cubic cells, removing two opposite faces from each, and linking sets of six together around a faceless
1053:
1046:
3973:
4246:
4189:
4063:
1586:
4108:
Dress, Andreas (1985). "A combinatorial theory of Grünbaum's new regular polyhedra, Part II: Complete enumeration".
1050:
1638:
90:
with Petrie and
Coxeters definition, discovering 31 regular skew apeirohedra with compact or paracompact symmetry.
3120:
1080:
1078:
1068:
1067:
1066:
1065:
1163:
1047:
122:
named them mucube, muoctahedron, and mutetrahedron respectively for multiple cube, octahedron, and tetrahedron.
1069:
1064:
1337:. Meaning that each can be constructed from any other by some combination of the Petrial and dual operations.
1063:
903:
For regular polytopes the last step is guaranteed to produce a unique result. This new polytope is called the
4050:, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge: Cambridge University Press,
395:
152:
3735:
twice giving a count of 18 paracompact cases and 32 total, but only listing 17 paracompact and 31 total.
329:
4081:
4205:
2910:
1062:
1061:
2599:
1382:
1191:
1139:
1058:
2655:
1717:
3704:
1768:
1057:
480:
166:
36:
4169:
1048:
4008:
2734:
2790:
170:
67:, polygons whose vertices are not all in the same plane, and extended it to polyhedra. While
538:, and uses them as the mirrors for the vertex figure of a polyhedron, the new vertex mirror
4073:
1375:
1345:
542:
is then a point located where the initial vertex of the polygon (or anywhere on the mirror
156:
118:
The three
Euclidean solutions in 3-space are {4,6|4}, {6,4|4}, and {6,6|3}.
64:
60:
2859:
1081:
8:
4211:
1592:
1648:
4271:
4145:
4125:
4044:
3996:
3126:
119:
87:
4174:, Discrete & Computational Geometry September 2007, Volume 38, Issue 2, pp 355–387
3969:
599:
94:
4242:
4198:
4185:
4153:
4129:
4059:
4000:
3699:
1363:
1184:
947:
792:
695:
443:
358:
285:
130:
1247:
Three additional pure apeirohedra can be formed with finite skew polygons as faces:
371:
4117:
4096:
4051:
4023:
3988:
3958:
1334:
965:
881:
Similarly add faces to every set of vertices all incident on the same face in both
743:
456:
182:
3684:
3629:
3542:
3487:
3400:
3345:
3258:
3208:
3037:
2987:
2456:
2399:
2310:
2255:
2168:
2111:
2022:
1972:
1895:
1845:
305:
4069:
1424: = 2, this represents the Coxeter group . It generates regular skews {2
1104:
1022:
134:
78:
Petrie discovered two regular skew apeirohedra, the mucube and the muoctahedron.
44:
4165:
4035:
3584:
3497:
3442:
3300:
3168:
2692:
2352:
2265:
1932:
1550:
186:
79:
68:
4100:
3646:
3639:
3591:
3504:
3449:
3362:
3355:
3307:
3225:
3218:
3175:
3086:
3079:
3004:
2997:
2954:
2947:
2876:
2869:
2829:
2822:
2751:
2744:
2699:
2616:
2609:
2564:
2557:
2416:
2409:
2359:
2272:
2217:
2210:
2128:
2121:
2071:
2064:
1989:
1982:
1939:
1862:
1855:
1812:
1805:
1734:
1727:
1687:
1680:
1609:
1602:
1557:
1190:
There are 12 regular pure apeirohedra in 3 dimensions. Three of these are the
449:
401:
364:
298:
291:
4265:
4055:
2471:
1464:
1389:
1378:
956:
936:
in orthogonal spaces and taking composing their generating mirrors pairwise.
647:
339:
210:
72:
48:
768:
4039:
3962:
1314:
1260:
998:
173:
cells, removing triangle faces, and linking sets of four around a faceless
40:
486:
159:
with their square faces removed and linking hole pairs of holes together.)
1374:
In 1967, C. W. L. Garner identified 31 hyperbolic skew apeirohedra with
1287:
514:
438:
174:
1151:
1084:
Some relationships between the 12 pure apeirohedra in 3D Euclidean space
353:
4121:
4027:
3992:
3749:
556:). The new initial vertex is placed at the intersection of the mirrors
878:
has no edges then add a virtual edge connecting its vertex to itself.)
4136:
3753:
3709:
1452:
1008:
4209:(Paper 2) H.S.M. Coxeter, "The Regular Sponges, or Skew Polyhedra",
893:
has no faces then add a virtual face connecting its edge to itself.)
3949:
Garner (1967), "Regular Skew
Polyhedra in Hyperbolic Three-Space",
1033:, are combinatrially equivalent to their non-blended counterparts.
748:
28:
280:
19:
4234:
4177:
1358:
1291:
1264:
1215:
1093:
754:
700:
524:
The apeir operation takes the generating mirrors of the polygon,
513:
of full rank, all three of these can be obtained by applying the
148:
706:
108:
1318:
731:
683:
658:
652:
635:
505:
There are 3 regular skew apeirohedra of full rank, also called
201:). The related honeycomb has the extended symmetry [].
83:
in 3-space as there
Coxeter showed there were no finite cases.
737:
641:
517:
to planar polytopes, in this case the three regular tilings.
899:
From the resulting polytope, select one connected component.
689:
181:
Coxeter gives these regular skew apeirohedra {2q,2r|p} with
138:
4139:
509:, that is skew apeirohedra in 2-dimensions. As with the
86:
In 1967 Garner investigated regular skew apeirohedra in
787:, a new polytope can be made by the following process:
928:
Equivalently the blend can be obtained by positioning
4254:
Regular Skew
Polyhedra in Three and Four Dimensions.
4142:, Isabel Hubard, Egon Schulte, Asia Ivic Weiss, 2005
165:: {6,6|3}: 6 hexagons about each vertex (related to
4195:Kaleidoscopes: Selected Writings of H.S.M. Coxeter
4043:
1444:}. All of these exist as a subset of faces of the
4263:
3777:
3775:
3773:
896:Repeat as such for all ranks of proper elements.
