4867:
4763:
4938:
4978:
4809:
4907:
4990:
4836:
492:
3651:
4824:
3096:
4340:
2721:
2714:
5043:
2870:
2818:
4329:
4450:
4426:
4415:
4391:
3103:
2863:
2811:
4855:
3117:
3028:
442:
3378:
61:
4922:
3035:
2986:
4272:
2240:
2877:
2825:
4461:
4439:
3300:
4380:
3014:
3000:
4693:
3007:
4686:
4250:
4404:
4318:
3627:
5027:
3775:
3110:
3397:
4883:
2499:
2490:
2472:
2461:
3089:
4958:
1006:
4261:
4228:
3390:
2220:
2517:
2508:
2481:
4679:
4672:
4239:
3371:
3786:
2993:
2804:
2086:
4497:
529:
4747:
2203:
2190:
2177:
2162:
2153:
2140:
2129:
2116:
2103:
3284:
3275:
3266:
3255:
3246:
3237:
5012:
4508:
3730:
4369:
3138:
4794:
2596:
2569:
4486:
4779:
3797:
2707:
2856:
4730:
4723:
4716:
4709:
3719:
3131:
3124:
3021:
748:
1219:
1001:{\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}}}
1912:
1704:
1539:
696:
4882:
2060:
1087:
4866:
3499:
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
753:
1271:
1710:
4937:
1092:
3934:
4808:
3515:
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.
737:
3874:
350:
3455:
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
5237:
1079:
1545:
1380:
3616:
601:
4906:
4762:
1942:
1372:
4921:
4835:
5348:
1345:
1325:
479:
4989:
392:. 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
1950:
1214:{\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}
4957:
6651:
6081:
3319:
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
5259:
4746:
5516:
5378:
5312:
5172:
4977:
6111:
5166:
3689:
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
3416:
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the
2669:
2605:
1907:{\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).}
2679:
2615:
1227:
4823:
4662:
4652:
4642:
4629:
4609:
4586:
4543:
4218:
4208:
4198:
4179:
4169:
4160:
4150:
4140:
4121:
4111:
4092:
4082:
4044:
4034:
4024:
4006:
3996:
2648:
2638:
2588:
2578:
141:
131:
113:
5026:
6646:
4619:
4596:
4576:
4563:
4553:
4454:
4430:
4419:
4395:
4189:
4131:
4102:
4016:
2764:
with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a
123:
1280:
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
2674:
2610:
149:
4854:
4657:
4647:
4624:
4614:
4591:
4581:
4558:
4548:
4213:
4203:
4184:
4174:
4155:
4145:
4126:
4116:
4097:
4087:
4039:
4029:
4011:
4001:
136:
118:
3674:
with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A
2643:
2583:
4635:
3359:
5408:
2338:
hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are
5042:
4465:
4443:
3892:
5011:
6234:
6214:
4384:
4276:
3981:
5346:
3439:
until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
105:
583:
circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base),
6656:
6209:
6166:
6141:
5474:
4928:
4913:
4793:
4602:
4408:
2975:
5197:
701:
6269:
5082:
4968:
448:
38:
31:
3839:
429:
are hexagonal for this reason and because the shape makes efficient use of space and building materials. The
324:
6194:
5509:
4254:
3977:
3431:(also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any
441:
6219:
6104:
5065:
4344:
3650:
3528:
2936:
3638:
is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the
1040:
6620:
6560:
6199:
6053:
6046:
6039:
5251:
5231:
5119:
4322:
4265:
4232:
3985:
3965:
3957:
3756:
3324:
2928:
2914:
2656:
2347:
377:
369:
281:
4339:
1699:{\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),}
1534:{\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),}
491:
6504:
6274:
6204:
6146:
5710:
5657:
5125:
5114:
4309:
4299:
4243:
4072:
4062:
3961:
3690:
3655:
3635:
3166:
3076:
2971:
2952:
2940:
2917:
2683:
are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
2346:
forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an
1274:
506:
457:
423:
397:
314:
284:
45:
5462:
3095:
2869:
2817:
691:{\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}
6641:
6610:
6585:
6555:
6550:
6509:
6224:
6065:
5964:
5714:
5400:
5303:
4948:
4893:
4814:
4778:
4569:
4501:
4449:
4425:
4414:
4390:
4328:
4304:
4285:
4067:
3819:
3428:
3312:
3102:
2862:
2810:
2720:
2224:
1295:
that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long
295:
291:
192:
3568:
3396:
2713:
6615:
6156:
5934:
5884:
5834:
5791:
5761:
5721:
5684:
5502:
5209:
5108:
5003:
3535:
3259:
3203:
2889:
2777:
2311:
1944:
are the distances from the vertices of a regular hexagon to any point on its circumcircle, then
307:
3377:
2903:
277:
95:
4460:
4438:
4271:
3116:
3027:
2876:
2824:
44:"Hexagonal" redirects here. For the FIFA World Cup qualifying tournament in North America, see
6595:
6189:
6097:
6073:
5429:
5255:
5162:
4873:
4845:
4512:
4379:
4373:
4055:
3953:
3686:
3034:
2985:
2956:
404:
299:
154:
85:
60:
5366:
5298:
6124:
6077:
5642:
5631:
5620:
5609:
5600:
5591:
5578:
5556:
5544:
5530:
5526:
5219:
5156:
5092:
5087:
5048:
4897:
4700:
4692:
4536:
3752:
3501:
3417:
3411:
3354:
2907:
2659:
2625:
2465:
2404:
2239:
396:
with a vertex at the center of the regular hexagon and sharing one side with the hexagon is
357:
4685:
4403:
4249:
3013:
2999:
2652:, are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
2376:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
1920:
1350:
6590:
6570:
6565:
6535:
6254:
6229:
6161:
5667:
5652:
5412:
5352:
5077:
5018:
4490:
4317:
3989:
3456:
3448:
3339:
3335:
3006:
2761:
2362:
2335:
2327:
2146:
2122:
430:
271:
200:
196:
81:
74:
3818:
of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex
3774:
3299:
2920:, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D
6600:
6580:
6545:
6540:
6171:
6151:
6017:
5481:
5096:
4333:
3945:
3790:
3748:
3626:
3512:
3452:
3316:
3185:
3088:
2765:
2663:
2632:
2628:
2498:
2489:
2471:
2460:
1330:
1310:
464:
385:
365:
303:
258:
188:
184:
170:
166:
5262:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
4260:
4227:
3389:
3109:
2516:
2507:
2480:
6635:
6575:
6426:
6319:
6239:
6181:
6034:
5922:
5915:
5908:
5872:
5865:
5858:
5822:
5815:
5539:
4353:
3436:
3432:
2381:
2243:
The dihedral symmetries are divided depending on whether they pass through vertices (
227:
4678:
4671:
4284:
There are other symmetry polyhedra with stretched or flattened hexagons, like these
4238:
3785:
3370:
2085:
2055:{\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}
6605:
6475:
6431:
6395:
6385:
6380:
5974:
4496:
3949:
3779:
3671:
3524:
3508:
3331:
3199:
3154:
2836:
2773:
2400:
2370:
2339:
2288:
2219:
2202:
2189:
2176:
2170:
2161:
2152:
2139:
2128:
2115:
2102:
1288:
568:
536:
510:
502:
482:
419:
318:
207:
5406:
2992:
2803:
2740:-gon whose opposite sides are parallel and of equal length) can be dissected into
1327:, whose distances to the centroid of the regular hexagon and its six vertices are
400:, and that the regular hexagon can be partitioned into six equilateral triangles.
