4856:
4752:
4927:
4967:
4798:
4896:
4979:
4825:
481:
3640:
4813:
3085:
4329:
2710:
2703:
5032:
2859:
2807:
4318:
4439:
4415:
4404:
4380:
3092:
2852:
2800:
4844:
3106:
3017:
431:
3367:
50:
4911:
3024:
2975:
4261:
2229:
2866:
2814:
4450:
4428:
3289:
4369:
3003:
2989:
4682:
2996:
4675:
4239:
4393:
4307:
3616:
5016:
3764:
3099:
3386:
4872:
2488:
2479:
2461:
2450:
3078:
4947:
995:
4250:
4217:
3379:
2209:
2506:
2497:
2470:
4668:
4661:
4228:
3360:
3775:
2982:
2793:
2075:
4486:
518:
4736:
2192:
2179:
2166:
2151:
2142:
2129:
2118:
2105:
2092:
3273:
3264:
3255:
3244:
3235:
3226:
5001:
4497:
3719:
4358:
3127:
4783:
2585:
2558:
4475:
4768:
3786:
2696:
2845:
4719:
4712:
4705:
4698:
3708:
3120:
3113:
3010:
737:
1208:
990:{\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}}
1901:
1693:
1528:
685:
4871:
2049:
1076:
4855:
3488:
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are
742:
1260:
1699:
4926:
1081:
3923:
4797:
3504:
at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.
726:
3863:
339:
3444:
hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its
5226:
1068:
1534:
1369:
3605:
590:
4895:
4751:
1931:
1361:
4910:
4824:
5337:
1334:
1314:
468:
4978:
381:. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a
1939:
1203:{\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}
4946:
6640:
6070:
3308:
each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less
5248:
4735:
5505:
5367:
5301:
5161:
4966:
6100:
5155:
3678:
with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a
3405:
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the
2658:
2594:
1896:{\displaystyle d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).}
2668:
2604:
1216:
4812:
4651:
4641:
4631:
4618:
4598:
4575:
4532:
4207:
4197:
4187:
4168:
4158:
4149:
4139:
4129:
4110:
4100:
4081:
4071:
4033:
4023:
4013:
3995:
3985:
2637:
2627:
2577:
2567:
130:
120:
102:
5015:
6635:
4608:
4585:
4565:
4552:
4542:
4443:
4419:
4408:
4384:
4178:
4120:
4091:
4005:
2753:
with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a
112:
1269:
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then
2663:
2599:
138:
4843:
4646:
4636:
4613:
4603:
4580:
4570:
4547:
4537:
4202:
4192:
4173:
4163:
4144:
4134:
4115:
4105:
4086:
4076:
4028:
4018:
4000:
3990:
125:
107:
3663:
with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A
2632:
2572:
4624:
3348:
5397:
2327:
hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are
5031:
4454:
4432:
3881:
5000:
6223:
6203:
4373:
4265:
3970:
5335:
3428:
until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
94:
572:
circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base),
6645:
6198:
6155:
6130:
5463:
4917:
4902:
4782:
4591:
4397:
2964:
5186:
690:
6258:
5071:
4957:
437:
27:
20:
3828:
418:
are hexagonal for this reason and because the shape makes efficient use of space and building materials. The
313:
6183:
5498:
4243:
3966:
3420:(also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any
430:
6208:
6093:
5054:
4333:
3639:
3517:
2925:
3627:
is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the
1029:
6609:
6549:
6188:
6042:
6035:
6028:
5240:
5220:
5108:
4311:
4254:
4221:
3974:
3954:
3946:
3745:
3313:
2917:
2903:
2645:
2336:
366:
358:
270:
4328:
1688:{\displaystyle d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),}
1523:{\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),}
480:
6493:
6263:
6193:
6135:
5699:
5646:
5114:
5103:
4298:
4288:
4232:
4061:
4051:
3950:
3679:
3644:
3624:
3155:
3065:
2960:
2941:
2929:
2906:
2672:
are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
2335:
forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an
1263:
495:
446:
412:
386:
303:
273:
34:
5451:
3084:
2858:
2806:
680:{\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}
6630:
6599:
6574:
6544:
6539:
6498:
6213:
6054:
5953:
5703:
5389:
5292:
4937:
4882:
4803:
4767:
4558:
4490:
4438:
4414:
4403:
4379:
4317:
4293:
4274:
4056:
3808:
3417:
3301:
3091:
2851:
2799:
2709:
2213:
1284:
that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long
284:
280:
181:
3557:
3385:
2702:
6604:
6145:
5923:
5873:
5823:
5780:
5750:
5710:
5673:
5491:
5198:
5097:
4992:
3524:
3248:
3192:
2878:
2766:
2300:
1933:
are the distances from the vertices of a regular hexagon to any point on its circumcircle, then
296:
3366:
2892:
266:
84:
4449:
4427:
4260:
3105:
3016:
2865:
2813:
33:"Hexagonal" redirects here. For the FIFA World Cup qualifying tournament in North America, see
6584:
6178:
6086:
6062:
5418:
5244:
5151:
4862:
4834:
4501:
4368:
4362:
4044:
3942:
3675:
3023:
2974:
2945:
393:
288:
143:
74:
49:
5355:
5287:
6113:
6066:
5631:
5620:
5609:
5598:
5589:
5580:
5567:
5545:
5533:
5519:
5515:
5208:
5145:
5081:
5076:
5037:
4886:
4689:
4681:
4525:
3741:
3490:
3406:
3400:
3343:
2896:
2648:
2614:
2454:
2393:
2228:
385:
with a vertex at the center of the regular hexagon and sharing one side with the hexagon is
346:
4674:
4392:
4238:
3002:
2988:
2641:, are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
2365:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
1909:
1339:
6579:
6559:
6554:
6524:
6243:
6218:
6150:
5656:
5641:
5401:
5341:
5066:
5007:
4479:
4306:
3978:
3445:
3437:
3328:
3324:
2995:
2750:
2351:
2324:
2316:
2135:
2111:
419:
260:
189:
185:
70:
63:
3807:
of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex
3763:
3288:
2909:, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D
6589:
6569:
6534:
6529:
6160:
6140:
6006:
5470:
5085:
4322:
3934:
3779:
3737:
3615:
3501:
3441:
3305:
3174:
3077:
2754:
2652:
2621:
2617:
2487:
2478:
2460:
2449:
1319:
1299:
453:
374:
354:
292:
247:
177:
173:
159:
155:
5251:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
4249:
4216:
3378:
3098:
2505:
2496:
2469:
6624:
6564:
6415:
6308:
6228:
6170:
6023:
5911:
5904:
5897:
5861:
5854:
5847:
5811:
5804:
5528:
4342:
3425:
3421:
2370:
2232:
The dihedral symmetries are divided depending on whether they pass through vertices (
216:
4667:
4660:
4273:
There are other symmetry polyhedra with stretched or flattened hexagons, like these
4227:
3774:
3359:
2074:
2044:{\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}
6594:
6464:
6420:
6384:
6374:
6369:
5963:
4485:
3938:
3768:
3660:
3513:
3497:
3320:
3188:
3143:
2825:
2762:
2389:
2359:
2328:
2277:
2208:
2191:
2178:
2165:
2159:
2150:
2141:
2128:
2117:
2104:
2091:
1277:
557:
525:
499:
491:
471:
408:
307:
196:
5395:
2981:
2792:
2729:-gon whose opposite sides are parallel and of equal length) can be dissected into
1316:, whose distances to the centroid of the regular hexagon and its six vertices are
389:, and that the regular hexagon can be partitioned into six equilateral triangles.
3520:
states that the three main diagonals AD, BE, and CF intersect at a single point.
6503:
6410:
6389:
6379:
5972:
5933:
5883:
5833:
5790:
5760:
5692:
5678:
5447:
4984:
4496:
4357:
3958:
3718:
3046:
517:
397:
5441:
3272:
3263:
3254:
3243:
3234:
3225:
6508:
6364:
6354:
6238:
5958:
5942:
5892:
5842:
5799:
5769:
5683:
5457:
4474:
3723:
3697:(same as triangular antiprism) have regular skew hexagons as petrie polygons.
3694:
3169:
3126:
2933:
5421:
2319:
hexagon constructed by three mirrors can alternate long and short edges, and
407:(three hexagons meeting at every vertex), and so are useful for constructing
6483:
6473:
6450:
6440:
6430:
6359:
6268:
6233:
6014:
5928:
5878:
5828:
5785:
5755:
5724:
5467:
5426:
5212:
5062:
5058:
4774:
3962:
3790:
3297:
3057:
2952:
2921:
2584:
2557:
569:
415:
5132:
3475:, then the three main diagonals intersect in a single point if and only if
3304:, hexagonal patterns are prevalent in nature due to their efficiency. In a
2331:
of each other and have half the symmetry order of the regular hexagon. The
1296:
For an arbitrary point in the plane of a regular hexagon with circumradius
422:
of a regular triangular lattice is the honeycomb tessellation of hexagons.
5437:
4718:
4711:
4704:
3785:
2844:
2695:
2264:
symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D
2240:
for perpendiculars) Cyclic symmetries in the middle column are labeled as
6488:
6478:
6435:
6394:
6323:
6313:
6303:
6122:
5988:
5743:
5739:
5666:
5474:
5334:
Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter",
5091:
4953:
4789:
4742:
4465:
3628:
3049:
2937:
1285:
1281:
577:
549:
545:
533:
400:
382:
208:
4697:
3707:
2299:
These symmetries express nine distinct symmetries of a regular hexagon.