4034:
4006:
3922:
3878:
3781:
1420:}. For the special case of linear graph groups
1044:
1359:Regular skew apeirohedra in hyperbolic 3-space
185:[] which he says is isomorphic to his
3770:
3743:
3741:
1446:convex uniform honeycombs in hyperbolic space
775:with the edges of one face highlighted in red
109:Regular skew apeirohedra in Euclidean 3-space
1392:graphs of the form [], These define
570:. Thus the apeir polyhedron is generated by
1021:Since the skeleton of the square tiling is
495:
113:
3738:
3727:
3725:
23:The mucube is a regular skew apeirohedron.
3826:The Symmetry of Things, 2008, Chapter 23
98:apeirohedra, and the other 11 were pure,
4256:Proc. London Math. Soc. 43, 33–62, 1937.
4184:, Third edition, (1973), Dover edition,
4148:, Heidi Burgiel, Chaim Goodman-Strauss,
4079:
3968:
3911:
3900:
3889:
3856:
3804:
2466:17 Paracompact regular skew apeirohedra
1362:
1040:
767:
18:
4007:McMullen, Peter; Schulte, Egon (1997).
3722:
4264:
3948:
3933:
3793:
1451:The skew apeirohedron shares the same
993:The only 1-dimensional polytopes are:
763:
4239:The Beauty of Geometry: Twelve Essays
4227:Regular and Semi-Regular Polytopes II
4107:
4016:Discrete & Computational Geometry
4009:"Regular Polytopes in Ordinary Space"
3867:
3845:Regular and Semi-Regular Polytopes II
3815:
4220:Regular and Semi Regular Polytopes I
1459:14 Compact regular skew apeirohedra
1146:
940:
1036:
806:Add edges between any two vertices
13:
4171:Four-Dimensional Regular Polyhedra
500:
14:
4283:
3974:"Regular polyhedra - old and new"
1218:of the Petrie-Coxeter polyhedra:
549:other than its intersection with
205:Compact regular skew apeirohedra
4082:"Regular Polytopes of Full Rank"
3683:
3673:
3668:
3663:
3658:
3653:
3645:
3638:
3628:
3618:
3613:
3608:
3603:
3598:
3590:
3583:
3571:
3566:
3561:
3556:
3551:
3541:
3531:
3526:
3521:
3516:
3511:
3503:
3496:
3486:
3476:
3471:
3466:
3461:
3456:
3448:
3441:
3429:
3424:
3419:
3414:
3409:
3399:
3389:
3384:
3379:
3374:
3369:
3361:
3354:
3344:
3334:
3329:
3324:
3319:
3314:
3306:
3299:
3287:
3282:
3277:
3272:
3267:
3257:
3247:
3242:
3237:
3232:
3224:
3217:
3207:
3197:
3192:
3187:
3182:
3174:
3167:
3155:
3150:
3145:
3140:
3135:
3125:
3113:
3108:
3103:
3098:
3093:
3085:
3078:
3066:
3061:
3056:
3051:
3046:
3036:
3026:
3021:
3016:
3011:
3003:
2996:
2986:
2976:
2971:
2966:
2961:
2953:
2946:
2934:
2929:
2924:
2919:
2909:
2893:
2888:
2883:
2875:
2868:
2858:
2846:
2841:
2836:
2828:
2821:
2809:
2804:
2799:
2789:
2773:
2768:
2763:
2758:
2750:
2743:
2733:
2721:
2716:
2711:
2706:
2698:
2691:
2679:
2674:
2669:
2664:
2654:
2638:
2633:
2628:
2623:
2615:
2608:
2598:
2586:
2581:
2576:
2571:
2563:
2556:
2544:
2539:
2534:
2529:
2455:
2443:
2438:
2433:
2428:
2423:
2415:
2408:
2398:
2386:
2381:
2376:
2371:
2366:
2358:
2351:
2339:
2334:
2329:
2324:
2319:
2309:
2299:
2294:
2289:
2284:
2279:
2271:
2264:
2254:
2244:
2239:
2234:
2229:
2224:
2216:
2209:
2197:
2192:
2187:
2182:
2177:
2167:
2155:
2150:
2145:
2140:
2135:
2127:
2120:
2110:
2098:
2093:
2088:
2083:
2078:
2070:
2063:
2051:
2046:
2041:
2036:
2031:
2021:
2011:
2006:
2001:
1996:
1988:
1981:
1971:
1961:
1956:
1951:
1946:
1938:
1931:
1919:
1914:
1909:
1904:
1894:
1884:
1879:
1874:
1869:
1861:
1854:
1844:
1834:
1829:
1824:
1819:
1811:
1804:
1792:
1787:
1782:
1777:
1767:
1751:
1746:
1741:
1733:
1726:
1716:
1704:
1699:
1694:
1686:
1679:
1667:
1662:
1657:
1647:
1631:
1626:
1621:
1616:
1608:
1601:
1591:
1579:
1574:
1569:
1564:
1556:
1549:
1537:
1532:
1527:
1522:
1306:
1279:
1252:
1150:
753:
736:
705:
688:
657:
640:
485:
473:
468:
463:
455:
448:
437:
420:
415:
410:
400:
388:
383:
378:
370:
363:
352:
338:
322:
317:
312:
304:
297:
290:
279:
262:
257:
252:
16:Infinite regular skew polyhedron
4089:Discrete Computational Geometry
3951:Canadian Journal of Mathematics
3942:
3927:
3916:
3905:
3894:
3883:
3872:
3861:
1321:arrangement form the vertex of
1294:arrangement form the vertex of
1267:arrangement form the vertex of
1214:Three more are obtained as the
779:For any two regular polytopes,
99:
3850:
3837:
3820:
3809:
3798:
3787:
1367:The compact skew apeirohedron
151:about each vertex (related to
133:about each vertex (related to
80:Harold Scott MacDonald Coxeter
1:
4137:Petrie–Coxeter Maps Revisited
3923:McMullen & Schulte (2002)
3879:McMullen & Schulte (1997)
3828:Objects with Primary Symmetry
3763:
1333:These 3 are closed under the
838:iff there is an edge between
4241:, Dover Publications, 1999,
3782:McMullen & Schulte (1997
3548:
3406:
3264:
3132:
3043:
2916:
2796:
2661:
2526:
2316:
2174:
2028:
1901:
1774:
1654:
1519:
407:
345:
249:
7:
4225:(Paper 23) H.S.M. Coxeter,
4218:(Paper 22) H.S.M. Coxeter,
3832:Infinite Platonic Polyhedra
3693:
974:Petrial triangular tiling:
153:bitruncated cubic honeycomb
10:
4288:
4046:Abstract Regular Polytopes
2500:
986:Petrial hexagonal tiling:
54:
4101:10.1007/s00454-004-0848-5
3731:Garner mistakenly counts
1493:
1381:, found by extending the
666:Petrial triangular tiling
612:
598:
362:
296:
250:
243:
33:regular skew apeirohedron
4150:The Symmetries of Things
4110:Aequationes Mathematicae
4080:McMullen, Peter (2004).