3531:
states that the three main diagonals AD, BE, and CF intersect at a single point.
6514:
6421:
6400:
6390:
5983:
5944:
5894:
5844:
5801:
5771:
5703:
5689:
5458:
4995:
4507:
4368:
3969:
3729:
3057:
528:
408:
5452:
3283:
3274:
3265:
3254:
3245:
3236:
6519:
6375:
6365:
6249:
5969:
5953:
5903:
5853:
5810:
5780:
5694:
5468:
4485:
3734:
3708:(same as triangular antiprism) have regular skew hexagons as petrie polygons.
3705:
3180:
3137:
2944:
5432:
2330:
hexagon constructed by three mirrors can alternate long and short edges, and
418:(three hexagons meeting at every vertex), and so are useful for constructing
6494:
6484:
6461:
6451:
6441:
6370:
6279:
6244:
6025:
5939:
5889:
5839:
5796:
5766:
5735:
5478:
5437:
5223:
5073:
5069:
4785:
3973:
3801:
3308:
3068:
2963:
2932:
2595:
2568:
580:
426:
5143:
3486:, then the three main diagonals intersect in a single point if and only if
3315:, hexagonal patterns are prevalent in nature due to their efficiency. In a
2342:
of each other and have half the symmetry order of the regular hexagon. The
1307:
For an arbitrary point in the plane of a regular hexagon with circumradius
433:
of a regular triangular lattice is the honeycomb tessellation of hexagons.
5448:
4729:
4722:
4715:
3796:
2855:
2706:
2275:
symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D
2251:
for perpendiculars) Cyclic symmetries in the middle column are labeled as
6499:
6489:
6446:
6405:
6334:
6324:
6314:
6133:
5999:
5754:
5750:
5677:
5485:
5345:
Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter",
5102:
4964:
4800:
4753:
4476:
3639:
3060:
2948:
1296:
1292:
588:
560:
556:
544:
411:
393:
219:
4708:
3718:
2310:
These symmetries express nine distinct symmetries of a regular hexagon.
2255:
for their central gyration orders. Full symmetry of the regular form is
447:
A step-by-step animation of the construction of a regular hexagon using
17:
6456:
6436:
6349:
6344:
6339:
6304:
6259:
6120:
6008:
5978:
5745:
5740:
5731:
5271:
5061:
4999:
4841:
2733:
2729:
2350:
1299:
of 1.0000000 will have a distance of 0.8660254 between parallel sides.
1016:
1012:
353:
254:
3130:
3123:
6264:
5948:
5898:
5848:
5805:
5775:
5726:
5662:
5299:"Dao's theorem on six circumcenters associated with a cyclic hexagon"
5033:
4889:
2967:
2694:
516:
four times on the circumscribed circle and connect the corner points.
452:
4900:; large masses must cool slowly to form a polygonal fracture pattern
3020:
501:
is given, drawing a circular arc from point A and point B gives the
5214:
6309:
6089:
5367:"Equilateral triangles and Kiepert perspectors in complex numbers"
4769:
3625:
3298:
2955:
by adding a center point. This pattern repeats within the regular
2218:
527:
361:
3334:
and can also tile the plane by translation. In three dimensions,
2369:
hexagons, with opposite sides parallel are also called hexagonal
5698:
5274:, Mathematical recreations and Essays, Thirteenth edition, p.141
3755:, uniform and dual polyhedra and polytopes, shown in these skew
3723:
3701:
3170:
2884:
2769:
6093:
4817:, a hexagonal cloud pattern around the north pole of the planet
3320:
248:
242:
236:
230:
2786:
Dissection of hexagons into three rhombs and parallelograms
1266:{\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}
3330:
Irregular hexagons with parallel opposite edges are called
2776:
and projective directions of the cube are dissected within
5122:: abstract board game played on a six-sided hexagonal grid
3630:
Equilateral triangles on the sides of an arbitrary hexagon
3622:
Equilateral triangles on the sides of an arbitrary hexagon
5198:"Cyclic Averages of Regular Polygons and Platonic Solids"
3642:
of opposite triangles form another equilateral triangle.
2403:
can tessellate the
Euclidean plane by translation. Other
2910:, {6,3}, with three hexagonal faces around each vertex.
2069:
595:. The maxima and minima are related by the same factor:
313:
The common length of the sides equals the radius of the
290:
A regular hexagon is defined as a hexagon that is both
3678:
has vertices alternating between two parallel planes.
2690:
2563:
2380:
subgroup has no degrees of freedom but can be seen as
1232:
329:
3895:
3842:
3571:
2361:
can be seen as horizontally and vertically elongated
1953:
1923:
1713:
1548:
1383:
1353:
1333:
1313:
1230:
1090:
1043:
751:
704:
604:
467:
327:
3948:
made of only regular hexagons, because the hexagons
3822:
hexagon (one with all sides equal) with common side
3083:
2980:
2947:
with equilateral triangles on its edges, creating a
2935:, {12}, alternating two types (colors) of edges. An
414:, regular hexagons fit together without any gaps to
6528:
6474:
6414:
6358:
6297:
6288:
6180:
6132:
206:
180:
165:
148:
104:
94:
80:
70:
53:
3929:{\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}
3928:
3868:
3654:A regular skew hexagon seen as edges (black) of a
3610:
2054:
1936:
1906:
1698:
1533:
1366:
1339:
1319:
1265:
1213:
1073:
1000:
731:
690:
473:
344:
5236:: CS1 maint: DOI inactive as of September 2024 (
3342:and these can tessellate 3-space by translation.
3210:Self-intersecting hexagons with regular vertices
2962:A regular hexagon can be extended into a regular
1015:, the area can also be expressed in terms of the
461:, Book IV, Proposition 15: this is possible as 6
3462:If the successive sides of a cyclic hexagon are
2906:{6}. A regular hexagon is a part of the regular
481:2 × 3, a product of a power of two and distinct
5111:: single path, six-sided star, within a hexagon
2760:parallelograms. In particular this is true for
5202:Communications in Mathematics and Applications
2951:. A regular hexagon can be dissected into six
579:. The minimal diameter or the diameter of the
6105:
5510:
1026:. For the regular hexagon these are given by
287:, t{3}, which alternates two types of edges.
8:
4931:mirror is composed of 18 hexagonal segments.
3952:, not allowing the result to "fold up". The
2974:around it. This pattern repeats within the
2913:A regular hexagon can also be created as a
6294:
6112:
6098:
6090:
5517:
5503:
5495:
5453:construction with compass and straightedge
4739:Gallery of natural and artificial hexagons
4519:
4472:
4358:
4290:
4050:
3323:to construct and gain much strength under
2782:
2314:labels these by a letter and group order.
732:{\displaystyle d={\frac {\sqrt {3}}{2}}D.}
257:. The total of the internal angles of any
253:, meaning "corner, angle") is a six-sided
5465:a website devoted to hexagon mathematics.