2244:
for their central gyration orders. Full symmetry of the regular form is
436:
A step-by-step animation of the construction of a regular hexagon using
6445:
6425:
6338:
6333:
6328:
6293:
6248:
6109:
5997:
5967:
5734:
5729:
5720:
5260:
5050:
4988:
4830:
2722:
2718:
2339:
1288:
of 1.0000000 will have a distance of 0.8660254 between parallel sides.
1005:
1001:
342:
243:
3119:
3112:
6253:
5937:
5887:
5837:
5794:
5764:
5715:
5651:
5288:"Dao's theorem on six circumcenters associated with a cyclic hexagon"
5022:
4878:
2956:
2683:
505:
four times on the circumscribed circle and connect the corner points.
441:
4889:; large masses must cool slowly to form a polygonal fracture pattern
3009:
490:
is given, drawing a circular arc from point A and point B gives the
5203:
6298:
6078:
5356:"Equilateral triangles and Kiepert perspectors in complex numbers"
4758:
3614:
3287:
2944:
by adding a center point. This pattern repeats within the regular
2207:
516:
350:
3323:
and can also tile the plane by translation. In three dimensions,
2358:
hexagons, with opposite sides parallel are also called hexagonal
5687:
5263:, Mathematical recreations and Essays, Thirteenth edition, p.141
3744:, uniform and dual polyhedra and polytopes, shown in these skew
3712:
3690:
3159:
2873:
2758:
6082:
4806:, a hexagonal cloud pattern around the north pole of the planet
3309:
237:
231:
225:
219:
2775:
Dissection of hexagons into three rhombs and parallelograms
1255:{\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}
3319:
Irregular hexagons with parallel opposite edges are called
2765:
and projective directions of the cube are dissected within
5111:: abstract board game played on a six-sided hexagonal grid
3619:
Equilateral triangles on the sides of an arbitrary hexagon
3611:
Equilateral triangles on the sides of an arbitrary hexagon
5187:"Cyclic Averages of Regular Polygons and Platonic Solids"
3631:
of opposite triangles form another equilateral triangle.
2392:
can tessellate the
Euclidean plane by translation. Other
2899:, {6,3}, with three hexagonal faces around each vertex.
2058:
584:. The maxima and minima are related by the same factor:
302:
The common length of the sides equals the radius of the
279:
A regular hexagon is defined as a hexagon that is both
3667:
has vertices alternating between two parallel planes.
2679:
2552:
2369:
subgroup has no degrees of freedom but can be seen as
1221:
318:
3884:
3831:
3560:
2350:
can be seen as horizontally and vertically elongated
1942:
1912:
1702:
1537:
1372:
1342:
1322:
1302:
1219:
1079:
1032:
740:
693:
593:
456:
316:
3937:
made of only regular hexagons, because the hexagons
3811:
hexagon (one with all sides equal) with common side
3072:
2969:
2936:
with equilateral triangles on its edges, creating a
2924:, {12}, alternating two types (colors) of edges. An
403:, regular hexagons fit together without any gaps to
6517:
6463:
6403:
6347:
6286:
6277:
6169:
6121:
195:
169:
154:
137:
93:
83:
69:
59:
42:
3918:{\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}
3917:
3857:
3643:A regular skew hexagon seen as edges (black) of a
3599:
2043:
1925:
1895:
1687:
1522:
1355:
1328:
1308:
1254:
1202:
1062:
989:
720:
679:
462:
333:
5225:: CS1 maint: DOI inactive as of September 2024 (
3331:and these can tessellate 3-space by translation.
3199:Self-intersecting hexagons with regular vertices
2951:A regular hexagon can be extended into a regular
1004:, the area can also be expressed in terms of the
450:, Book IV, Proposition 15: this is possible as 6
3451:If the successive sides of a cyclic hexagon are
2895:{6}. A regular hexagon is a part of the regular
470:2 × 3, a product of a power of two and distinct
5100:: single path, six-sided star, within a hexagon
2749:parallelograms. In particular this is true for
5191:Communications in Mathematics and Applications
2940:. A regular hexagon can be dissected into six
568:. The minimal diameter or the diameter of the
6094:
5499:
1015:. For the regular hexagon these are given by
276:, t{3}, which alternates two types of edges.
8:
4920:mirror is composed of 18 hexagonal segments.
3941:, not allowing the result to "fold up". The
2963:around it. This pattern repeats within the
2902:A regular hexagon can also be created as a
6283:
6101:
6087:
6079:
5506:
5492:
5484:
5442:construction with compass and straightedge
4728:Gallery of natural and artificial hexagons
4508:
4461:
4347:
4279:
4039:
3312:to construct and gain much strength under
2771:
2303:labels these by a letter and group order.
721:{\displaystyle d={\frac {\sqrt {3}}{2}}D.}
246:. The total of the internal angles of any
242:, meaning "corner, angle") is a six-sided
5454:a website devoted to hexagon mathematics.
5202:
5150:, Cambridge University Press, p. 9,
5094:: six-sided star within a regular hexagon
3905:
3891:
3885:
3883:
3838:
3832:
3830:
3559:
2032:
2027:
2017:
2006:
1990:
1979:
1974:
1964:
1953:
1941:
1917:
1911:
1879:
1869:
1853:
1842:
1829:
1802:
1797:
1784:
1779:
1766:
1761:
1748:
1743:
1730:
1725:
1712:
1707:
1701:
1671:
1658:
1637:
1632:
1619:
1614:
1601:
1596:
1583:
1578:
1565:
1560:
1547:
1542:
1536:
1506:
1493:
1472:
1467:
1454:
1449:
1436:
1431:
1418:
1413:
1400:
1395:
1382:
1377:
1371:
1347:
1341:
1321:
1301:
1227:
1220:
1218:
1187:
1163:
1157:
1134:
1119:
1094:
1080:
1078:
1053:
1033:
1031:
974:
958:
935:
919:
896:
880:
861:
852:
835:
829:
813:
802:
778:
761:
755:
741:
739:
700:
692:
662:
644:
629:
594:
592:
455:
317:
315:
250:(non-self-intersecting) hexagon is 720°.
5325:, Dover Publications, 2007 (orig. 1960).
5180:
5178:
3858:{\displaystyle {\frac {d_{1}}{a}}\leq 2}
3750:
3699:
3638:
3333:
3327:with parallel opposite faces are called
3197:
2398:
2227:
2067:
6071:List of regular polytopes and compounds
5125:
4731:
1213:The regular hexagon fills the fraction
334:{\displaystyle {\tfrac {2}{\sqrt {3}}}}
5438:Definition and properties of a hexagon
5218:
3512:Let ABCDEF be a hexagon formed by six
39:
5448:An Introduction to Hexagonal Geometry
4865:composed of hexagonal aromatic rings.
4849:Hexagonal order of bubbles in a foam.
3508:Hexagon tangential to a conic section
269:{6} and can also be constructed as a
7:
5057:figure which, like the hexagon, has
4901:An aerial view of Fort Jefferson in
3815:, there exists a principal diagonal
3413:Hexagon inscribed in a conic section
5272:Cartensen, Jens, "About hexagons",
5243:, (2008) The Symmetries of Things,
4741:The ideal crystalline structure of
3973:. These hexagons can be considered
1063:{\displaystyle {}=6R=4r{\sqrt {3}}}
3945:with some hexagonal faces are the
2761:, with 3 of 6 square faces. Other
14:
4940:for its vaguely hexagonal shape.
3496:If a hexagon has vertices on the
2394:hexagon shapes can tile the plane
576:, is twice the minimal radius or
556:, is twice the maximal radius or
295:(has a circumscribed circle) and
5030:
5014:
4999:
4977:
4965:
4945:
4925:
4909:
4894:
4870:
4854:
4842:
4823:
4811:
4796:
4781:
4766:
4750:
4734:
4717:
4710:
4703:
4696:
4680:
4673:
4666:
4659:
4649:
4644:
4639:
4634:
4629:
4616:
4611:
4606:
4601:
4596:
4583:
4578:
4573:
4568:
4563:
4550:
4545:
4540:
4535:
4530:
4495:
4484:
4473:
4448:
4437:
4426:
4413:
4402:
4391:
4378:
4367:
4356:
4327:
4316:
4305:
4259:
4248:
4237:
4226:
4215:
4205:
4200:
4195:
4190:
4185:
4176:
4171:
4166:
4161:
4156:
4147:
4142:
4137:
4132:
4127:
4118:
4113:
4108:
4103:
4098:
4089:
4084:
4079:
4074:
4069:
4031:
4026:
4021:
4016:
4011:
4003:
3998:
3993:
3988:
3983:
3784:
3773:
3762:
3736:The regular skew hexagon is the
3717:
3706:
3384:
3377:
3365:
3358:
3271:
3262:
3253:
3242:
3233:
3224:
3125:
3118:
3111:
3104:
3097:
3090:
3083:
3076:
3022:
3015:
3008:
3001:
2994:
2987:
2980:
2973:
2932:, {3}. A regular hexagon can be
2864:
2857:
2850:
2843:
2812:
2805:
2798:
2791:
2708:
2701:
2694:
2666:
2661:
2656:
2635:
2630:
2625:
2602:
2597:
2592:
2583:
2575:
2570:
2565:
2556:
2504:
2495:
2486:
2477:
2468:
2459:
2448:
2190:
2177:
2164:
2149:
2140:
2127:
2116:
2103:
2090:
2073:
564:, which equals the side length,
479:
429:
363:rotational symmetry of order six
128:
123:
118:
110:
105:
100:
48:
6641:Polygons by the number of sides
5440:with interactive animation and
5370:from the original on 2015-07-05
5304:from the original on 2014-12-05
5239:John H. Conway, Heidi Burgiel,
5164:from the original on 2016-01-02
4444:augmented truncated tetrahedron
4420:metabiaugmented hexagonal prism
4409:parabiaugmented hexagonal prism
4385:gyroelongated triangular cupola
4283:Hexagons in Goldberg polyhedra
3527:and that has consecutive sides
548:(which corresponds to the long
4512:Tilings with regular hexagons
3335:Hexagonal prism tessellations
2216:of a regular hexagon, with Dih
731:The area of a regular hexagon
635:
622:
1:
5185:Meskhishvili, Mamuka (2020).