4056:10.1017/CBO9780511546686
3981:Aequationes Mathematicae
3715:
1383:Petrie-Coxeter polyhedra
1192:Petrie-Coxeter polyhedra
714:Petrial hexagonal tiling
496:Grünbaum-Dress polyhedra
183:extended chiral symmetry
114:Petrie-Coxeter polyhedra
3705:Regular skew polyhedron
1239:{6,6 | 3} = {∞, 6}
1231:{6,4 | 4} = {∞, 6}
1223:{4,6 | 4} = {∞, 4}
1025:, two of these blends,
980:Petrial square tiling:
507:regular skew honeycombs
167:quarter cubic honeycomb
37:regular skew polyhedron
3963:10.4153/CJM-1967-106-9
1394:regular skew polyhedra
1371:
1144:
776:
63:took the concept of a
24:
1385:to hyperbolic space.
1366:
1083:
799:with the vertices of
771:
618:Petrial square tiling
511:finite skew polyhedra
171:truncated tetrahedron
65:regular skew polygons
22:
1376:regular skew polygon
1122:represents facetting
856:and an edge between
157:truncated octahedron
61:John Flinders Petrie
4212:Scripta Mathematica
2467:
1460:
915:and is represented
795:of the vertices of
764:Blended apeirohedra
206:
39:. They have either
4252:Coxeter, H. S. M.
4122:10.1007/BF02189831
4028:10.1007/PL00009304
3993:10.1007/BF01836414
2465:
1458:
1372:
1162:. You can help by
1145:
1130:represents skewing
1114:represents halving
777:
204:
88:hyperbolic 3-space
25:
4215:6 (1939) 240–244.
4203:978-0-471-01003-6
4182:Regular Polytopes
4158:978-1-56881-220-5
3700:Skew apeirohedron
3691:
3690:
2463:
2462:
1355:is self-Petrial.
1335:Wilson operations
1185:subrepresentation
1180:
1179:
948:Triangular tiling
793:Cartesian product
761:
760:
696:Triangular tiling
493:
492:
169:, constructed by
155:, constructed by
4279:
4133:
4104:
4086:
4076:
4049:
4031:
4013:
4003:
3978:
3970:Grünbaum, Branko
3965:
3936:
3931:
3925:
3920:
3914:
3909:
3903:
3898:
3892:
3887:
3881:
3876:
3870:
3865:
3859:
3854:
3848:
3841:
3835:
3824:
3818:
3813:
3807:
3802:
3796:
3791:
3785:
3779:
3757:
3745:
3736:
3734:
3729:
3687:
3678:
3677:
3676:
3672:
3671:
3667:
3666:
3662:
3661:
3657:
3656:
3649:
3642:
3632:
3623:
3622:
3621:
3617:
3616:
3612:
3611:
3607:
3606:
3602:
3601:
3594:
3587:
3576:
3575:
3574:
3570:
3569:
3565:
3564:
3560:
3559:
3555:
3554:
3545:
3536:
3535:
3534:
3530:
3529:
3525:
3524:
3520:
3519:
3515:
3514:
3507:
3500:
3490:
3481:
3480:
3479:
3475:
3474:
3470:
3469:
3465:
3464:
3460:
3459:
3452:
3445:
3434:
3433:
3432:
3428:
3427:
3423:
3422:
3418:
3417:
3413:
3412:
3403:
3394:
3393:
3392:
3388:
3387:
3383:
3382:
3378:
3377:
3373:
3372:
3365:
3358:
3348:
3339:
3338:
3337:
3333:
3332:
3328:
3327:
3323:
3322:
3318:
3317:
3310:
3303:
3292:
3291:
3290:
3286:
3285:
3281:
3280:
3276:
3275:
3271:
3270:
3261:
3252:
3251:
3250:
3246:
3245:
3241:
3240:
3236:
3235:
3228:
3221:
3211:
3202:
3201:
3200:
3196:
3195:
3191:
3190:
3186:
3185:
3178:
3171:
3160:
3159:
3158:
3154:
3153:
3149:
3148:
3144:
3143:
3139:
3138:
3129:
3118:
3117:
3116:
3112:
3111:
3107:
3106:
3102:
3101:
3097:
3096:
3089:
3082:
3071:
3070:
3069:
3065:
3064:
3060:
3059:
3055:
3054:
3050:
3049:
3040:
3031:
3030:
3029:
3025:
3024:
3020:
3019:
3015:
3014:
3007:
3000:
2990:
2981:
2980:
2979:
2975:
2974:
2970:
2969:
2965:
2964:
2957:
2950:
2939:
2938:
2937:
2933:
2932:
2928:
2927:
2923:
2922:
2913:
2898:
2897:
2896:
2892:
2891:
2887:
2886:
2879:
2872:
2862:
2851:
2850:
2849:
2845:
2844:
2840:
2839:
2832:
2825:
2814:
2813:
2812:
2808:
2807:
2803:
2802:
2793:
2778:
2777:
2776:
2772:
2771:
2767:
2766:
2762:
2761:
2754:
2747:
2737:
2726:
2725:
2724:
2720:
2719:
2715:
2714:
2710:
2709:
2702:
2695:
2684:
2683:
2682:
2678:
2677:
2673:
2672:
2668:
2667:
2658:
2643:
2642:
2641:
2637:
2636:
2632:
2631:
2627:
2626:
2619:
2612:
2602:
2591:
2590:
2589:
2585:
2584:
2580:
2579:
2575:
2574:
2567:
2560:
2549:
2548:
2547:
2543:
2542:
2538:
2537:
2533:
2532:
2468:
2464:
2459:
2448:
2447:
2446:
2442:
2441:
2437:
2436:
2432:
2431:
2427:
2426:
2419:
2412:
2402:
2391:
2390:
2389:
2385:
2384:
2380:
2379:
2375:
2374:
2370:
2369:
2362:
2355:
2344:
2343:
2342:
2338:
2337:
2333:
2332:
2328:
2327:
2323:
2322:
2313:
2304:
2303:
2302:
2298:
2297:
2293:
2292:
2288:
2287:
2283:
2282:
2275:
2268:
2258:
2249:
2248:
2247:
2243:
2242:
2238:
2237:
2233:
2232:
2228:
2227:
2220:
2213:
2202:
2201:
2200:
2196:
2195:
2191:
2190:
2186:
2185:
2181:
2180:
2171:
2160:
2159:
2158:
2154:
2153:
2149:
2148:
2144:
2143:
2139:
2138:
2131:
2124:
2114:
2103:
2102:
2101:
2097:
2096:
2092:
2091:
2087:
2086:
2082:
2081:
2074:
2067:
2056:
2055:
2054:
2050:
2049:
2045:
2044:
2040:
2039:
2035:
2034:
2025:
2016:
2015:
2014:
2010:
2009:
2005:
2004:
2000:
1999:
1992:
1985:
1975:
1966:
1965:
1964:
1960:
1959:
1955:
1954:
1950:
1949:
1942:
1935:
1924:
1923:
1922:
1918:
1917:
1913:
1912:
1908:
1907:
1898:
1889:
1888:
1887:
1883:
1882:
1878:
1877:
1873:
1872:
1865:
1858:
1848:
1839:
1838:
1837:
1833:
1832:
1828:
1827:
1823:
1822:
1815:
1808:
1797:
1796:
1795:
1791:
1790:
1786:
1785:
1781:
1780:
1771:
1756:
1755:
1754:
1750:
1749:
1745:
1744:
1737:
1730:
1720:
1709:
1708:
1707:
1703:
1702:
1698:
1697:
1690:
1683:
1672:
1671:
1670:
1666:
1665:
1661:
1660:
1651:
1636:
1635:
1634:
1630:
1629:
1625:
1624:
1620:
1619:
1612:
1605:
1595:
1584:
1583:
1582:
1578:
1577:
1573:
1572:
1568:
1567:
1560:
1553:
1542:
1541:
1540:
1536:
1535:
1531:
1530:
1526:
1525:
1461:
1457:
1370:
1354:
1343:
1327:
1310:
1300:
1283:
1273:
1256:
1243:
1235:
1227:
1210:
1205:
1200:
1175:
1172:
1154:
1147:
1137:
1136:
1129:
1128:
1121:
1120:
1113:
1112:
1102:
1101:
1091:
1090:
1043:
1037:Pure apeirohedra
1032:
1028:
1014:
1004:
989:
983:
977:
971:
966:Hexagonal tiling
962:
953:
935:
931:
924:
923:
919:
914:
910:
892:
888:
884:
877:
873:
866:
859:
855:
848:
841:
837:
833:
826:
821:
817:
810:
802:
798:
786:
782:
774:
757:
744:Hexagonal tiling
740:
727:
719:
709:
692:
679:
671:
661:
644:
631:
623:
593:
592:
589:
588:
586:
579:
569:
562:
555:
548:
541:
537:
530:
489:
478:
477:
476:
472:
471:
467:
466:
459:
452:
441:
425:
424:
423:
419:
418:
414:
413:
404:
393:
392:
391:
387:
386:
382:
381:
374:
367:
356:
342:
327:
326:
325:
321:
320:
316:
315:
308:
301:
294:
283:
267:
266:
265:
261:
260:
256:
255:
207:
203:
101:
47:or skew regular
4287:
4286:
4282:
4281:
4280:
4278:
4277:
4276:
4262:
4261:
4084:
4066:
4036:McMullen, Peter
4022:(47): 449–478.