5213:
5161:, Cambridge University Press, p. 9,
5105:: six-sided star within a regular hexagon
3916:
3902:
3896:
3894:
3849:
3843:
3841:
3570:
2043:
2038:
2028:
2017:
2001:
1990:
1985:
1975:
1964:
1952:
1928:
1922:
1890:
1880:
1864:
1853:
1840:
1813:
1808:
1795:
1790:
1777:
1772:
1759:
1754:
1741:
1736:
1723:
1718:
1712:
1682:
1669:
1648:
1643:
1630:
1625:
1612:
1607:
1594:
1589:
1576:
1571:
1558:
1553:
1547:
1517:
1504:
1483:
1478:
1465:
1460:
1447:
1442:
1429:
1424:
1411:
1406:
1393:
1388:
1382:
1358:
1352:
1332:
1312:
1238:
1231:
1229:
1198:
1174:
1168:
1145:
1130:
1105:
1091:
1089:
1064:
1044:
1042:
985:
969:
946:
930:
907:
891:
872:
863:
846:
840:
824:
813:
789:
772:
766:
752:
750:
711:
703:
673:
655:
640:
605:
603:
466:
328:
326:
261:(non-self-intersecting) hexagon is 720°.
5336:, Dover Publications, 2007 (orig. 1960).
5191:
5189:
3869:{\displaystyle {\frac {d_{1}}{a}}\leq 2}
3761:
3710:
3649:
3344:
3338:with parallel opposite faces are called
3208:
2409:
2238:
2078:
6082:List of regular polytopes and compounds
5136:
4742:
1224:The regular hexagon fills the fraction
345:{\displaystyle {\tfrac {2}{\sqrt {3}}}}
5449:Definition and properties of a hexagon
5229:
3523:Let ABCDEF be a hexagon formed by six
50:
5459:An Introduction to Hexagonal Geometry
4876:composed of hexagonal aromatic rings.
4860:Hexagonal order of bubbles in a foam.
3519:Hexagon tangential to a conic section
280:{6} and can also be constructed as a
7:
5068:figure which, like the hexagon, has
4912:An aerial view of Fort Jefferson in
3826:, there exists a principal diagonal
3424:Hexagon inscribed in a conic section
5283:Cartensen, Jens, "About hexagons",
5254:, (2008) The Symmetries of Things,
4752:The ideal crystalline structure of
3984:. These hexagons can be considered
1074:{\displaystyle {}=6R=4r{\sqrt {3}}}
3956:with some hexagonal faces are the
2772:, with 3 of 6 square faces. Other
25:
4951:for its vaguely hexagonal shape.
3507:If a hexagon has vertices on the
2405:hexagon shapes can tile the plane
587:, is twice the minimal radius or
567:, is twice the maximal radius or
306:(has a circumscribed circle) and
5041:
5025:
5010:
4988:
4976:
4956:
4936:
4920:
4905:
4881:
4865:
4853:
4834:
4822:
4807:
4792:
4777:
4761:
4745:
4728:
4721:
4714:
4707:
4691:
4684:
4677:
4670:
4660:
4655:
4650:
4645:
4640:
4627:
4622:
4617:
4612:
4607:
4594:
4589:
4584:
4579:
4574:
4561:
4556:
4551:
4546:
4541:
4506:
4495:
4484:
4459:
4448:
4437:
4424:
4413:
4402:
4389:
4378:
4367:
4338:
4327:
4316:
4270:
4259:
4248:
4237:
4226:
4216:
4211:
4206:
4201:
4196:
4187:
4182:
4177:
4172:
4167:
4158:
4153:
4148:
4143:
4138:
4129:
4124:
4119:
4114:
4109:
4100:
4095:
4090:
4085:
4080:
4042:
4037:
4032:
4027:
4022:
4014:
4009:
4004:
3999:
3994:
3795:
3784:
3773:
3747:The regular skew hexagon is the
3728:
3717:
3395:
3388:
3376:
3369:
3282:
3273:
3264:
3253:
3244:
3235:
3136:
3129:
3122:
3115:
3108:
3101:
3094:
3087:
3033:
3026:
3019:
3012:
3005:
2998:
2991:
2984:
2943:, {3}. A regular hexagon can be
2875:
2868:
2861:
2854:
2823:
2816:
2809:
2802:
2719:
2712:
2705:
2677:
2672:
2667:
2646:
2641:
2636:
2613:
2608:
2603:
2594:
2586:
2581:
2576:
2567:
2515:
2506:
2497:
2488:
2479:
2470:
2459:
2201:
2188:
2175:
2160:
2151:
2138:
2127:
2114:
2101:
2084:
575:, which equals the side length,
490:
440:
374:rotational symmetry of order six
139:
134:
129:
121:
116:
111:
59:
6652:Polygons by the number of sides
5451:with interactive animation and
5381:from the original on 2015-07-05
5315:from the original on 2014-12-05
5250:John H. Conway, Heidi Burgiel,
5175:from the original on 2016-01-02
4455:augmented truncated tetrahedron
4431:metabiaugmented hexagonal prism
4420:parabiaugmented hexagonal prism
4396:gyroelongated triangular cupola
4294:Hexagons in Goldberg polyhedra
3538:and that has consecutive sides
559:(which corresponds to the long
4523:Tilings with regular hexagons
3346:Hexagonal prism tessellations
2227:of a regular hexagon, with Dih
742:The area of a regular hexagon
646:
633:
1:
5196:Meskhishvili, Mamuka (2020).
5155:Wenninger, Magnus J. (1974),
4362:Johnson solids with hexagons
3751:for these higher dimensional
3712:Skew hexagons on 3-fold axes
3360:Hexagonal prismatic honeycomb
3153:A self-intersecting hexagon (
2407:with different orientations.
2073:Example hexagons by symmetry
1287:It follows from the ratio of
5297:Dergiades, Nikolaos (2014).
4466:triangular hebesphenorotunda
4444:triaugmented hexagonal prism
3611:{\displaystyle a+c+e=b+d+f.}
2898:Related polygons and tilings
368:. A regular hexagon has six
249:
237:
30:For the crystal system, see
5334:Advanced Euclidean Geometry
4385:elongated triangular cupola
4277:truncated icosidodecahedron
3982:truncated icosidodecahedron
2259:and no symmetry is labeled
1282:PE + PF = PA + PB + PC + PD
310:(has an inscribed circle).
6673:
6071:
5498:
5470:Hexagons are the Bestagons
5399:Inequalities proposed in "
4929:James Webb Space Telescope
4914:Dry Tortugas National Park
4522:
4482:
4475:
4435:
4400:
4365:
4361:
4293:
4224:
4053:
3810:Convex equilateral hexagon
3771:
3409:
3233:
2882:
2852:
2830:
2800:
2785:
2524:
2184:
2112:
2080:
302:, meaning that it is both
243:
231:
43:
36:
29:
5287:33(2) (2000–2001), 37–40.