5144:Wenninger, Magnus J. (1974),
4351:Johnson solids with hexagons
3740:for these higher dimensional
3701:Skew hexagons on 3-fold axes
3349:Hexagonal prismatic honeycomb
3142:A self-intersecting hexagon (
2396:with different orientations.
2062:Example hexagons by symmetry
1276:It follows from the ratio of
5286:Dergiades, Nikolaos (2014).
4455:triangular hebesphenorotunda
4433:triaugmented hexagonal prism
3600:{\displaystyle a+c+e=b+d+f.}
2887:Related polygons and tilings
357:. A regular hexagon has six
238:
226:
19:For the crystal system, see
5323:Advanced Euclidean Geometry
4374:elongated triangular cupola
4266:truncated icosidodecahedron
3971:truncated icosidodecahedron
2248:and no symmetry is labeled
1271:PE + PF = PA + PB + PC + PD
299:(has an inscribed circle).
6662:
6060:
5487:
5459:Hexagons are the Bestagons
5388:Inequalities proposed in "
4918:James Webb Space Telescope
4903:Dry Tortugas National Park
4511:
4471:
4464:
4424:
4389:
4354:
4350:
4282:
4213:
4042:
3799:Convex equilateral hexagon
3760:
3398:
3222:
2871:
2841:
2819:
2789:
2774:
2513:
2173:
2101:
2069:
291:, meaning that it is both
232:
220:
32:
25:
18:
5276:33(2) (2000–2001), 37–40.
4818:Micrograph of a snowflake
4788:The scutes of a turtle's
4688:
4519:
4398:augmented hexagonal prism
4060:
4055:
3868:and a principal diagonal
3753:
3516:of a conic section. Then
3209:
3203:
3041:Hypertruncated triangles
3040:
2965:rhombitrihexagonal tiling
2877:
2872:
2837:
2834:
2831:
2823:
2785:
2779:
2688:
2532:
2433:
2085:
47:
5072:Hexagonal crystal system
4958:hexagonal crystal system
3424:, and pairs of opposite
3395:Tesselations by hexagons
3292:Giant's Causeway closeup
3195:of the regular hexagon:
2236:for diagonal) or edges (
438:compass and straightedge
28:Hexagon (disambiguation)
21:Hexagonal crystal family
5213:10.26713/cma.v11i3.1420
4861:Crystal structure of a
4345:with regular hexagons:
4244:truncated cuboctahedron
3967:truncated cuboctahedron
3929:Polyhedra with hexagons
3686:, symmetry, order 12.
687: and, similarly,
95:Coxeter–Dynkin diagrams
6636:Constructible polygons
5354:Dao Thanh Oai (2015).
5344:, Accessed 2012-04-17.
5215:(inactive 2024-09-12).
5088:of hexagons in a plane
4334:Chamfered dodecahedron
3919:
3859:
3746:orthogonal projections
3652:
3620:
3601:
3525:tangential to a circle
3293:
3189:self-crossing hexagons
3183:Self-crossing hexagons
2955:by adding alternating
2891:A regular hexagon has
2307:is full symmetry, and
2296:) and the trivial (e)
2253:
2225:
2045:
2022:
1969:
1927:
1897:
1689:
1524:
1363:respectively, we have
1357:
1330:
1310:
1256:
1204:
1064:
991:
722:
681:
541:
464:
335:
5274:Mathematical Spectrum
5241:Chaim Goodman-Strauss
4956:crystal, one of many
4837:with hexagonal shape.
4312:Chamfered tetrahedron
4255:truncated icosahedron
4222:truncated tetrahedron
3955:truncated icosahedron
3947:truncated tetrahedron
3920:
3860:
3642:
3618:
3602:
3523:In a hexagon that is
3409:will tile the plane.
3291:
2961:equilateral triangles
2942:equilateral triangles
2928:hexagon, h{6}, is an
2646:Exceptional Lie group
2376:Hexagons of symmetry
2231:
2211:
2046:
2002:
1949:
1928:
1926:{\displaystyle d_{i}}
1898:
1690:
1525:
1358:
1356:{\displaystyle d_{i}}
1331:
1311:
1257:
1205:
1065:
992:
723:
682:
520:
494:M, the center of the
486:When the side length
465:
371:six lines of symmetry
367:reflection symmetries
359:rotational symmetries
336:
230:, meaning "six", and
6334:Nonagon/Enneagon (9)
6264:Tangential trapezoid
5115:Central place theory
5104:Honeycomb conjecture
5006:Władysław Gliński's
4745:is a hexagonal grid.
4233:truncated octahedron
3951:truncated octahedron
3882:
3829:
3680:triangular antiprism
3672:regular skew hexagon
3665:skew zig-zag hexagon
3651:, , (2*3), order 12.
3645:triangular antiprism
3625:equilateral triangle
3558:
3284:Hexagonal structures
3151:Central {6} in {12}
2930:equilateral triangle
2920:hexagon, t{6}, is a
2907:equilateral triangle
2689:12 rhomb dissection
2644:The 12 roots of the
1940:
1910:
1700:
1535:
1370:
1340:
1320:
1300:
1264:circumscribed circle
1217:
1077:
1030:
738:
691:
591:
496:circumscribed circle
454:
314:
304:circumscribed circle
274:equilateral triangle
35:Hexagonal (CONCACAF)
26:For other uses, see
16:Shape with six sides
6446:Megagon (1,000,000)
6214:Isosceles trapezoid
6055:pentagonal polytope
5954:Uniform 10-polytope
5514:Fundamental convex
5390:Crux Mathematicorum
5360:Forum Geometricorum
5321:Johnson, Roger A.,
5293:Forum Geometricorum
4938:Metropolitan France
4491:Hexagonal antiprism
4275:Goldberg polyhedron
3702:
3518:Brianchon's theorem
3336:
3200:
2767:rectangular cuboids
2651:, represented by a
2620:, represented by a
2613:The 6 roots of the
2224:symmetry, order 12.
2037:
1984:
1807:
1789:
1771:
1753:
1735:
1717:
1642:
1624:
1606:
1588:
1570:
1552:
1477:
1459:
1441:
1423:
1405:
1387:
6416:Icositetragon (24)
5924:Uniform 9-polytope
5874:Uniform 8-polytope
5824:Uniform 7-polytope
5781:Uniform 6-polytope
5751:Uniform 5-polytope
5711:Uniform polychoron
5674:Uniform polyhedron
5522:in dimensions 2–10
5473:about hexagons by
5419:Weisstein, Eric W.