4011:
3976:
3945:
3940:
3939:
3932:
3928:
3921:
3917:
3912:McMullen (2004)
3910:
3906:
3901:McMullen (2004)
3899:
3895:
3890:McMullen (2004)
3888:
3884:
3877:
3873:
3866:
3862:
3857:Grünbaum (1977)
3855:
3851:
3842:
3838:
3825:
3821:
3814:
3810:
3805:Grünbaum (1977)
3803:
3799:
3792:
3788:
3780:
3771:
3766:
3761:
3760:
3746:
3739:
3732:
3730:
3723:
3718:
3696:
3679:
3674:
3669:
3664:
3659:
3654:
3652:
3624:
3619:
3614:
3609:
3604:
3599:
3597:
3577:
3572:
3567:
3562:
3557:
3552:
3550:
3537:
3532:
3527:
3522:
3517:
3512:
3510:
3482:
3477:
3472:
3467:
3462:
3457:
3455:
3435:
3430:
3425:
3420:
3415:
3410:
3408:
3395:
3390:
3385:
3380:
3375:
3370:
3368:
3340:
3335:
3330:
3325:
3320:
3315:
3313:
3293:
3288:
3283:
3278:
3273:
3268:
3266:
3253:
3248:
3243:
3238:
3233:
3231:
3203:
3198:
3193:
3188:
3183:
3181:
3161:
3156:
3151:
3146:
3141:
3136:
3134:
3119:
3114:
3109:
3104:
3099:
3094:
3092:
3072:
3067:
3062:
3057:
3052:
3047:
3045:
3032:
3027:
3022:
3017:
3012:
3010:
2982:
2977:
2972:
2967:
2962:
2960:
2940:
2935:
2930:
2925:
2920:
2918:
2904:
2899:
2894:
2889:
2884:
2882:
2852:
2847:
2842:
2837:
2835:
2815:
2810:
2805:
2800:
2798:
2784:
2779:
2774:
2769:
2764:
2759:
2757:
2727:
2722:
2717:
2712:
2707:
2705:
2685:
2680:
2675:
2670:
2665:
2663:
2649:
2644:
2639:
2634:
2629:
2624:
2622:
2592:
2587:
2582:
2577:
2572:
2570:
2550:
2545:
2540:
2535:
2530:
2528:
2522:
2514:
2509:
2504:
2497:
2489:
2484:
2479:
2473:
2449:
2444:
2439:
2434:
2429:
2424:
2422:
2392:
2387:
2382:
2377:
2372:
2367:
2365:
2345:
2340:
2335:
2330:
2325:
2320:
2318:
2305:
2300:
2295:
2290:
2285:
2280:
2278:
2250:
2245:
2240:
2235:
2230:
2225:
2223:
2203:
2198:
2193:
2188:
2183:
2178:
2176:
2161:
2156:
2151:
2146:
2141:
2136:
2134:
2104:
2099:
2094:
2089:
2084:
2079:
2077:
2057:
2052:
2047:
2042:
2037:
2032:
2030:
2017:
2012:
2007:
2002:
1997:
1995:
1967:
1962:
1957:
1952:
1947:
1945:
1925:
1920:
1915:
1910:
1905:
1903:
1890:
1885:
1880:
1875:
1870:
1868:
1840:
1835:
1830:
1825:
1820:
1818:
1798:
1793:
1788:
1783:
1778:
1776:
1762:
1757:
1752:
1747:
1742:
1740:
1710:
1705:
1700:
1695:
1693:
1673:
1668:
1663:
1658:
1656:
1642:
1637:
1632:
1627:
1622:
1617:
1615:
1585:
1580:
1575:
1570:
1565:
1563:
1543:
1538:
1533:
1528:
1523:
1521:
1515:
1507:
1502:
1497:
1490:
1482:
1477:
1472:
1466:
1368:
1361:
1353:
1349:
1342:
1338:
1329:
1326:
1322:
1311:
1302:
1299:
1295:
1284:
1275:
1272:
1268:
1257:
1242:
1238:
1234:
1230:
1226:
1222:
1208:
1203:
1198:
1176:
1170:
1167:
1160:needs expansion
1134:
1133:
1126:
1125:
1118:
1117:
1110:
1109:
1103:represents the
1099:
1098:
1092:represents the
1088:
1087:
1082:
1041:
1039:
1030:
1026:
1012:
1002:
987:
981:
975:
969:
960:
951:
941:skew honeycombs
933:
929:
921:
917:
916:
912:
908:
890:
886:
882:
875:
871:
869:
864:
862:
857:
853:
851:
846:
844:
839:
836:
831:
829:
824:
823:
820:
815:
813:
808:
807:
800:
796:
791:Start with the
784:
780:
772:
766:
726:
722:
717:
678:
674:
669:
630:
626:
621:
600:Schläfli symbol
596:Skew honeycombs
585:
581:
578:
574:
572:
571:
568:
564:
561:
557:
554:
550:
547:
543:
539:
536:
532:
529:
525:
503:
501:Skew honeycombs
498:
479:
474:
469:
464:
462:
442:
433:
428:
426:
421:
416:
411:
409:
394:
389:
384:
379:
377:
357:
348:
333:
328:
323:
318:
313:
311:
284:
275:
270:
268:
263:
258:
253:
251:
245:
240:
231:
226:
218:
213:
135:cubic honeycomb
116:
111:
57:
35:is an infinite
17:
12:
11:
5:
4285:
4275:
4274:
4260:
4259:
4258:
4257:
4232:
4231:
4230:
4223:
4216:
4192:
4175:
4166:Peter McMullen
4163:
4146:John H. Conway
4143:
4134:
4105:
4077:
4064:
4032:
4004:
3966:
3944:
3941:
3938:
3937:
3926:
3915:
3904:
3893:
3882:
3871:
3860:
3849:
3836:
3819:
3808:
3797:
3786:
3768:
3767:
3765:
3762:
3759:
3758:
3737:
3720:
3719:
3717:
3714:
3713:
3712:
3707:
3702:
3695:
3692:
3689:
3688:
3681:
3650:
3643:
3636:
3633:
3626:
3595:
3588:
3581:
3580:{12,12|3}
3578:
3547:
3546:
3539:
3508:
3501:
3494:
3493:{10,12|3}
3491:
3484:
3453:
3446:
3439:
3438:{12,10|3}
3436:
3405:
3404:
3397:
3366:
3359:
3352:
3349:
3342:
3311:
3304:
3297:
3294:
3263:
3262:
3255:
3229:
3222:
3215:
3212:
3205:
3179:
3172:
3165:
3162:
3131:
3130:
3123:
3090:
3083:
3076:
3073:
3042:
3041:
3034:
3008:
3001:
2994:
2991:
2984:
2958:
2951:
2944:
2941:
2915:
2914:
2907:
2902:
2880:
2873:
2866:
2863:
2856:
2833:
2826:
2819:
2816:
2795:
2794:
2787:
2782:
2755:
2748:
2741:
2738:
2731:
2703:
2696:
2689:
2686:
2660:
2659:
2652:
2647:
2620:
2613:
2606:
2603:
2596:
2568:
2561:
2554:
2551:
2525:
2524:
2519:
2516:
2511:
2506:
2501:
2499:
2494:
2491:
2486:
2481:
2476:
2461:
2460:
2453:
2420:
2413:
2406:
2403:
2396:
2363:
2356:
2349:
2348:{10,10|3}
2346:
2315:
2314:
2307:
2276:
2269:
2262:
2259:
2252:
2221:
2214:
2207:
2204:
2173:
2172:
2165:
2132:
2125:
2118:
2115:
2108:
2075:
2068:
2061:
2058:
2027:
2026:
2019:
1993:
1986:
1979:
1976:
1969:
1943:
1936:
1929:
1926:
1900:
1899:
1892:
1866:
1859:
1852:
1849:
1842:
1816:
1809:
1802:
1799:
1773:
1772:
1765:
1760:
1738:
1731:
1724:
1721:
1714:
1691:
1684:
1677:
1674:
1653:
1652:
1645:
1640:
1613:
1606:
1599:
1596:
1589:
1561:
1554:
1547:
1544:
1518:
1517:
1512:
1509:
1504:
1499:
1494:
1492:
1487:
1484:
1479:
1474:
1469:
1390:Coxeter groups
1379:vertex figures
1369:{4,6 | 5}
1360:
1357:
1351:
1340:
1331:
1330:
1324:
1312:
1305:
1303:
1297:
1285:
1278:
1276:
1270:
1258:
1251:
1245:
1244:
1240:
1236:
1232:
1228:
1224:
1212:
1211:
1209:{6,6 | 3}
1206:
1204:{6,4 | 4}
1201:
1199:{4,6 | 4}
1178:
1177:
1157:
1155:
1143:
1142:
1131:
1123:
1115:
1107:
1096:
1038:
1035:
1016:
1015:
1005:
991:
990:
984:
978:
972:
963:
954:
901:
900:
897:
894:
879:
867:
860:
849:
842:
834:
827:
818:
811:
804:
765:
762:
759:
758:
751:
746:
741:
734:
728:
724:
720:
715:
711:
710:
703:
698:
693:
686:
680:
676:
672:
667:
663:
662:
655:
650:
645:
638:
632:
628:
624:
619:
615:
614:
611:
608:
605:
602:
597:
583:
576:
566:
559:
552:
545:
534:
527:
502:
499:
497:
494:
491:
490:
483:
460:
453:
446:
435:
430:
406:
405:
398:
375:
368:
361:
350:
344:
343:
336:
331:
309:
302:
295:
288:
277:
272:
248:
247:
242:
237:
228:
223:
220:
215:
187:abstract group
179:
178:
160:
142:
115:
112:
110:
107:
56:
53:
49:vertex figures
15:
9:
6:
4:
3:
2:
4284:
4273:
4270:
4269:
4267:
4255:
4251:
4250:
4248:
4247:0-486-40919-8
4244:
4240:
4236:
4233:
4228:
4224:
4221:
4217:
4214:
4213:
4208:
4207:
4206:
4204:
4200:
4196:
4193:
4191:
4190:0-486-61480-8
4187:
4183:
4179:
4176:
4173:
4172:
4167:
4164:
4162:
4159:
4155:
4151:
4147:
4144:
4141:
4138:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4106:
4102:
4098:
4094:
4090:
4083:
4078:
4075:
4071:
4067:
4065:0-521-81496-0
4061:
4057:
4053:
4048:
4047:
4041:
4040:Schulte, Egon
4037:
4033:
4029:
4025:
4021:
4017:
4010:
4005:
4002:
3998:
3994:
3990:
3987:(1–2): 1–20,
3986:
3982:
3975:
3971:
3967:
3964:
3960:
3957:: 1179–1186,
3956:
3952:
3947:
3946:
3935:
3934:Garner (1967)
3930:
3924:
3919:
3913:
3908:
3902:
3897:
3891:
3886:
3880:
3875:
3869:
3864:
3858:
3853:
3846:
3840:
3834:, pp. 333–335
3833:
3829:
3823:
3817:
3812:
3806:
3801:
3795:
3794:Garner (1967)
3790:
3783:
3778:
3776:
3774:
3769:
3756:between them.
3755:
3751:
3744:
3742:
3728:
3726:
3721:
3711:
3708:
3706:
3703:
3701:
3698:
3697:
3686:
3682:
3680:ct{(3,6,3,6)}
3651:
3648:
3644:
3641:
3637:
3634:
3631:
3627:
3625:ct{(6,3,6,3)}
3596:
3593:
3589:
3586:
3582:
3579:
3549:
3544:
3540:
3538:ct{(5,3,6,3)}
3509:
3506:
3502:
3499:
3495:
3492:
3489:
3485:
3483:ct{(6,3,5,3)}
3454:
3451:
3447:
3444:
3440:
3437:
3407:
3402:
3398:
3396:ct{(4,3,6,3)}
3367:
3364:
3360:
3357:
3353:
3351:{8,12|3}
3350:
3347:
3343:
3341:ct{(6,3,4,3)}
3312:
3309:
3305:
3302:
3298:
3296:{12,8|3}
3295:
3265:
3260:
3256:
3254:ct{(3,3,6,3)}
3230:
3227:
3223:
3220:
3216:
3214:{6,12|3}
3213:
3210:
3206:
3204:ct{(6,3,3,3)}
3180:
3177:
3173:
3170:
3166:
3164:{12,6|3}
3163:
3133:
3128:
3124:
3122:
3091:
3088:
3084:
3081:
3077:
3074:
3044:
3039:
3035:
3033:ct{(3,4,4,4)}
3009:
3006:
3002:
2999:
2995:
2992:
2989:
2985:
2983:ct{(4,4,3,4)}
2959:
2956:
2952:
2949:
2945:
2942:
2917:
2912:
2908:
2906:
2881:
2878:
2874:
2871:
2867:
2864:
2861:
2857:
2855:
2834:
2831:
2827:
2824:
2820:
2817:
2797:
2792:
2788:
2786:
2756:
2753:
2749:
2746:
2742:
2740:{4,12|3}
2739:
2736:
2732:
2730:
2704:
2701:
2697:
2694:
2690:
2688:{12,4|3}
2687:
2662:
2657:
2653:
2651:
2621:
2618:
2614:
2611:
2607:
2604:
2601:
2597:
2595:
2569:
2566:
2562:
2559:
2555:
2552:
2527:
2520:
2517:
2512:
2507:
2502:
2495:
2492:
2487:
2482:
2477:
2475:
2470:
2469:
2458:
2454:
2452:
2451:ct{(3,5,3,5)}
2421:
2418:
2414:
2411:
2407:
2404:
2401:
2397:
2395:
2394:ct{(5,3,5,3)}
2364:
2361:
2357:
2354:
2350:
2347:
2317:
2312:
2308:
2306:ct{(5,3,4,3)}
2277:
2274:
2270:
2267:
2263:
2261:{10,8|3}
2260:
2257:
2253:
2251:ct{(4,3,5,3)}
2222:
2219:
2215:
2212:
2208:
2206:{8,10|3}
2205:
2175:
2170:
2166:
2164:
2163:ct{(3,4,3,4)}
2133:
2130:
2126:
2123:
2119:
2116:
2113:
2109:
2107:
2106:ct{(4,3,4,3)}
2076:
2073:
2069:
2066:
2062:
2059:
2029:
2024:
2020:
2018:ct{(3,3,5,3)}
1994:
1991:
1987:
1984:
1980:
1978:{6,10|3}
1977:
1974:
1970:
1968:ct{(5,3,3,3)}
1944:
1941:
1937:
1934:
1930:
1928:{10,6|3}
1927:
1902:
1897:
1893:
1891:ct{(3,3,4,3)}
1867:
1864:
1860:
1857:
1853:
1850:
1847:
1843:
1841:ct{(4,3,3,3)}
1817:
1814:
1810:
1807:
1803:
1800:
1775:
1770:
1766:
1764:
1739:
1736:
1732:
1729:
1725:
1722:
1719:
1715:
1713:
1692:
1689:
1685:
1682:
1678:
1675:
1655:
1650:
1646:
1644:
1614:
1611:
1607:
1604:
1600:
1598:{4,10|3}
1597:
1594:
1590:
1588:
1562:
1559:
1555:
1552:
1548:
1546:{10,4|3}
1545:
1520:
1513:
1510:
1505:
1500:
1495:
1488:
1485:
1480:
1475:
1470:
1468:
1463:
1462:
1456:
1454:
1449:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1408:} and dual {2
1407:
1403:
1399:
1395:
1391:
1386:
1384:
1380:
1377:
1365:
1356:
1347:
1336:
1320:
1316:
1315:skew hexagons
1309:
1304:
1293:
1289:
1282:
1277:
1266:
1262:
1261:skew hexagons
1255:
1250:
1249:
1248:
1237:
1229:
1221:
1220:
1219:
1217:
1207:
1202:
1197:
1196:
1195:
1193:
1188:
1186:
1174:
1171:February 2024
1165:
1161:
1158:This section
1156:
1153:
1149:
1148:
1141:
1140:rectification
1132:
1124:
1116:
1108:
1106:
1097:
1095:
1086:
1085:
1034:
1024:
1019:
1010:
1006:
1000:
996:
995:
994:
985:
979:
973:
967:
964:
958:
957:Square tiling
955:
949:
946:
945:
944:
942:
937:
926:
906:
898:
895:
880:
805:
794:
790:
789:
788:
770:
756:
752:
750:
747:
745:
742:
739:
735:
733:
729:
721:
716:
713:
712:
708:
704:
702:
699:
697:
694:
691:
687:
685:
681:
673:
668:
665:
664:
660:
656:
654:
651:
649:
648:Square tiling
646:
643:
639:
637:
633:
625:
620:
617:
616:
609:
606:
603:
601:
595:
594:
591:
522:
518:
516:
512:
508:
488:
484:
482:
461:
458:
454:
451:
447:
445:
440:
436:
434:Mutetrahedron
431:
408:
403:
399:
397:
376:
373:
369:
366:
360:
355:
351:
346:
341:
337:
335:
310:
307:
303:
300:
293:
289:
287:
282:
278:
273:
238:
235:
229:
224:
221:
216:
212:
211:Coxeter group
209:
208:
202:
200:
196:
192:
188:
184:
176:
172:
168:
164:
163:Mutetrahedron
161:
158:
154:
150:
147:: {6,4|4}: 4
146:
143:
140:
136:
132:
129:: {4,6|4}: 6
128:
125:
124:
123:
121:
106:
103:
96:
91:
89:
84:
81:
76:
74:
73:vertex figure
70:
66:
62:
52:
50:
46:
42:
38:
34:
30:
21:
4253:
4238:
4226:
4219:
4210:
4194:
4181:
4170:
4161:
4149:
4113:
4109:
4092:
4088:
4045:
4019:
4015:
3984:
3980:
3954:
3950:
3943:Bibliography
3929:
3918:
3907:
3896:
3885:
3874:
3868:Dress (1985)
3863:
3852:
3844:
3839:
3831:
3827:
3822:
3816:Dress (1985)
3811:
3800:
3789:
3733:{8,8|4}
3635:{6,6|6}
3075:{8,8|4}
2993:{6,8|4}
2943:{8,6|4}
2865:{4,6|6}
2818:{6,4|6}
2605:{4,8|4}
2553:{8,4|4}
2505:{p,q|l}
2503:Apeirohedron
2480:{p,q|l}
2478:Apeirohedron
2405:{6,6|5}
2117:{6,6|4}
2060:{8,8|3}
1851:{6,8|3}
1801:{8,6|3}
1723:{4,6|5}
1676:{6,4|5}
1498:{p,q|l}
1496:Apeirohedron
1473:{p,q|l}
1471:Apeirohedron
1450:
1441:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1387:
1373:
1332:
1288:skew squares
1246:
1213:
1189:
1181:
1168:
1164:adding to it
1159:
1020:
1017:
999:line segment
992:
938:
927:
904:
902:
778:
523:
519:
506:
504:
432:{6,6|3}
349:Muoctahedron
347:{6,4|4}
274:{4,6|4}
233:
219:{p,q|l}
217:Apeirohedron
198:
194:
190:
180:
162:
145:Muoctahedron
144:
126:
117:
104:
92:
85:
77:
58:
32:
26:
4116:: 222–243.