5400:2017-08-30 at the
5340:2012-05-11 at the
5098:Unicursal hexagram
4993:Reading, Berkshire
4045:Archimedean solids
3943:Archimedean solids
3915:
3855:
3805:principal diagonal
3700:
3653:
3621:
3597:
3426:sides are extended
3334:
3294:
3198:
3193:vertex arrangement
3139:A concave hexagon
2879:Rectangular cuboid
2838:Rectangular faces
2721:states that every
2254:
2226:
2041:
2023:
1970:
1923:
1893:
1793:
1775:
1757:
1739:
1721:
1703:
1685:
1628:
1610:
1592:
1574:
1556:
1538:
1520:
1463:
1445:
1427:
1409:
1391:
1373:
1353:
1326:
1306:
1252:
1244:
1200:
1198:
1060:
1011:and the perimeter
987:
985:
718:
677:
542:
460:
331:
329:
6646:Elementary shapes
6618:
6617:
6459:
6458:
6436:Myriagon (10,000)
6421:Triacontagon (30)
6385:Heptadecagon (17)
6375:Pentadecagon (15)
6370:Tetradecagon (14)
6309:Quadrilateral (4)
6179:Antiparallelogram
6076:
6075:
6063:Polytope families
5520:uniform polytopes
5249:978-1-56881-220-5
5147:Polyhedron Models
5025:Botanical Gardens
4877:Naturally formed
4863:molecular hexagon
4835:aromatic compound
4725:
4724:
4690:2-uniform tilings
4507:
4506:
4502:Hexagonal pyramid
4460:
4459:
4363:triangular cupola
4341:There are also 9
4339:
4338:
4271:
4270:
3910:
3900:
3847:
3796:
3795:
3729:
3728:
3676:vertex-transitive
3392:
3391:
3281:
3280:
3180:
3179:
3071:
3070:
2946:triangular tiling
2884:
2883:
2716:
2715:
2611:
2610:
2546:
2545:
2268:), 2 dihedral: (D
2212:The six lines of
2206:
2205:
2202:
2201:
1329:{\displaystyle L}
1309:{\displaystyle R}
1243:
1232:
1168:
1145:
1139:
1107:
1058:
890:
886:
869:
846:
840:
807:
772:
766:
710:
706:
672:
668:
654:
650:
602:
552:of the hexagon),
463:{\displaystyle =}
411:. The cells of a
373:), making up the
328:
327:
205:
204:
54:A regular hexagon
6653:
6431:Chiliagon (1000)
6411:Icositrigon (23)
6390:Octadecagon (18)
6380:Hexadecagon (16)
6284:
6103:
6096:
6089:
6080:
6067:Regular polytope
5628:
5617:
5606:
5565:
5508:
5501:
5494:
5485:
5460:
5432:
5431:
5405:
5385:
5379:
5378:
5376:
5375:
5351:
5345:
5332:
5326:
5319:
5313:
5312:
5310:
5309:
5283:
5277:
5270:
5264:
5258:
5252:
5237:
5231:
5230:
5224:
5216:
5206:
5182:
5173:
5171:
5170:
5169:
5141:
5135:
5130:
5082:Hexagonal tiling
5077:Hexagonal number
5065:and tessellates
5055:four-dimensional
5038:Hexagonal window
5034:
5021:Pavilion in the
5018:
5003:
4981:
4969:
4949:
4929:
4913:
4898:
4887:Northern Ireland
4883:Giant's Causeway
4874:
4858:
4846:
4827:
4815:
4804:Saturn's hexagon
4800:
4785:
4770:
4754:
4738:
4721:
4714:
4707:
4700:
4684:
4677:
4670:
4663:
4654:
4653:
4652:
4648:
4647:
4643:
4642:
4638:
4637:
4633:
4632:
4621:
4620:
4619:
4615:
4614:
4610:
4609:
4605:
4604:
4600:
4599:
4588:
4587:
4586:
4582:
4581:
4577:
4576:
4572:
4571:
4567:
4566:
4555:
4554:
4553:
4549:
4548:
4544:
4543:
4539:
4538:
4534:
4533:
4509:
4499:
4488:
4477:
4462:
4452:
4441:
4430:
4417:
4406:
4395:
4382:
4371:
4360:
4348:
4331:
4320:
4309:
4280:
4263:
4252:
4241:
4230:
4219:
4210:
4209:
4208:
4204:
4203:
4199:
4198:
4194:
4193:
4189:
4188:
4181:
4180:
4179:
4175:
4174:
4170:
4169:
4165:
4164:
4160:
4159:
4152:
4151:
4150:
4146:
4145:
4141:
4140:
4136:
4135:
4131:
4130:
4123:
4122:
4121:
4117:
4116:
4112:
4111:
4107:
4106:
4102:
4101:
4094:
4093:
4092:
4088:
4087:
4083:
4082:
4078:
4077:
4073:
4072:
4040:
4036:
4035:
4034:
4030:
4029:
4025:
4024:
4020:
4019:
4015:
4014:
4008:
4007:
4006:
4002:
4001:
3997:
3996:
3992:
3991:
3987:
3986:
3979:Coxeter diagrams
3977:triangles, with
3924:
3922:
3921:
3916:
3911:
3906:
3901:
3896:
3895:
3886:
3864:
3862:
3861:
3856:
3848:
3843:
3842:
3833:
3788:
3777:
3766:
3751:
3721:
3710:
3703:
3606:
3604:
3603:
3598:
3484:
3418:Pascal's theorem
3407:Conway criterion
3401:Hexagonal tiling
3388:
3381:
3369:
3362:
3344:Hexagonal tiling
3337:
3329:parallelohedrons
3325:hexagonal prisms
3302:Giant's Causeway
3275:
3266:
3257:
3246:
3237:
3228:
3201:
3129:
3122:
3115:
3108:
3101:
3094:
3087:
3080:
3073:
3026:
3019:
3012:
3005:
2998:
2991:
2984:
2977:
2970:
2897:hexagonal tiling
2868:
2861:
2854:
2847:
2816:
2809:
2802:
2795:
2772:
2757:projection of a
2751:regular polygons
2748:
2739:
2738:
2734:
2712:
2705:
2698:
2680:
2671:
2670:
2669:
2665:
2664:
2660:
2659:
2640:
2639:
2638:
2634:
2633:
2629:
2628:
2615:simple Lie group
2607:
2606:
2605:
2601:
2600:
2596:
2595:
2587:
2580:
2579:
2578:
2574:
2573:
2569:
2568:
2560:
2553:
2549:A2 and G2 groups
2508:
2499:
2490:
2481:
2472:
2463:
2452:
2399:
2311:is no symmetry.
2194:
2181:
2168:
2153:
2144:
2131:
2120:
2107:
2094:
2077:
2068:
2059:
2050:
2048:
2047:
2042:
2036:
2031:
2021:
2016:
1995:
1994:
1989:
1985:
1983:
1978:
1968:
1963:
1932:
1930:
1929:
1924:
1922:
1921:
1902:
1900:
1899:
1894:
1889:
1885:
1884:
1883:
1874:
1873:
1858:
1857:
1852:
1848:
1847:
1846:
1834:
1833:
1806:
1801:
1788:
1783:
1770:
1765:
1752:
1747:
1734:
1729:
1716:
1711:
1694:
1692:
1691:
1686:
1681:
1677:
1676:
1675:
1663:
1662:
1641:
1636:
1623:
1618:
1605:
1600:
1587:
1582:
1569:
1564:
1551:
1546:
1529:
1527:
1526:
1521:
1516:
1512:
1511:
1510:
1498:
1497:
1476:
1471:
1458:
1453:
1440:
1435:
1422:
1417:
1404:
1399:
1386:
1381:
1362:
1360:
1359:
1354:
1352:
1351:
1335:
1333:
1332:
1327:
1315:
1313:
1312:
1307:
1272:
1261:
1259:
1258:
1253:
1245:
1242:
1234:
1233:
1228:
1222:
1209:
1207:
1206:
1201:
1199:
1192:
1191:
1173:
1169:
1164:
1162:
1161:
1146:
1141:
1140:
1135:
1120:
1112:
1108:
1103:
1095:
1069:
1067:
1066:
1061:
1059:
1054:
1034:
1000:For any regular
996:
994:
993:
988:
986:
979:
978:
963:
962:
944:
940:
939:
924:
923:
905:
901:
900:
891:
882:
881:
870:
862:
857:
856:
847:
842:
841:
836:
830:
822:
818:
817:
808:
803:
783:
782:
773:
768:
767:
762:
756:
727:
725:
724:
719:
711:
702:
701:
686:
684:
683:
678:
673:
664:
663:
655:
646:
645:
634:
633:
603:
595:
504:
489:
483:
469:
467:
466:
461:
433:
349:). All internal
347:inscribed circle
340:
338:
337:
332:
330:
323:
319:
241:
235:
234:
229:
223:
222:
133:
132:
131:
127:
126:
122:
121:
115:
114:
113:
109:
108:
104:
103:
52:
40:
6661:
6660:
6656:
6655:
6654:
6652:
6651:
6650:
6621:
6620:
6619:
6614:
6513:
6467:
6455:
6399:
6365:Tridecagon (13)
6355:Hendecagon (11)
6343:
6279:
6273:
6244:Right trapezoid
6165:
6117:
6107:
6077:
6046:
6039:
6032:
5915:
5908:
5901:
5865:
5858:
5851:
5815:
5808:
5642:Regular polygon
5635:
5626:
5619:
5615:
5608:
5604:
5595:
5586:
5579:
5575:
5563:
5557:
5553:
5541:
5523:
5512:
5481:
5458:
5417:
5416:
5413:
5408:
5402:Wayback Machine
5386:
5382:
5373:
5371:
5353:
5352:
5348:
5342:Wayback Machine
5333:
5329:
5320:
5316:
5307:
5305:
5285:
5284:
5280:
5271:
5267:
5259:
5255:
5238:
5234:
5217:
5184:
5183:
5176:
5167:
5165:
5158:
5143:
5142:
5138:
5131:
5127:
5123:
5067:Euclidean space
5047:
5040:
5035:
5026:
5019:
5010:
5008:hexagonal chess
5004:
4995:
4982:
4973:
4970:
4961:
4950:
4941:
4930:
4921:
4914:
4905:
4899:
4890:
4875:
4866:
4859:
4850:
4847:
4838:
4833:, the simplest
4828:
4819:
4816:
4807:
4801:
4792:
4786:
4777:
4771:
4762:
4761:mirror segments