1138:represents
610:Petrie dual
573:⟨w,
515:Petrie dual
271:[]
175:tetrahedron
120:John Conway
69:apeirohedra
3764:References
3750:affine map
1436:} and {4,2
429:[]
427:[]
269:[]
246:honeycomb
4272:Polyhedra
4130:121260389
4001:125049930
3843:Coxeter,
3784::449–450)
3754:full rank
3710:Tetrastix
2854:2t{6,3,6}
2729:2t{3,6,3}
2594:2t{4,4,4}
2518:Honeycomb
2493:Honeycomb
1712:2t{5,3,5}
1587:2t{3,5,3}
1511:Honeycomb
1486:Honeycomb
1453:antiprism
1346:self-dual
1292:hexagonal
1265:hexagonal
1031:{4, 4}#{}
1027:{4, 4}#{}
1023:bipartite
1009:apeirogon
613:Apeir of
444:animation
396:2t{4,3,4}
359:animation
286:animation
214:symmetry
4266:Category
4095:: 1–35.
4042:(2002),
3972:(1977),
3694:See also
3121:q{4,4,4}
1216:Petrials
773:{3,6}#{}
749:Triangle
587:⟩
481:q{4,3,4}
149:hexagons
95:Grünbaum
93:In 1977
59:In 1926
43:regular
29:geometry
4235:Coxeter
4178:Coxeter
4074:1965665
2905:{6,3,6}
2785:{3,6,3}
2650:{4,4,4}
2523:figure
2472:Coxeter
1763:{5,3,5}
1643:{3,5,3}
1516:figure
1465:Coxeter
1094:Petrial
732:zigzags
701:Hexagon
684:zigzags
636:zigzags
334:{4,3,4}
244:Related
131:squares
55:History
4245:
4201:
4188:
4156:
4152:2008,
4128:
4072:
4062:
3999:
2521:Vertex
2498:figure
2496:Vertex
1514:Vertex
1491:figure
1489:Vertex
1319:square
988:{6, 3}
982:{4, 4}
976:{3, 6}
970:{6, 3}
961:{4, 4}
952:{3, 6}
889:. (If
874:. (If
653:Square
582:ρ
575:ρ
565:ρ
558:ρ
551:ρ
544:ρ
533:ρ
526:ρ
276:Mucube
241:figure
239:Vertex
127:Mucube
4126:S2CID
4085:(PDF)
4012:(PDF)
3997:S2CID
3977:(PDF)
3847:2.34)
3716:Notes
2474:group
1467:group
1350:{6,4}
1339:{6,6}
1323:{6,4}
1317:in a
1296:{4,6}
1290:in a
1269:{6,6}
1263:in a
905:blend
723:{∞,3}
718:{6,3}
675:{∞,6}
670:{3,6}
627:{∞,4}
622:{4,4}
607:Image
604:Faces
222:Image
45:faces
4243:ISBN
4199:ISBN
4186:ISBN
4154:ISBN
4060:ISBN
2513:Hole
2508:Face
2488:Hole
2483:Face
1506:Hole
1501:Face
1481:Hole
1476:Face
1348:and
1105:dual
1029:and
1007:The
997:The
932:and
911:and
885:and
863:and
845:and
822:and
783:and
563:and
531:and
230:Hole
225:Face
139:cube
100:i.e.
41:skew
31:, a
4140:PDF
4118:doi
4097:doi
4052:doi
4024:doi
3989:doi
3959:doi
3752:of
2903:0,3
2783:0,3
2648:0,3
2515:{l}
2510:{p}
2490:{l}
2485:{p}
1761:0,3
1641:0,3
1508:{l}
1503:{p}
1483:{l}
1478:{p}
1344:is
1241:6,3
1233:4,4
1225:6,4
1166:.
1013:{∞}
943:):
907:of
870:in
852:in
332:0,3
227:{p}
197:|2,
27:In
4268::
4237:,
4229:,
4222:,
4180:,
4168:,
4124:.
4114:29
4112:.
4093:32
4091:.
4087:.
4070:MR
4068:,
4058:,
4038:;
4020:17
4018:.
4014:.
3995:,
3985:16
3983:,
3979:,
3955:19
3953:,
3830:,
3772:^
3740:^
3724:^
1448:.
1412:,2
1400:,2
1396:{2
1313:4
1286:6
1259:6
1194::
1187:.
1011::
1003:{}
1001::
968::
959::
950::
925:.
830:×
814:×
730:∞
682:∞
634:∞
590:.
580:,
193:,2
189:(2
177:.)
141:.)
75:.
51:.
4160:,
4132:.
4120::
4103:.
4099::
4054::
4030:.
4026::
3991::
3961::
2901:t
2781:t
2646:t
1759:t
1639:t
1442:p
1440:|
1438:q
1434:p
1432:|
1430:4
1428:,
1426:q
1422:r
1418:p
1416:|
1414:q
1410:r
1406:p
1404:|
1402:r
1398:q
1352:6
1341:4
1328:.
1325:6
1301:.
1298:6
1274:.
1271:4
1173:)
1169:(
1135:r
1127:σ
1119:φ
1111:η
1100:δ
1089:π
934:Q
930:P
922:Q
920:#
918:P
913:Q
909:P
891:Q
887:Q
883:P
876:Q
872:Q
868:1
865:q
861:0
858:q
854:P
850:1
847:p
843:0
840:p
835:1
832:q
828:1
825:p
819:0
816:q
812:0
809:p
803:.
801:Q
797:P
785:Q
781:P
725:6
677:3
629:4
584:0
577:0
567:1
560:0
553:0
546:1
540:w
535:1
528:0
330:t
236:}
234:l
232:{
199:p
195:r
191:q
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