4755:
4746:
4739:
4730:
4650:
4645:
4640:
4635:
4630:
4628:
4627:
4617:
4612:
4607:
4602:
4597:
4595:
4594:
4584:
4579:
4574:
4569:
4564:
4562:
4561:
4551:
4546:
4541:
4536:
4531:
4529:
4528:
4500:
4489:
4480:Hexagonal prism
4478:
4453:
4442:
4431:
4418:
4407:
4396:
4383:
4372:
4361:
4332:
4321:
4310:
4264:
4253:
4242:
4231:
4220:
4206:
4201:
4196:
4191:
4186:
4184:
4177:
4172:
4167:
4162:
4157:
4155:
4148:
4143:
4138:
4133:
4128:
4126:
4119:
4114:
4109:
4104:
4099:
4097:
4090:
4085:
4080:
4075:
4070:
4068:
4032:
4027:
4022:
4017:
4012:
4010:
4004:
3999:
3994:
3989:
3984:
3982:
3931:
3887:
3880:
3879:
3874:
3834:
3827:
3826:
3821:
3801:
3789:
3778:
3767:
3734:
3732:Petrie polygons
3722:
3711:
3685:
3682:with the same D
3650:
3637:
3613:
3556:
3555:
3510:
3476:
3446:symmedian point
3438:Lemoine hexagon
3434:
3415:
3403:
3397:
3374:Parallelogonal
3286:
3276:
3267:
3258:
3247:
3238:
3229:
3219:
3213:
3207:
3185:
3168:
3150:
3135:
3063:
3055:
3045:
3037:
3032:
2912:
2893:Schläfli symbol
2889:
2786:Parallelograms
2736:
2732:
2731:
2730:
2678:
2667:
2662:
2657:
2655:
2636:
2631:
2626:
2624:
2603:
2598:
2593:
2591:
2590:
2588:
2576:
2571:
2566:
2564:
2563:
2561:
2551:
2542:
2536:
2530:
2524:
2518:
2509:
2500:
2491:
2482:
2473:
2464:
2453:
2295:
2291:
2287:
2283:
2275:
2271:
2267:
2263:
2258:regular hexagon
2219:
2195:
2182:
2169:
2158:
2156:
2154:
2145:
2134:
2132:
2123:
2121:
2110:
2108:
2095:
2080:
2078:
2057:
1948:
1944:
1943:
1938:
1937:
1913:
1908:
1907:
1875:
1865:
1838:
1825:
1824:
1820:
1819:
1818:
1814:
1698:
1697:
1667:
1654:
1653:
1649:
1533:
1532:
1502:
1489:
1488:
1484:
1368:
1367:
1343:
1338:
1337:
1318:
1317:
1298:
1297:
1294:
1270:
1235:
1223:
1215:
1214:
1197:
1196:
1183:
1171:
1170:
1153:
1121:
1110:
1109:
1096:
1087:
1075:
1074:
1028:
1027:
984:
983:
970:
954:
942:
941:
931:
915:
903:
902:
892:
848:
831:
820:
819:
809:
774:
757:
748:
736:
735:
689:
688:
625:
589:
588:
515:
510:
509:
508:
507:
506:
502:
498:. Transfer the
487:
484:
476:
475:
452:
451:
434:
420:Voronoi diagram
380:
345:(radius of the
312:
311:
310:, which equals
267:Schläfli symbol
256:
254:Regular hexagon
149:
129:
124:
119:
117:
116:
111:
106:
101:
99:
85:Schläfli symbol
64:Regular polygon
55:
43:Regular hexagon
38:
31:
24:
17:
12:
11:
5:
6659:
6657:
6649:
6648:
6643:
6638:
6633:
6623:
6622:
6616:
6615:
6613:
6612:
6607:
6602:
6597:
6592:
6587:
6582:
6577:
6572:
6570:Pseudotriangle
6567:
6562:
6557:
6552:
6547:
6542:
6537:
6532:
6527:
6521:
6519:
6515:
6514:
6512:
6511:
6506:
6501:
6496:
6491:
6486:
6481:
6476:
6470:
6468:
6461:
6460:
6457:
6456:
6454:
6453:
6448:
6443:
6438:
6433:
6428:
6423:
6418:
6413:
6407:
6405:
6401:
6400:
6398:
6397:
6392:
6387:
6382:
6377:
6372:
6367:
6362:
6360:Dodecagon (12)
6357:
6351:
6349:
6345:
6344:
6342:
6341:
6336:
6331:
6326:
6321:
6316:
6311:
6306:
6301:
6296:
6290:
6288:
6281:
6275:
6274:
6272:
6271:
6266:
6261:
6256:
6251:
6246:
6241:
6236:
6231:
6226:
6221:
6216:
6211:
6206:
6201:
6196:
6191:
6186:
6181:
6175:
6173:
6171:Quadrilaterals
6167:
6166:
6164:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6127:
6125:
6119:
6118:
6108:
6106:
6105:
6098:
6091:
6083:
6074:
6073:
6058:
6057:
6048:
6044:
6037:
6030:
6026:
6017:
6000:
5991:
5980:
5979:
5977:
5975:
5970:
5961:
5956:
5950:
5949:
5947:
5945:
5940:
5931:
5926:
5920:
5919:
5917:
5913:
5906:
5899:
5895:
5890:
5881:
5876:
5870:
5869:
5867:
5863:
5856:
5849:
5845:
5840:
5831:
5826:
5820:
5819:
5817:
5813:
5806:
5802:
5797:
5788:
5783:
5777:
5776:
5774:
5772:
5767:
5758:
5753:
5747:
5746:
5737:
5732:
5727:
5718:
5713:
5707:
5706:
5697:
5695:
5690:
5681:
5676:
5670:
5669:
5664:
5659:
5654:
5649:
5644:
5638:
5637:
5633:
5629:
5624:
5613:
5602:
5593:
5584:
5577:
5571:
5561:
5555:
5549:
5543:
5537:
5531:
5525:
5524:
5513:
5511:
5510:
5503:
5496:
5488:
5483:
5479:
5478:
5471:internet video
5455:
5445:
5434:
5433:
5412:
5411:External links
5409:
5407:
5406:
5380:
5346:
5327:
5314:
5278:
5265:
5253:
5232:
5174:
5156:
5136:
5124:
5122:
5119:
5118:
5117:
5112:
5106:
5101:
5095:
5089:
5086:regular tiling
5079:
5074:
5069:
5046:
5043:
5042:
5041:
5036:
5029:
5027:
5020:
5013:
5011:
5005:
4998:
4996:
4987:, a hexagonal
4983:
4976:
4974:
4972:Hexagonal barn
4971:
4964:
4962:
4951:
4944:
4942:
4931:
4924:
4922:
4915:
4908:
4906:
4900:
4893:
4891:
4876:
4869:
4867:
4860:
4853:
4851:
4848:
4841:
4839:
4829:
4822:
4820:
4817:
4810:
4808:
4802:
4795:
4793:
4787:
4780:
4778:
4772:
4765:
4763:
4756:
4749:
4747:
4740:
4733:
4729:
4726:
4723:
4722:
4715:
4708:
4701:
4693:
4692:
4686:
4685:
4678:
4671:
4664:
4656:
4655:
4622:
4589:
4556:
4522:
4521:
4518:
4514:
4513:
4505:
4504:
4493:
4482:
4470:
4469:
4468:with hexagons
4458:
4457:
4446:
4435:
4423:
4422:
4411:
4400:
4388:
4387:
4376:
4365:
4353:
4352:
4343:Johnson solids
4337:
4336:
4325:
4323:Chamfered cube
4314:
4302:
4301:
4296:
4291:
4285:
4284:
4269:
4268:
4257:
4246:
4235:
4224:
4212:
4211:
4182:
4153:
4124:
4095:
4065:
4064:
4059:
4054:
4048:
4047:
3935:Platonic solid
3930:
3927:
3926:
3925:
3914:
3909:
3904:
3899:
3894:
3890:
3872:
3866:
3865:
3854:
3851:
3846:
3841:
3837:
3819:
3800:
3797:
3794:
3793:
3782:
3780:3-3 duopyramid
3771:
3759:
3758:
3755:
3738:Petrie polygon
3733:
3730:
3727:
3726:
3715:
3683:
3648:
3636:
3633:
3612:
3609:
3608:
3607:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3509:
3506:
3502:acute triangle
3433:
3432:Cyclic hexagon
3430:
3414:
3411:
3399:Main article:
3396:
3393:
3390:
3389:
3382:
3375:
3371:
3370:
3363:
3356:
3352:
3351:
3346:
3341:
3306:hexagonal grid
3285:
3282:
3279:
3278:
3269:
3260:
3251:
3240:
3231:
3221:
3220:
3217:
3214:
3211:
3208:
3205:
3187:There are six
3184:
3181:
3178:
3177:
3175:Complete graph
3172:
3165:
3164:Dissected {6}
3162:
3152:
3147:
3140:
3137:
3131:
3130:
3123:
3116:
3109:
3102:
3095:
3088:
3081:
3069:
3068:
3060:
3052:
3042:
3039:
3034:
3028:
3027:
3020:
3013:
3006:
2999:
2992:
2985:
2978:
2910:
2888:
2885:
2882:
2881:
2876:
2870:
2869:
2862:
2855:
2848:
2840:
2839:
2836:
2833:
2829:
2828:
2822:
2818:
2817:
2810:
2803:
2796:
2788:
2787:
2784:
2781:
2777:
2776:
2755:Petrie polygon
2714:
2713:
2706:
2699:
2691:
2690:
2687:
2677:
2674:
2653:Dynkin diagram
2622:Dynkin diagram
2609:
2608:
2589:G2 group roots
2581:
2562:A2 group roots
2550:
2547:
2544:
2543:
2540:
2537:
2534:
2531:
2528:
2525:
2522:
2519:
2516:
2512:
2511:
2502:
2493:
2484:
2475:
2466:
2457:
2445:
2444:
2438:
2432:
2422:
2416:
2410:
2371:directed edges
2293:
2289:
2285:
2281:
2273:
2269:
2265:
2261:
2217:
2204:
2203:
2200:
2199:
2197:
2188:
2186:
2184:
2175:
2172:
2171:
2162:
2147:
2138:
2125:
2114:
2100:
2099:
2097:
2088:
2086:
2084:
2082:
2071:
2064:
2063:
2056:
2053:
2052:
2051:
2040:
2035:
2030:
2026:
2020:
2015:
2012:
2009:
2005:
2001:
1998:
1993:
1988:
1982:
1977:
1973:
1967:
1962:
1959:
1956:
1952:
1947:
1920:
1916:
1904:
1903:
1892:
1888:
1882:
1878:
1872:
1868:
1864:
1861:
1856:
1851:
1845:
1841:
1837:
1832:
1828:
1823:
1817:
1813:
1810:
1805:
1800:
1796:
1792:
1787:
1782:
1778:
1774:
1769:
1764:
1760:
1756:
1751:
1746:
1742:
1738:
1733:
1728:
1724:
1720:
1715:
1710:
1706:
1695:
1684:
1680:
1674:
1670:
1666:
1661:
1657:
1652:
1648:
1645:
1640:
1635:
1631:
1627:
1622:
1617:
1613:
1609:
1604:
1599:
1595:
1591:
1586:
1581:
1577:
1573:
1568:
1563:
1559:
1555:
1550:
1545:
1541:
1530:
1519:
1515:
1509:
1505:
1501:
1496:
1492:
1487:
1483:
1480:
1475:
1470:
1466:
1462:
1457:
1452:
1448:
1444:
1439:
1434:
1430:
1426:
1421:
1416:
1412:
1408:
1403:
1398:
1394:
1390:
1385:
1380:
1376:
1350:
1346:
1325:
1305:
1293:
1292:Point in plane
1290:
1251:
1248:
1241:
1238:
1231:
1226:
1211:
1210:
1195:
1190:
1186:
1182:
1179:
1176:
1174:
1172:
1167:
1160:
1156:
1152:
1149:
1144:
1138:
1133:
1130:
1127:
1124:
1118:
1115:
1113:
1111:
1106:
1102:
1099:
1093:
1090:
1088:
1086:
1083:
1082:
1057:
1052:
1049:
1046:
1043:
1040:
1037:
998:
997:
982:
977:
973:
969:
966:
961:
957:
953:
950:
947:
945:
943:
938:
934:
930:
927:
922:
918:
914:
911:
908:
906:
904:
899:
895:
889:
885:
879:
876:
873:
868:
865:
860:
855:
851:
845:
839:
834:
828:
825:
823:
821:
816:
812:
806:
801:
798:
795:
792:
789:
786:
781:
777:
771:
765:
760:
754:
751:
749:
747:
744:
743:
729:
728:
717:
714:
709:
705:
699:
696:
676:
671:
667:
661:
658:
653:
649:
643:
640:
637:
632:
628:
624:
621:
618:
615:
612:
609:
606:
601:
598:
514:
511:
485:
478:
477:
459:
435:
428:
427:
426:
425:
424:
405:tile the plane
378:
375:dihedral group
326:
322:
255:
252:
203:
202:
199:
193:
192:
171:
167:
166:
163:
156:Internal angle
152:
151:
147:
141:
139:Symmetry group
135:
134:
97:
91:
90:
87:
81:
80:
77:
67:
66:
61:
57:
56:
53:
45:
44:
15:
13:
10:
9:
6:
4:
3:
2:
6658:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6629:
6628:
6626:
6611:
6610:Weakly simple
6608:
6606:
6603:
6601:
6598:
6596:
6593:
6591:
6588:
6586:
6583:
6581:
6578:
6576:
6573:
6571:
6568:
6566:
6563:
6561:
6558:
6556:
6553:
6551:
6550:Infinite skew
6548:
6546:
6543:
6541:
6538:
6536:
6533:
6531:
6528:
6526:
6523:
6522:
6520:
6516:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6471:
6469:
6466:
6465:Star polygons
6462:
6452:
6451:Apeirogon (∞)
6449:
6447:
6444:
6442:
6439:
6437:
6434:
6432:
6429:
6427:
6424:
6422:
6419:
6417:
6414:
6412:
6409:
6408:
6406:
6402:
6396:
6395:Icosagon (20)
6393:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6352:
6350:
6346:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6291:
6289:
6285:
6282:
6276:
6270:
6267:
6265:
6262:
6260:
6257:
6255:
6252:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6230:
6229:Parallelogram
6227:
6225:
6224:Orthodiagonal
6222:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6204:Ex-tangential
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6180:
6177:
6176:
6174:
6172:
6168:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6128:
6126:
6124:
6120:
6115:
6111:
6104:
6099:
6097:
6092:
6090:
6085:
6084:
6081:
6072:
6068:
6064:
6059:
6056:
6052:
6049:
6047:
6040:
6033:
6027:
6025:
6021:
6018:
6016:
6012:
6008:
6004:
6001:
5999:
5995:
5992:
5990:
5986:
5982:
5981:
5978:
5976:
5974:
5971:
5969:
5965:
5962:
5960:
5957:
5955:
5952:
5951:
5948:
5946:
5944:
5941:
5939:
5935:
5932:
5930:
5927:
5925:
5922:
5921:
5918:
5916:
5909:
5902:
5896:
5894:
5891:
5889:
5885:
5882:
5880:
5877:
5875:
5872:
5871:
5868:
5866:
5859:
5852:
5846:
5844:
5841:
5839:
5835:
5832:
5830:
5827:
5825:
5822:
5821:
5818:
5816:
5809:
5803:
5801:
5798:
5796:
5792:
5789:
5787:
5784:
5782:
5779:
5778:
5775:
5773:
5771:
5768:
5766:
5762:
5759:
5757:
5754:
5752:
5749:
5748:
5745:
5741:
5738:
5736:
5733:
5731:
5730:Demitesseract
5728:
5726:
5722:
5719:
5717:
5714:
5712:
5709:
5708:
5705:
5701:
5698:
5696:
5694:
5691:
5689:
5685:
5682:
5680:
5677:
5675:
5672:
5671:
5668:
5665:
5663:
5660:
5658:
5655:
5653:
5650:
5648:
5645:
5643:
5640:
5639:
5636:
5630:
5627:
5623:
5616:
5612:
5605:
5601:
5596:
5592:
5587:
5583:
5578:
5576:
5574:
5570:
5560:
5556:
5554:
5552:
5548:
5544:
5542:
5540:
5536:
5532:
5530:
5527:
5526:
5521:
5517:
5509:
5504:
5502:
5497:
5495:
5490:
5489:
5486:
5482:
5476:
5472:
5469:
5465:
5461:
5456:
5453:
5449:
5446:
5443:
5439:
5436:
5435:
5429:
5428:
5423:
5420:
5415:
5414:
5410:
5403:
5399:
5396:
5393:
5391:
5384:
5381:
5369:
5365:
5361:
5357:
5350:
5347:
5343:
5339:
5336:
5331:
5328:
5324:
5318:
5315:
5303:
5299:
5295:
5294:
5289:
5282:
5279:
5275:
5269:
5266:
5262:
5257:
5254:
5250:
5246:
5242:
5236:
5233:
5228:
5222:
5214:
5210:
5205:
5200:
5196:
5192:
5188:
5181:
5179:
5175:
5163:
5159:
5157:9780521098595
5153:
5149:
5148:
5140:
5137:
5134:
5129:
5126:
5120:
5116:
5113:
5110:
5107:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5064:
5060:
5056:
5052:
5049:
5048:
5044:
5039:
5033:
5028:
5024:
5017:
5012:
5009:
5002:
4997:
4994:
4990:
4986:
4980:
4975:
4968:
4963:
4959:
4955:
4948:
4943:
4939:
4935:
4928:
4923:
4919:
4912:
4907:
4904:
4897:
4892:
4888:
4884:
4881:columns from
4880:
4873:
4868:
4864:
4857:
4852:
4845:
4840:
4836:
4832:
4826:
4821:
4814:
4809:
4805:
4799:
4794:
4791:
4784:
4779:
4776:
4769:
4764:
4760:
4753:
4748:
4744:
4737:
4732:
4727:
4720:
4716:
4713:
4709:
4706:
4702:
4699:
4695:
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4691:
4687:
4683:
4679:
4676:
4672:
4669:
4665:
4662:
4658:
4657:
4626:
4623:
4593:
4590:
4560:
4557:
4527:
4524:
4523:
4516:
4515:
4510:
4503:
4498:
4494:
4492:
4487:
4483:
4481:
4476:
4472:
4467:
4463:
4456:
4451:
4447:
4445:
4440:
4436:
4434:
4429:
4425:
4421:
4416:
4412:
4410:
4405:
4401:
4399:
4394:
4390:
4386:
4381:
4377:
4375:
4370:
4366:
4364:
4359:
4355:
4349:
4346:
4344:
4335:
4330:
4326:
4324:
4319:
4315:
4313:
4308:
4304:
4303:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4281:
4278:
4276:
4267:
4262:
4258:
4256:
4251:
4247:
4245:
4240:
4236:
4234:
4229:
4225:
4223:
4218:
4214:
4183:
4154:
4125:
4096:
4067:
4066:
4063:
4058:
4053:
4050:
4049:
4046:
4041:
4038:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3948:
3944:
3940:
3936:
3928:
3912:
3907:
3902:
3897:
3892:
3888:
3878:
3877:
3876:
3871:
3852:
3849:
3844:
3839:
3835:
3825:
3824:
3823:
3818:
3814:
3810:
3806:
3798:
3792:
3787:
3783:
3781:
3776:
3772:
3770:
3765:
3761:
3756:
3752:
3749:
3747:
3743:
3739:
3731:
3725:
3720:
3716:
3714:
3709:
3705:
3704:
3698:
3696:
3692:
3687:
3681:
3677:
3673:
3668:
3666:
3662:
3658:
3646:
3641:
3634:
3632:
3630:
3626:
3617:
3610:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3554:
3553:
3552:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3521:
3519:
3515:
3514:tangent lines
3507:
3505:
3503:
3499:
3494:
3492:
3486:
3483:
3479:
3474:
3470:
3466:
3462:
3458:
3454:
3449:
3447:
3443:
3439:
3431:
3429:
3427:
3423:
3422:conic section
3419:
3412:
3410:
3408:
3402:
3394:
3387:
3383:
3380:
3376:
3373:
3372:
3368:
3364:
3361:
3357:
3354:
3353:
3350:
3347:
3345:
3342:
3339:
3338:
3332:
3330:
3326:
3322:
3321:parallelogons
3317:
3315:
3311:
3307:
3303:
3299:
3290:
3283:
3274:
3270:
3265:
3261:
3256:
3252:
3250:
3245:
3241:
3236:
3232:
3230:Figure-eight
3227:
3223:
3215:
3202:
3196:
3194:
3190:
3182:
3176:
3173:
3171:
3166:
3163:
3161:
3157:
3153:
3148:
3145:
3141:
3138:
3133:
3132:
3128:
3124:
3121:
3117:
3114:
3110:
3107:
3103:
3100:
3096:
3093:
3089:
3086:
3082:
3079:
3075:
3074:
3067:
3061:
3059:
3053:
3051:
3048:
3043:
3035:
3030:
3029:
3025:
3021:
3018:
3014:
3011:
3007:
3004:
3000:
2997:
2993:
2990:
2986:
2983:
2979:
2976:
2972:
2971:
2968:
2966:
2962:
2958:
2954:
2949:
2947:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2914:
2908:
2905:
2900:
2898:
2894:
2886:
2880:
2875:
2867:
2863:
2860:
2856:
2853:
2849:
2846:
2842:
2835:Square faces
2830:
2827:
2826:parallelogons
2820:
2815:
2811:
2808:
2804:
2801:
2797:
2794:
2790:
2782:
2778:
2773:
2770:
2768:
2764:
2763:parallelogons
2760:
2756:
2752:
2746:
2742:
2728:
2724:
2720:
2711:
2707:
2704:
2700:
2697:
2693:
2692:
2685:
2682:
2681:
2675:
2673:
2654:
2650:
2647:
2642:
2623:
2619:
2616:
2586:
2582:
2559:
2555:
2554:
2548:
2538:
2526:
2520:
2514:
2507:
2503:
2498:
2494:
2489:
2485:
2480:
2476:
2471:
2467:
2462:
2458:
2456:
2451:
2447:
2446:
2442:
2439:
2436:
2430:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2404:
2401:
2400:
2397:
2395:
2391:
2390:parallelogons
2387:
2383:
2379:
2374:
2372:
2368:
2363:
2361:
2360:parallelogons
2357:
2353:
2349:
2345:
2341:
2338:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2297:
2279:
2259:
2251:
2247:
2243:
2239:
2235:
2230:
2223:
2215:
2210:
2198:
2193:
2189:
2187:
2185:
2180:
2176:
2174:
2167:
2163:
2161:
2152:
2148:
2143:
2139:
2137:
2130:
2126:
2119:
2115:
2113:
2106:
2102:
2098:
2093:
2089:
2087:
2083:
2076:
2072:
2070:
2066:
2065:
2061:
2060:
2054:
2038:
2033:
2028:
2024:
2018:
2013:
2010:
2007:
2003:
1999:
1996:
1991:
1986:
1980:
1975:
1971:
1965:
1960:
1957:
1954:
1950:
1945:
1936:
1935:
1934:
1918:
1914:
1890:
1886:
1880:
1876:
1870:
1866:
1862:
1859:
1854:
1849:
1843:
1839:
1835:
1830:
1826:
1821:
1815:
1811:
1808:
1803:
1798:
1794:
1790:
1785:
1780:
1776:
1772:
1767:
1762:
1758:
1754:
1749:
1744:
1740:
1736:
1731:
1726:
1722:
1718:
1713:
1708:
1704:
1696:
1682:
1678:
1672:
1668:
1664:
1659:
1655:
1650:
1646:
1643:
1638:
1633:
1629:
1625:
1620:
1615:
1611:
1607:
1602:
1597:
1593:
1589:
1584:
1579:
1575:
1571:
1566:
1561:
1557:
1553:
1548:
1543:
1539:
1531:
1517:
1513:
1507:
1503:
1499:
1494:
1490:
1485:
1481:
1478:
1473:
1468:
1464:
1460:
1455:
1450:
1446:
1442:
1437:
1432:
1428:
1424:
1419:
1414:
1410:
1406:
1401:
1396:
1392:
1388:
1383:
1378:
1374:
1366:
1365:
1364:
1348:
1344:
1323:
1303:
1291:
1289:
1287:
1283:
1279:
1274:
1267:
1265:
1249:
1246:
1239:
1236:
1229:
1224:
1193:
1188:
1184:
1180:
1177:
1175:
1165:
1158:
1154:
1150:
1147:
1142:
1136:
1131:
1128:
1125:
1122:
1116:
1114:
1104:
1100:
1097:
1091:
1089:
1084:
1073:
1072:
1071:
1055:
1050:
1047:
1044:
1041:
1038:
1035:
1026:
1022:
1018:
1014:
1010:
1007:
1003:
980:
975:
971:
967:
964:
959:
955:
951:
948:
946:
936:
932:
928:
925:
920:
916:
912:
909:
907:
897:
893:
887:
883:
877:
874:
871:
866:
863:
858:
853:
849:
843:
837:
832:
826:
824:
814:
810:
804:
799:
796:
793:
790:
787:
784:
779:
775:
769:
763:
758:
752:
750:
745:
734:
733:
732:
715:
712:
707:
703:
697:
694:
674:
669:
665:
659:
656:
651:
647:
641:
638:
630:
626:
619:
616:
613:
610:
607:
604:
599:
596:
587:
586:
585:
583:
579:
575:
571:
567:
563:
559:
555:
551:
547:
540:= side length
539:
535:
531:
527:
523:
519:
512:
501:
497:
493:
482:
473:
472:Fermat primes
457:
449:
448:
443:
439:
432:
423:
421:
417:
414:
410:
409:tessellations
406:
402:
399:
395:
390:
388:
384:
376:
372:
368:
364:
360:
356:
352:
348:
344:
324:
320:
309:
305:
300:
298:
294:
290:
286:
282:
277:
275:
272:
268:
264:
262:
253:
251:
249:
245:
240:
228:
218:
214:
210:
200:
198:
194:
191:
187:
183:
179:
175:
172:
168:
164:
161:
157:
153:
145:
142:
140:
136:
98:
96:
92:
88:
86:
82:
78:
76:
72:
68:
65:
62:
58:
51:
46:
41:
36:
29:
22:
6404:>20 sides
6339:Decagon (10)
6324:Heptagon (7)
6318:
6314:Pentagon (5)
6304:Triangle (3)
6199:Equidiagonal
6050:
6019:
6010:
6002:
5993:
5984:
5964:10-orthoplex
5700:Dodecahedron
5661:
5621:
5610:
5599:
5590:
5581:
5572:
5568:
5558:
5550:
5546:
5538:
5534:
5480:
5425:
5387:
5383:
5372:. Retrieved
5363:
5359:
5349:
5330:
5322:
5317:
5306:. Retrieved
5297:
5291:
5281:
5273:
5268:
5256:
5235:
5221:cite journal
5194:
5190:
5166:, retrieved
5146:
5139:
5133:Cube picture
5128:
4933:
4340:
4272:
4043:Hexagons in
3981:of the form
3933:There is no
3932:
3869:
3867:
3816:
3812:
3804:
3802:
3769:3-3 duoprism
3735:
3688:
3671:
3669:
3664:
3661:skew polygon
3657:skew hexagon
3656:
3654:
3647:, symmetry D
3635:Skew hexagon
3622:
3548:
3544:
3540:
3536:
3532:
3528:
3522:
3511:
3498:circumcircle
3495:
3487:
3481:
3477:
3472:
3468:
3464:
3460:
3456:
3452:
3450:
3435:
3416:
3404:
3318:
3295:
3277:Triple-tail
3268:Double-tail
3239:Center-flip
3186:
3156:skew hexagon
3144:star polygon
2950:
2915:
2901:
2890:
2821:Regular {6}
2744:
2740:
2726:
2717:
2643:
2612:
2440:
2434:
2428:
2424:
2418:
2412:
2406:
2402:
2385:
2381:
2377:
2375:
2366:
2364:
2355:
2347:
2343:
2332:
2320:
2312:
2308:
2304:
2298:
2257:
2255:
2249:
2245:
2241:
2237:
2233:
2221:
2160:parallelogon
1905:
1295:
1278:circumradius
1275:
1268:
1212:
1024:
1020:
1016:
1012:
1008:
999:
730:
581:
573:
565:
561:
558:circumradius
553:
544:The maximal
543:
537:
529:
526:Circumradius
521:
500:line segment
492:intersection
445:
404:
391:
370:
362:
308:circumcircle
301:
278:
259:
257:
212:
206:
197:Dual polygon
150:), order 2×6
6600:Star-shaped
6575:Rectilinear
6545:Equilateral
6540:Equiangular
6504:Hendecagram
6348:11–20 sides
6329:Octagon (8)
6319:Hexagon (6)
6294:Monogon (1)
6136:Equilateral
5973:10-demicube
5934:9-orthoplex
5884:8-orthoplex
5834:7-orthoplex
5791:6-orthoplex
5761:5-orthoplex
5716:Pentachoron
5704:Icosahedron
5679:Tetrahedron
5366:: 105–114.
5300:: 243–246.
5197:: 335–355.
5061:facets, is
4985:The Hexagon
4932:In French,
4299:Icosahedral
4289:Tetrahedral
4062:Icosahedral
4052:Tetrahedral
3959:soccer ball
3809:equilateral
3314:compression
3296:From bees'
3047:Star figure
3038:t{3} = {6}
2686:projection
2301:John Conway
440:, given by
398:equilateral
387:equilateral
285:equiangular
281:equilateral
182:equilateral
6631:6 (number)
6625:Categories
6605:Tangential
6509:Dodecagram
6287:1–10 sides
6278:By number
6259:Tangential
6239:Right kite
5959:10-simplex
5943:9-demicube
5893:8-demicube
5843:7-demicube
5800:6-demicube
5770:5-demicube
5684:Octahedron
5374:2015-04-12
5308:2014-11-17
5204:2010.12340
5168:2015-11-06
5121:References
4952:Hexagonal
4936:refers to
4934:l'Hexagone
4773:A beehive
4757:Assembled
4520:1-uniform
4294:Octahedral
4057:Octahedral
3939:tessellate
3875:such that
3822:such that
3724:Octahedron
3695:octahedron
3491:concurrent
3298:honeycombs
3259:Fish-tail
3170:octahedron
3167:projection
3062:Alternated
2926:alternated
2913:symmetry.
2824:Hexagonal
2676:Dissection
2214:reflection
513:Parameters
365:) and six
341:times the
297:tangential
170:Properties
6585:Reinhardt
6494:Enneagram
6484:Heptagram
6474:Pentagram
6441:65537-gon
6299:Digon (2)
6269:Trapezoid
6234:Rectangle
6184:Bicentric
6146:Isosceles
6123:Triangles
6007:orthoplex
5929:9-simplex
5879:8-simplex
5829:7-simplex
5786:6-simplex
5756:5-simplex
5725:Tesseract
5427:MathWorld
5422:"Hexagon"
5063:self-dual
5059:orthoplex
4775:honeycomb
4466:Prismoids
3975:truncated
3963:fullerene
3850:≤
3791:5-simplex
3629:centroids
3249:Unicursal
3191:with the
3158:, within
3054:Truncated
3044:Stellated
3036:Truncated
2953:dodecagon
2934:stellated
2922:dodecagon
2918:truncated
2904:truncated
2421:2 (2222)
2337:elongated
2124:directed
2004:∑
1951:∑
1247:≈
1240:π
1178:≈
1126:⋅
965:≈
949:≈
926:≈
910:≈
631:∘
620:
570:inscribed
416:honeycomb
401:triangles
289:bicentric
271:truncated
89:{6}, t{3}
6560:Isotoxal
6555:Isogonal
6499:Decagram
6489:Octagram
6479:Hexagram
6280:of sides
6209:Harmonic
6110:Polygons
6061:Topics:
6024:demicube
5989:polytope
5983:Uniform
5744:600-cell
5740:120-cell
5693:Demicube
5667:Pentagon
5647:Triangle
5475:CGP Grey
5468:animated
5398:Archived
5368:Archived
5338:Archived
5302:Archived
5162:archived
5109:Havannah
5092:Hexagram
5045:See also
4960:minerals
4954:Hanksite
4790:carapace
4743:graphene
4517:Regular
4277:G(2,0):
3969:and the
3355:Regular
3149:Extended
3136:hexagon
2938:hexagram
2342:, while
2325:isotoxal
2317:isogonal
2136:isogonal
2112:isotoxal
2081:regular
2055:Symmetry
1286:diagonal
1282:inradius
578:inradius
550:diagonal
546:diameter
534:Inradius
447:Elements
383:triangle
353:are 120
287:. It is
209:geometry
190:isotoxal
186:isogonal
144:Dihedral
75:vertices
6580:Regular
6525:Concave
6518:Classes
6426:257-gon
6249:Rhombus
6189:Crossed
5998:simplex
5968:10-cube
5735:24-cell
5721:16-cell
5662:Hexagon
5516:regular
5464:YouTube
5261:Coxeter
5051:24-cell
4989:theatre
4831:Benzene
4625:tr{6,3}
4592:rr{6,3}
3965:fame),
3742:regular
3300:to the
3134:Crossed
3064:h{6} =
3056:t{6} =
3031:Regular
2957:squares
2783:Rhombs
2735:⁄
2723:zonogon
2719:Coxeter
2415:(2*22)
2409:(*632)
2340:rhombus
2157:general
1262:of its
1006:apothem
1002:polygon
413:beehive
394:squares
355:degrees
343:apothem
263:hexagon
261:regular
244:polygon
213:hexagon
160:degrees
6590:Simple
6535:Cyclic
6530:Convex
6254:Square
6194:Cyclic
6156:Obtuse
6151:Kepler
5938:9-cube
5888:8-cube
5838:7-cube
5795:6-cube
5765:5-cube
5652:Square
5529:Family
5452:Hexnet
5247:
5154:
5023:Taiwan
4879:basalt
4559:r{6,3}
3623:If an
3547:, and
3500:of an
3442:cyclic
2684:6-cube
2437:(22*)
2431:(3*3)
2384:, and
2278:cyclic
1250:0.8270
1023:, and
952:0.6495
442:Euclid
351:angles
293:cyclic
248:simple
215:(from
178:cyclic
174:Convex
6565:Magic
6161:Right
6141:Ideal
6131:Acute
5657:p-gon
5466:– an
5199:arXiv
4759:E-ELT
4526:{6,3}
3659:is a
3440:is a
3340:Form
2443:(××)
2388:, as
2352:kites
2329:duals
2323:, an
2315:, an
2276:), 4
2260:has D
1181:3.464
1070:, so
968:0.866
929:3.464
913:2.598
392:Like
239:gonía
233:γωνία
217:Greek
71:Edges
6595:Skew
6219:Kite
6114:List
6015:cube
5688:Cube
5518:and
5245:ISBN
5227:link
5152:ISBN
5084:: a
5053:: a
4916:The
4009:and
3961:and
3957:(of
3903:>
3713:Cube
3693:and
3691:cube
3689:The
3436:The
3160:cube
3058:{12}
3050:2{3}
3033:{6}
2959:and
2874:Cube
2759:cube
2747:− 1)
2725:(a 2
2346:and
2280:: (Z
2256:The
1336:and
396:and
283:and
265:has
211:, a
201:Self
165:120°
73:and
60:Type
5564:(p)
5462:on
5450:on
5209:doi
4991:in
4885:in
3757:5D
3754:4D
3674:is
3482:bdf
3478:ace
3310:wax
3216:Dih
3210:Dih
3204:Dih
3066:{3}
2832:3D
2780:2D
2533:Dih
2521:Dih
2515:Dih
2510:a1
2501:p2
2492:d2
2483:d2
2474:g2
2465:i4
2455:r12
2435:pmg
2413:cmm
2386:r12
2305:r12
2292:, Z
2288:, Z
2284:, Z
2246:r12
2222:r12
2220:or
2196:a1
2183:g3
2170:p2
2146:d2
2096:i4
2079:r12
1906:If
1280:to
617:cos
444:'s
306:or
227:hex
207:In
6627::
6069:•
6065:•
6045:21
6041:•
6038:k1
6034:•
6031:k2
6009:•
5966:•
5936:•
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5910:•
5907:41
5903:•
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5886:•
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5860:•
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5853:•
5850:32
5836:•
5814:21
5810:•
5807:22
5793:•
5763:•
5742:•
5723:•
5702:•
5686:•
5618:/
5607:/
5597:/
5588:/
5566:/
5424:.
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5364:15
5362:.
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5298:14
5296:.
5290:.
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5219:{{
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3467:,
3463:,
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1273:.
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1019:=
627:30
580:,
560:,
536:;
532:=
528:;
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503:AB
488:AB
258:A
236:,
224:,
221:ἕξ
188:,
184:,
180:,
176:,
146:(D
6116:)
6112:(
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6051:n
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5634:n
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3586:d
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3577:=
3574:e
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2523:2
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2429:m
2425:p
2419:p
2407:m
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2252:.
2242:g
2238:p
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2039:.
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2029:i
2025:d
2019:6
2014:1
2011:=
2008:i
2000:4
1997:=
1992:2
1987:)
1981:2
1976:i
1972:d
1966:6
1961:1
1958:=
1955:i
1946:(
1919:i
1915:d
1891:.
1887:)
1881:2
1877:L
1871:2
1867:R
1863:2
1860:+
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1844:2
1840:L
1836:+
1831:2
1827:R
1822:(
1816:(
1812:3
1809:=
1804:4
1799:6
1795:d
1791:+
1786:4
1781:4
1777:d
1773:+
1768:4
1763:2
1759:d
1755:=
1750:4
1745:5
1741:d
1737:+
1732:4
1727:3
1723:d
1719:+
1714:4
1709:1
1705:d
1683:,
1679:)
1673:2
1669:L
1665:+
1660:2
1656:R
1651:(
1647:3
1644:=
1639:2
1634:6
1630:d
1626:+
1621:2
1616:4
1612:d
1608:+
1603:2
1598:2
1594:d
1590:=
1585:2
1580:5
1576:d
1572:+
1567:2
1562:3
1558:d
1554:+
1549:2
1544:1
1540:d
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1514:)
1508:2
1504:L
1500:+
1495:2
1491:R
1486:(
1482:2
1479:=
1474:2
1469:6
1465:d
1461:+
1456:2
1451:3
1447:d
1443:=
1438:2
1433:5
1429:d
1425:+
1420:2
1415:2
1411:d
1407:=
1402:2
1397:4
1393:d
1389:+
1384:2
1379:1
1375:d
1349:i
1345:d
1324:L
1304:R
1237:2
1230:3
1225:3
1194:.
1189:2
1185:r
1166:3
1159:2
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1148:=
1143:2
1137:3
1132:r
1129:4
1123:r
1117:=
1105:2
1101:p
1098:a
1092:=
1085:A
1056:3
1051:r
1048:4
1045:=
1042:R
1039:6
1036:=
1025:p
1021:r
1017:a
1013:p
1009:a
981:.
976:2
972:d
960:2
956:D
937:2
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898:2
894:d
888:2
884:3
878:=
875:d
872:D
867:4
864:3
859:=
854:2
850:D
844:8
838:3
833:3
827:=
815:2
811:r
805:3
800:2
797:=
794:r
791:R
788:3
785:=
780:2
776:R
770:2
764:3
759:3
753:=
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716:.
713:D
708:2
704:3
698:=
695:d
675:t
670:2
666:3
660:=
657:R
652:2
648:3
642:=
639:R
636:)
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614:=
611:r
608:=
605:d
600:2
597:1
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562:R
554:D
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474:.
458:=
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