4829:Micrograph of a snowflake
4799:The scutes of a turtle's
4699:
4530:
4409:augmented hexagonal prism
4071:
4066:
3879:and a principal diagonal
3764:
3527:of a conic section. Then
3220:
3214:
3052:Hypertruncated triangles
3051:
2976:rhombitrihexagonal tiling
2888:
2883:
2848:
2845:
2842:
2834:
2796:
2790:
2699:
2543:
2444:
2096:
58:
5083:Hexagonal crystal system
4969:hexagonal crystal system
3435:, and pairs of opposite
3406:Tesselations by hexagons
3303:Giant's Causeway closeup
3206:of the regular hexagon:
2247:for diagonal) or edges (
449:compass and straightedge
39:Hexagon (disambiguation)
32:Hexagonal crystal family
5224:10.26713/cma.v11i3.1420
4872:Crystal structure of a
4356:with regular hexagons:
4255:truncated cuboctahedron
3978:truncated cuboctahedron
3940:Polyhedra with hexagons
3697:, symmetry, order 12.
698: and, similarly,
106:Coxeter–Dynkin diagrams
6647:Constructible polygons
5365:Dao Thanh Oai (2015).
5355:, Accessed 2012-04-17.
5226:(inactive 2024-09-12).
5099:of hexagons in a plane
4345:Chamfered dodecahedron
3930:
3870:
3757:orthogonal projections
3663:
3631:
3612:
3536:tangential to a circle
3304:
3200:self-crossing hexagons
3194:Self-crossing hexagons
2966:by adding alternating
2902:A regular hexagon has
2318:is full symmetry, and
2307:) and the trivial (e)
2264:
2236:
2056:
2033:
1980:
1938:
1908:
1700:
1535:
1374:respectively, we have
1368:
1341:
1321:
1267:
1215:
1075:
1002:
733:
692:
552:
475:
346:
5285:Mathematical Spectrum
5252:Chaim Goodman-Strauss
4967:crystal, one of many
4848:with hexagonal shape.
4323:Chamfered tetrahedron
4266:truncated icosahedron
4233:truncated tetrahedron
3966:truncated icosahedron
3958:truncated tetrahedron
3931:
3871:
3653:
3629:
3613:
3534:In a hexagon that is
3420:will tile the plane.
3302:
2972:equilateral triangles
2953:equilateral triangles
2939:hexagon, h{6}, is an
2657:Exceptional Lie group
2387:Hexagons of symmetry
2242:
2222:
2057:
2013:
1960:
1939:
1937:{\displaystyle d_{i}}
1909:
1701:
1536:
1369:
1367:{\displaystyle d_{i}}
1342:
1322:
1268:
1216:
1076:
1003:
734:
693:
531:
505:M, the center of the
497:When the side length
476:
382:six lines of symmetry
378:reflection symmetries
370:rotational symmetries
347:
241:, meaning "six", and
6345:Nonagon/Enneagon (9)
6275:Tangential trapezoid
5126:Central place theory
5115:Honeycomb conjecture
5017:Władysław Gliński's
4756:is a hexagonal grid.
4244:truncated octahedron
3962:truncated octahedron
3893:
3840:
3691:triangular antiprism
3683:regular skew hexagon
3676:skew zig-zag hexagon
3662:, , (2*3), order 12.
3656:triangular antiprism
3636:equilateral triangle
3569:
3295:Hexagonal structures
3162:Central {6} in {12}
2941:equilateral triangle
2931:hexagon, t{6}, is a
2918:equilateral triangle
2700:12 rhomb dissection
2655:The 12 roots of the
1951:
1921:
1711:
1546:
1381:
1351:
1331:
1311:
1275:circumscribed circle
1228:
1088:
1041:
749:
702:
602:
507:circumscribed circle
465:
325:
315:circumscribed circle
285:equilateral triangle
46:Hexagonal (CONCACAF)
37:For other uses, see
27:Shape with six sides
6457:Megagon (1,000,000)
6225:Isosceles trapezoid
6066:pentagonal polytope
5965:Uniform 10-polytope
5525:Fundamental convex
5401:Crux Mathematicorum
5371:Forum Geometricorum
5332:Johnson, Roger A.,
5304:Forum Geometricorum
4949:Metropolitan France
4502:Hexagonal antiprism
4286:Goldberg polyhedron
3713:
3529:Brianchon's theorem
3347:
3211:
2778:rectangular cuboids
2662:, represented by a
2631:, represented by a
2624:The 6 roots of the
2235:symmetry, order 12.
2048:
1995:
1818:
1800:
1782:
1764:
1746:
1728:
1653:
1635:
1617:
1599:
1581:
1563:
1488:
1470:
1452:
1434:
1416:
1398:
6427:Icositetragon (24)
5935:Uniform 9-polytope
5885:Uniform 8-polytope
5835:Uniform 7-polytope
5792:Uniform 6-polytope
5762:Uniform 5-polytope
5722:Uniform polychoron
5685:Uniform polyhedron
5533:in dimensions 2–10
5484:about hexagons by
5430:Weisstein, Eric W.
5411:2017-08-30 at the
5351:2012-05-11 at the
5109:Unicursal hexagram
5004:Reading, Berkshire
4056:Archimedean solids
3954:Archimedean solids
3926:
3866:
3816:principal diagonal
3711:
3664:
3632:
3608:
3437:sides are extended
3345:
3305:
3209:
3204:vertex arrangement
3150:A concave hexagon
2890:Rectangular cuboid
2849:Rectangular faces
2732:states that every
2265:
2237:
2052:
2034:
1981:
1934:
1904:
1804:
1786:
1768:
1750:
1732:
1714:
1696:
1639:
1621:
1603:
1585:
1567:
1549:
1531:
1474:
1456:
1438:
1420:
1402:
1384:
1364:
1337:
1317:
1263:
1255:
1211:
1209:
1071:
1022:and the perimeter
998:
996:
729:
688:
553:
471:
342:
340:
6657:Elementary shapes
6629:
6628:
6470:
6469:
6447:Myriagon (10,000)
6432:Triacontagon (30)
6396:Heptadecagon (17)
6386:Pentadecagon (15)
6381:Tetradecagon (14)
6320:Quadrilateral (4)
6190:Antiparallelogram
6087:
6086:
6074:Polytope families
5531:uniform polytopes
5260:978-1-56881-220-5
5158:Polyhedron Models
5036:Botanical Gardens
4888:Naturally formed
4874:molecular hexagon
4846:aromatic compound
4736:
4735:
4701:2-uniform tilings
4518:
4517:
4513:Hexagonal pyramid
4471:
4470:
4374:triangular cupola
4352:There are also 9
4350:
4349:
4282:
4281:
3921:
3911:
3858:
3807:
3806:
3740:
3739:
3687:vertex-transitive
3403:
3402:
3292:
3291:
3191:
3190:
3082:
3081:
2957:triangular tiling
2895:
2894:
2727:
2726:
2622:
2621:
2557:
2556:
2279:), 2 dihedral: (D
2223:The six lines of
2217:
2216:
2213:
2212:
1340:{\displaystyle L}
1320:{\displaystyle R}
1254:
1243:
1179:
1156:
1150:
1118:
1069:
901:
897:
880:
857:
851:
818:
783:
777:
721:
717:
683:
679:
665:
661:
613:
563:of the hexagon),
474:{\displaystyle =}
422:. The cells of a
384:), making up the
339:
338:
216:
215:
65:A regular hexagon
16:(Redirected from
6664:
6442:Chiliagon (1000)
6422:Icositrigon (23)
6401:Octadecagon (18)
6391:Hexadecagon (16)
6295:
6114:
6107:
6100:
6091:
6078:Regular polytope
5639:
5628:
5617:
5576:
5519:
5512:
5505:
5496:
5471:
5443:
5442:
5416:
5396:
5390:
5389:
5387:
5386:
5362:
5356:
5343:
5337:
5330:
5324:
5323:
5321:
5320:
5294:
5288:
5281:
5275:
5269:
5263:
5248:
5242:
5241:
5235:
5227:
5217:
5193:
5184:
5182:
5181:
5180:
5152:
5146:
5141:
5093:Hexagonal tiling
5088:Hexagonal number
5076:and tessellates
5066:four-dimensional
5049:Hexagonal window
5045:
5032:Pavilion in the
5029:
5014:
4992:
4980:
4960:
4940:
4924:
4909:
4898:Northern Ireland
4894:Giant's Causeway
4885:
4869:
4857:
4838:
4826:
4815:Saturn's hexagon
4811:
4796:
4781:
4765:
4749:
4732:
4725:
4718:
4711:
4695:
4688:
4681:
4674:
4665:
4664:
4663:
4659:
4658:
4654:
4653:
4649:
4648:
4644:
4643:
4632:
4631:
4630:
4626:
4625:
4621:
4620:
4616:
4615:
4611:
4610:
4599:
4598:
4597:
4593:
4592:
4588:
4587:
4583:
4582:
4578:
4577:
4566:
4565:
4564:
4560:
4559:
4555:
4554:
4550:
4549:
4545:
4544:
4520:
4510:
4499:
4488:
4473:
4463:
4452:
4441:
4428:
4417:
4406:
4393:
4382:
4371:
4359:
4342:
4331:
4320:
4291:
4274:
4263:
4252:
4241:
4230:
4221:
4220:
4219:
4215:
4214:
4210:
4209:
4205:
4204:
4200:
4199:
4192:
4191:
4190:
4186:
4185:
4181:
4180:
4176:
4175:
4171:
4170:
4163:
4162:
4161:
4157:
4156:
4152:
4151:
4147:
4146:
4142:
4141:
4134:
4133:
4132:
4128:
4127:
4123:
4122:
4118:
4117:
4113:
4112:
4105:
4104:
4103:
4099:
4098:
4094:
4093:
4089:
4088:
4084:
4083:
4051:
4047:
4046:
4045:
4041:
4040:
4036:
4035:
4031:
4030:
4026:
4025:
4019:
4018:
4017:
4013:
4012:
4008:
4007:
4003:
4002:
3998:
3997:
3990:Coxeter diagrams
3988:triangles, with
3935:
3933:
3932:
3927:
3922:
3917:
3912:
3907:
3906:
3897:
3875:
3873:
3872:
3867:
3859:
3854:
3853:
3844:
3799:
3788:
3777:
3762:
3732:
3721:
3714:
3617:
3615:
3614:
3609:
3495:
3429:Pascal's theorem
3418:Conway criterion
3412:Hexagonal tiling
3399:
3392:
3380:
3373:
3355:Hexagonal tiling
3348:
3340:parallelohedrons
3336:hexagonal prisms
3313:Giant's Causeway
3286:
3277:
3268:
3257:
3248:
3239:
3212:
3140:
3133:
3126:
3119:
3112:
3105:
3098:
3091:
3084:
3037:
3030:
3023:
3016:
3009:
3002:
2995:
2988:
2981:
2908:hexagonal tiling
2879:
2872:
2865:
2858:
2827:
2820:
2813:
2806:
2783:
2768:projection of a
2762:regular polygons
2759:
2750:
2749:
2745:
2723:
2716:
2709:
2691:
2682:
2681:
2680:
2676:
2675:
2671:
2670:
2651:
2650:
2649:
2645:
2644:
2640:
2639:
2626:simple Lie group
2618:
2617:
2616:
2612:
2611:
2607:
2606:
2598:
2591:
2590:
2589:
2585:
2584:
2580:
2579:
2571:
2564:
2560:A2 and G2 groups
2519:
2510:
2501:
2492:
2483:
2474:
2463:
2410:
2322:is no symmetry.
2205:
2192:
2179:
2164:
2155:
2142:
2131:
2118:
2105:
2088:
2079:
2070:
2061:
2059:
2058:
2053:
2047:
2042:
2032:
2027:
2006:
2005:
2000:
1996:
1994:
1989:
1979:
1974:
1943:
1941:
1940:
1935:
1933:
1932:
1913:
1911:
1910:
1905:
1900:
1896:
1895:
1894:
1885:
1884:
1869:
1868:
1863:
1859:
1858:
1857:
1845:
1844:
1817:
1812:
1799:
1794:
1781:
1776:
1763:
1758:
1745:
1740:
1727:
1722:
1705:
1703:
1702:
1697:
1692:
1688:
1687:
1686:
1674:
1673:
1652:
1647:
1634:
1629:
1616:
1611:
1598:
1593:
1580:
1575:
1562:
1557:
1540:
1538:
1537:
1532:
1527:
1523:
1522:
1521:
1509:
1508:
1487:
1482:
1469:
1464:
1451:
1446:
1433:
1428:
1415:
1410:
1397:
1392:
1373:
1371:
1370:
1365:
1363:
1362:
1346:
1344:
1343:
1338:
1326:
1324:
1323:
1318:
1283:
1272:
1270:
1269:
1264:
1256:
1253:
1245:
1244:
1239:
1233:
1220:
1218:
1217:
1212:
1210:
1203:
1202:
1184:
1180:
1175:
1173:
1172:
1157:
1152:
1151:
1146:
1131:
1123:
1119:
1114:
1106:
1080:
1078:
1077:
1072:
1070:
1065:
1045:
1011:For any regular
1007:
1005:
1004:
999:
997:
990:
989:
974:
973:
955:
951:
950:
935:
934:
916:
912:
911:
902:
893:
892:
881:
873:
868:
867:
858:
853:
852:
847:
841:
833:
829:
828:
819:
814:
794:
793:
784:
779:
778:
773:
767:
738:
736:
735:
730:
722:
713:
712:
697:
695:
694:
689:
684:
675:
674:
666:
657:
656:
645:
644:
614:
606:
515:
500:
494:
480:
478:
477:
472:
444:
360:). All internal
358:inscribed circle
351:
349:
348:
343:
341:
334:
330:
252:
246:
245:
240:
234:
233:
144:
143:
142:
138:
137:
133:
132:
126:
125:
124:
120:
119:
115:
114:
63:
51:
21:
6672:
6671:
6667:
6666:
6665:
6663:
6662:
6661:
6632:
6631:
6630:
6625:
6524:
6478:
6466:
6410:
6376:Tridecagon (13)
6366:Hendecagon (11)
6354:
6290:
6284:
6255:Right trapezoid
6176:
6128:
6118:
6088:
6057:
6050:
6043:
5926:
5919:
5912:
5876:
5869:
5862:
5826:
5819:
5653:Regular polygon
5646:
5637:
5630:
5626:
5619:
5615:
5606:
5597:
5590:
5586:
5574:
5568:
5564:
5552:
5534:
5523:
5492:
5469:
5428:
5427:
5424:
5419:
5413:Wayback Machine
5397:
5393:
5384:
5382:
5364:
5363:
5359:
5353:Wayback Machine
5344:
5340:
5331:
5327:
5318:
5316:
5296:
5295:
5291:
5282:
5278:
5270:
5266:
5249:
5245:
5228:
5195:
5194:
5187:
5178:
5176:
5169:
5154:
5153:
5149:
5142:
5138:
5134:
5078:Euclidean space
5058:
5051:
5046:
5037:
5030:
5021:
5019:hexagonal chess
5015:
5006:
4993:
4984:
4981:
4972:
4961:
4952:
4941:
4932:
4925:
4916:
4910:
4901:
4886:
4877:
4870:
4861:
4858:
4849:
4844:, the simplest
4839:
4830:
4827:
4818:
4812:
4803:
4797:
4788:
4782:
4773:
4772:mirror segments
4766:
4757:
4750:
4741:
4661:
4656:
4651:
4646:
4641:
4639:
4638:
4628:
4623:
4618:
4613:
4608:
4606:
4605:
4595:
4590:
4585:
4580:
4575:
4573:
4572:
4562:
4557:
4552:
4547:
4542:
4540:
4539:
4511:
4500:
4491:Hexagonal prism
4489:
4464:
4453:
4442:
4429:
4418:
4407:
4394:
4383:
4372:
4343:
4332:
4321:
4275:
4264:
4253:
4242:
4231:
4217:
4212:
4207:
4202:
4197:
4195:
4188:
4183:
4178:
4173:
4168:
4166:
4159:
4154:
4149:
4144:
4139:
4137:
4130:
4125:
4120:
4115:
4110:
4108:
4101:
4096:
4091:
4086:
4081:
4079:
4043:
4038:
4033:
4028:
4023:
4021:
4015:
4010:
4005:
4000:
3995:
3993:
3942:
3898:
3891:
3890:
3885:
3845:
3838:
3837:
3832:
3812:
3800:
3789:
3778:
3745:
3743:Petrie polygons
3733:
3722:
3696:
3693:with the same D
3661:
3648:
3624:
3567:
3566:
3521:
3487:
3457:symmedian point
3449:Lemoine hexagon
3445:
3426:
3414:
3408:
3385:Parallelogonal
3297:
3287:
3278:
3269:
3258:
3249:
3240:
3230:
3224:
3218:
3196:
3179:
3161:
3146:
3074:
3066:
3056:
3048:
3043:
2923:
2904:Schläfli symbol
2900:
2797:Parallelograms
2747:
2743:
2742:
2741:
2689:
2678:
2673:
2668:
2666:
2647:
2642:
2637:
2635:
2614:
2609:
2604:
2602:
2601:
2599:
2587:
2582:
2577:
2575:
2574:
2572:
2562:
2553:
2547:
2541:
2535:
2529:
2520:
2511:
2502:
2493:
2484:
2475:
2464:
2306:
2302:
2298:
2294:
2286:
2282:
2278:
2274:
2269:regular hexagon
2230:
2206:
2193:
2180:
2169:
2167:
2165:
2156:
2145:
2143:
2134:
2132:
2121:
2119:
2106:
2091:
2089:
2068:
1959:
1955:
1954:
1949:
1948:
1924:
1919:
1918:
1886:
1876:
1849:
1836:
1835:
1831:
1830:
1829:
1825:
1709:
1708:
1678:
1665:
1664:
1660:
1544:
1543:
1513:
1500:
1499:
1495:
1379:
1378:
1354:
1349:
1348:
1329:
1328:
1309:
1308:
1305:
1281:
1246:
1234:
1226:
1225:
1208:
1207:
1194:
1182:
1181:
1164:
1132:
1121:
1120:
1107:
1098:
1086:
1085:
1039:
1038:
995:
994:
981:
965:
953:
952:
942:
926:
914:
913:
903:
859:
842:
831:
830:
820:
785:
768:
759:
747:
746:
700:
699:
636:
600:
599:
526:
521:
520:
519:
518:
517:
513:
509:. Transfer the
498:
495:
487:
486:
463:
462:
445:
431:Voronoi diagram
391:
356:(radius of the
323:
322:
321:, which equals
278:Schläfli symbol
267:
265:Regular hexagon
160:
140:
135:
130:
128:
127:
122:
117:
112:
110:
96:Schläfli symbol
75:Regular polygon
66:
54:Regular hexagon
49:
42:
35:
28:
23:
22:
15:
12:
11:
5:
6670:
6668:
6660:
6659:
6654:
6649:
6644:
6634:
6633:
6627:
6626:
6624:
6623:
6618:
6613:
6608:
6603:
6598:
6593:
6588:
6583:
6581:Pseudotriangle
6578:
6573:
6568:
6563:
6558:
6553:
6548:
6543:
6538:
6532:
6530:
6526:
6525:
6523:
6522:
6517:
6512:
6507:
6502:
6497:
6492:
6487:
6481:
6479:
6472:
6471:
6468:
6467:
6465:
6464:
6459:
6454:
6449:
6444:
6439:
6434:
6429:
6424:
6418:
6416:
6412:
6411:
6409:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6371:Dodecagon (12)
6368:
6362:
6360:
6356:
6355:
6353:
6352:
6347:
6342:
6337:
6332:
6327:
6322:
6317:
6312:
6307:
6301:
6299:
6292:
6286:
6285:
6283:
6282:
6277:
6272:
6267:
6262:
6257:
6252:
6247:
6242:
6237:
6232:
6227:
6222:
6217:
6212:
6207:
6202:
6197:
6192:
6186:
6184:
6182:Quadrilaterals
6178:
6177:
6175:
6174:
6169:
6164:
6159:
6154:
6149:
6144:
6138:
6136:
6130:
6129:
6119:
6117:
6116:
6109:
6102:
6094:
6085:
6084:
6069:
6068:
6059:
6055:
6048:
6041:
6037:
6028:
6011:
6002:
5991:
5990:
5988:
5986:
5981:
5972:
5967:
5961:
5960:
5958:
5956:
5951:
5942:
5937:
5931:
5930:
5928:
5924:
5917:
5910:
5906:
5901:
5892:
5887:
5881:
5880:
5878:
5874:
5867:
5860:
5856:
5851:
5842:
5837:
5831:
5830:
5828:
5824:
5817:
5813:
5808:
5799:
5794:
5788:
5787:
5785:
5783:
5778:
5769:
5764:
5758:
5757:
5748:
5743:
5738:
5729:
5724:
5718:
5717:
5708:
5706:
5701:
5692:
5687:
5681:
5680:
5675:
5670:
5665:
5660:
5655:
5649:
5648:
5644:
5640:
5635:
5624:
5613:
5604:
5595:
5588:
5582:
5572:
5566:
5560:
5554:
5548:
5542:
5536:
5535:
5524:
5522:
5521:
5514:
5507:
5499:
5494:
5490:
5489:
5482:internet video
5466:
5456:
5445:
5444:
5423:
5422:External links
5420:
5418:
5417:
5391:
5357:
5338:
5325:
5289:
5276:
5264:
5243:
5185:
5167:
5147:
5135:
5133:
5130:
5129:
5128:
5123:
5117:
5112:
5106:
5100:
5097:regular tiling
5090:
5085:
5080:
5057:
5054:
5053:
5052:
5047:
5040:
5038:
5031:
5024:
5022:
5016:
5009:
5007:
4998:, a hexagonal
4994:
4987:
4985:
4983:Hexagonal barn
4982:
4975:
4973:
4962:
4955:
4953:
4942:
4935:
4933:
4926:
4919:
4917:
4911:
4904:
4902:
4887:
4880:
4878:
4871:
4864:
4862:
4859:
4852:
4850:
4840:
4833:
4831:
4828:
4821:
4819:
4813:
4806:
4804:
4798:
4791:
4789:
4783:
4776:
4774:
4767:
4760:
4758:
4751:
4744:
4740:
4737:
4734:
4733:
4726:
4719:
4712:
4704:
4703:
4697:
4696:
4689:
4682:
4675:
4667:
4666:
4633:
4600:
4567:
4533:
4532:
4529:
4525:
4524:
4516:
4515:
4504:
4493:
4481:
4480:
4479:with hexagons
4469:
4468:
4457:
4446:
4434:
4433:
4422:
4411:
4399:
4398:
4387:
4376:
4364:
4363:
4354:Johnson solids
4348:
4347:
4336:
4334:Chamfered cube
4325:
4313:
4312:
4307:
4302:
4296:
4295:
4280:
4279:
4268:
4257:
4246:
4235:
4223:
4222:
4193:
4164:
4135:
4106:
4076:
4075:
4070:
4065:
4059:
4058:
3946:Platonic solid
3941:
3938:
3937:
3936:
3925:
3920:
3915:
3910:
3905:
3901:
3883:
3877:
3876:
3865:
3862:
3857:
3852:
3848:
3830:
3811:
3808:
3805:
3804:
3793:
3791:3-3 duopyramid
3782:
3770:
3769:
3766:
3749:Petrie polygon
3744:
3741:
3738:
3737:
3726:
3694:
3659:
3647:
3644:
3623:
3620:
3619:
3618:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3520:
3517:
3513:acute triangle
3444:
3443:Cyclic hexagon
3441:
3425:
3422:
3410:Main article:
3407:
3404:
3401:
3400:
3393:
3386:
3382:
3381:
3374:
3367:
3363:
3362:
3357:
3352:
3317:hexagonal grid
3296:
3293:
3290:
3289:
3280:
3271:
3262:
3251:
3242:
3232:
3231:
3228:
3225:
3222:
3219:
3216:
3198:There are six
3195:
3192:
3189:
3188:
3186:Complete graph
3183:
3176:
3175:Dissected {6}
3173:
3163:
3158:
3151:
3148:
3142:
3141:
3134:
3127:
3120:
3113:
3106:
3099:
3092:
3080:
3079:
3071:
3063:
3053:
3050:
3045:
3039:
3038:
3031:
3024:
3017:
3010:
3003:
2996:
2989:
2921:
2899:
2896:
2893:
2892:
2887:
2881:
2880:
2873:
2866:
2859:
2851:
2850:
2847:
2844:
2840:
2839:
2833:
2829:
2828:
2821:
2814:
2807:
2799:
2798:
2795:
2792:
2788:
2787:
2766:Petrie polygon
2725:
2724:
2717:
2710:
2702:
2701:
2698:
2688:
2685:
2664:Dynkin diagram
2633:Dynkin diagram
2620:
2619:
2600:G2 group roots
2592:
2573:A2 group roots
2561:
2558:
2555:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2523:
2522:
2513:
2504:
2495:
2486:
2477:
2468:
2456:
2455:
2449:
2443:
2433:
2427:
2421:
2382:directed edges
2304:
2300:
2296:
2292:
2284:
2280:
2276:
2272:
2228:
2215:
2214:
2211:
2210:
2208:
2199:
2197:
2195:
2186:
2183:
2182:
2173:
2158:
2149:
2136:
2125:
2111:
2110:
2108:
2099:
2097:
2095:
2093:
2082:
2075:
2074:
2067:
2064:
2063:
2062:
2051:
2046:
2041:
2037:
2031:
2026:
2023:
2020:
2016:
2012:
2009:
2004:
1999:
1993:
1988:
1984:
1978:
1973:
1970:
1967:
1963:
1958:
1931:
1927:
1915:
1914:
1903:
1899:
1893:
1889:
1883:
1879:
1875:
1872:
1867:
1862:
1856:
1852:
1848:
1843:
1839:
1834:
1828:
1824:
1821:
1816:
1811:
1807:
1803:
1798:
1793:
1789:
1785:
1780:
1775:
1771:
1767:
1762:
1757:
1753:
1749:
1744:
1739:
1735:
1731:
1726:
1721:
1717:
1706:
1695:
1691:
1685:
1681:
1677:
1672:
1668:
1663:
1659:
1656:
1651:
1646:
1642:
1638:
1633:
1628:
1624:
1620:
1615:
1610:
1606:
1602:
1597:
1592:
1588:
1584:
1579:
1574:
1570:
1566:
1561:
1556:
1552:
1541:
1530:
1526:
1520:
1516:
1512:
1507:
1503:
1498:
1494:
1491:
1486:
1481:
1477:
1473:
1468:
1463:
1459:
1455:
1450:
1445:
1441:
1437:
1432:
1427:
1423:
1419:
1414:
1409:
1405:
1401:
1396:
1391:
1387:
1361:
1357:
1336:
1316:
1304:
1303:Point in plane
1301:
1262:
1259:
1252:
1249:
1242:
1237:
1222:
1221:
1206:
1201:
1197:
1193:
1190:
1187:
1185:
1183:
1178:
1171:
1167:
1163:
1160:
1155:
1149:
1144:
1141:
1138:
1135:
1129:
1126:
1124:
1122:
1117:
1113:
1110:
1104:
1101:
1099:
1097:
1094:
1093:
1068:
1063:
1060:
1057:
1054:
1051:
1048:
1009:
1008:
993:
988:
984:
980:
977:
972:
968:
964:
961:
958:
956:
954:
949:
945:
941:
938:
933:
929:
925:
922:
919:
917:
915:
910:
906:
900:
896:
890:
887:
884:
879:
876:
871:
866:
862:
856:
850:
845:
839:
836:
834:
832:
827:
823:
817:
812:
809:
806:
803:
800:
797:
792:
788:
782:
776:
771:
765:
762:
760:
758:
755:
754:
740:
739:
728:
725:
720:
716:
710:
707:
687:
682:
678:
672:
669:
664:
660:
654:
651:
648:
643:
639:
635:
632:
629:
626:
623:
620:
617:
612:
609:
525:
522:
496:
489:
488:
470:
446:
439:
438:
437:
436:
435:
416:tile the plane
389:
386:dihedral group
337:
333:
266:
263:
214:
213:
210:
204:
203:
182:
178:
177:
174:
167:Internal angle
163:
162:
158:
152:
150:Symmetry group
146:
145:
108:
102:
101:
98:
92:
91:
88:
78:
77:
72:
68:
67:
64:
56:
55:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6669:
6658:
6655:
6653:
6650:
6648:
6645:
6643:
6640:
6639:
6637:
6622:
6621:Weakly simple
6619:
6617:
6614:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6592:
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6561:Infinite skew
6559:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6533:
6531:
6527:
6521:
6518:
6516:
6513:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6482:
6480:
6477:
6476:Star polygons
6473:
6463:
6462:Apeirogon (∞)
6460:
6458:
6455:
6453:
6450:
6448:
6445:
6443:
6440:
6438:
6435:
6433:
6430:
6428:
6425:
6423:
6420:
6419:
6417:
6413:
6407:
6406:Icosagon (20)
6404:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6372:
6369:
6367:
6364:
6363:
6361:
6357:
6351:
6348:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6302:
6300:
6296:
6293:
6287:
6281:
6278:
6276:
6273:
6271:
6268:
6266:
6263:
6261:
6258:
6256:
6253:
6251:
6248:
6246:
6243:
6241:
6240:Parallelogram
6238:
6236:
6235:Orthodiagonal
6233:
6231:
6228:
6226:
6223:
6221:
6218:
6216:
6215:Ex-tangential
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6187:
6185:
6183:
6179:
6173:
6170:
6168:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6139:
6137:
6135:
6131:
6126:
6122:
6115:
6110:
6108:
6103:
6101:
6096:
6095:
6092:
6083:
6079:
6075:
6070:
6067:
6063:
6060:
6058:
6051:
6044:
6038:
6036:
6032:
6029:
6027:
6023:
6019:
6015:
6012:
6010:
6006:
6003:
6001:
5997:
5993:
5992:
5989:
5987:
5985:
5982:
5980:
5976:
5973:
5971:
5968:
5966:
5963:
5962:
5959:
5957:
5955:
5952:
5950:
5946:
5943:
5941:
5938:
5936:
5933:
5932:
5929:
5927:
5920:
5913:
5907:
5905:
5902:
5900:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5879:
5877:
5870:
5863:
5857:
5855:
5852:
5850:
5846:
5843:
5841:
5838:
5836:
5833:
5832:
5829:
5827:
5820:
5814:
5812:
5809:
5807:
5803:
5800:
5798:
5795:
5793:
5790:
5789:
5786:
5784:
5782:
5779:
5777:
5773:
5770:
5768:
5765:
5763:
5760:
5759:
5756:
5752:
5749:
5747:
5744:
5742:
5741:Demitesseract
5739:
5737:
5733:
5730:
5728:
5725:
5723:
5720:
5719:
5716:
5712:
5709:
5707:
5705:
5702:
5700:
5696:
5693:
5691:
5688:
5686:
5683:
5682:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5650:
5647:
5641:
5638:
5634:
5627:
5623:
5616:
5612:
5607:
5603:
5598:
5594:
5589:
5587:
5585:
5581:
5571:
5567:
5565:
5563:
5559:
5555:
5553:
5551:
5547:
5543:
5541:
5538:
5537:
5532:
5528:
5520:
5515:
5513:
5508:
5506:
5501:
5500:
5497:
5493:
5487:
5483:
5480:
5476:
5472:
5467:
5464:
5460:
5457:
5454:
5450:
5447:
5446:
5440:
5439:
5434:
5431:
5426:
5425:
5421:
5414:
5410:
5407:
5404:
5402:
5395:
5392:
5380:
5376:
5372:
5368:
5361:
5358:
5354:
5350:
5347:
5342:
5339:
5335:
5329:
5326:
5314:
5310:
5306:
5305:
5300:
5293:
5290:
5286:
5280:
5277:
5273:
5268:
5265:
5261:
5257:
5253:
5247:
5244:
5239:
5233:
5225:
5221:
5216:
5211:
5207:
5203:
5199:
5192:
5190:
5186:
5174:
5170:
5168:9780521098595
5164:
5160:
5159:
5151:
5148:
5145:
5140:
5137:
5131:
5127:
5124:
5121:
5118:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5075:
5071:
5067:
5063:
5060:
5059:
5055:
5050:
5044:
5039:
5035:
5028:
5023:
5020:
5013:
5008:
5005:
5001:
4997:
4991:
4986:
4979:
4974:
4970:
4966:
4959:
4954:
4950:
4946:
4939:
4934:
4930:
4923:
4918:
4915:
4908:
4903:
4899:
4895:
4892:columns from
4891:
4884:
4879:
4875:
4868:
4863:
4856:
4851:
4847:
4843:
4837:
4832:
4825:
4820:
4816:
4810:
4805:
4802:
4795:
4790:
4787:
4780:
4775:
4771:
4764:
4759:
4755:
4748:
4743:
4738:
4731:
4727:
4724:
4720:
4717:
4713:
4710:
4706:
4705:
4702:
4698:
4694:
4690:
4687:
4683:
4680:
4676:
4673:
4669:
4668:
4637:
4634:
4604:
4601:
4571:
4568:
4538:
4535:
4534:
4527:
4526:
4521:
4514:
4509:
4505:
4503:
4498:
4494:
4492:
4487:
4483:
4478:
4474:
4467:
4462:
4458:
4456:
4451:
4447:
4445:
4440:
4436:
4432:
4427:
4423:
4421:
4416:
4412:
4410:
4405:
4401:
4397:
4392:
4388:
4386:
4381:
4377:
4375:
4370:
4366:
4360:
4357:
4355:
4346:
4341:
4337:
4335:
4330:
4326:
4324:
4319:
4315:
4314:
4311:
4308:
4306:
4303:
4301:
4298:
4297:
4292:
4289:
4287:
4278:
4273:
4269:
4267:
4262:
4258:
4256:
4251:
4247:
4245:
4240:
4236:
4234:
4229:
4225:
4194:
4165:
4136:
4107:
4078:
4077:
4074:
4069:
4064:
4061:
4060:
4057:
4052:
4049:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3963:
3959:
3955:
3951:
3947:
3939:
3923:
3918:
3913:
3908:
3903:
3899:
3889:
3888:
3887:
3882:
3863:
3860:
3855:
3850:
3846:
3836:
3835:
3834:
3829:
3825:
3821:
3817:
3809:
3803:
3798:
3794:
3792:
3787:
3783:
3781:
3776:
3772:
3767:
3763:
3760:
3758:
3754:
3750:
3742:
3736:
3731:
3727:
3725:
3720:
3716:
3715:
3709:
3707:
3703:
3698:
3692:
3688:
3684:
3679:
3677:
3673:
3669:
3657:
3652:
3645:
3643:
3641:
3637:
3628:
3621:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3565:
3564:
3563:
3561:
3557:
3553:
3549:
3545:
3541:
3537:
3532:
3530:
3526:
3525:tangent lines
3518:
3516:
3514:
3510:
3505:
3503:
3497:
3494:
3490:
3485:
3481:
3477:
3473:
3469:
3465:
3460:
3458:
3454:
3450:
3442:
3440:
3438:
3434:
3433:conic section
3430:
3423:
3421:
3419:
3413:
3405:
3398:
3394:
3391:
3387:
3384:
3383:
3379:
3375:
3372:
3368:
3365:
3364:
3361:
3358:
3356:
3353:
3350:
3349:
3343:
3341:
3337:
3333:
3332:parallelogons
3328:
3326:
3322:
3318:
3314:
3310:
3301:
3294:
3285:
3281:
3276:
3272:
3267:
3263:
3261:
3256:
3252:
3247:
3243:
3241:Figure-eight
3238:
3234:
3226:
3213:
3207:
3205:
3201:
3193:
3187:
3184:
3182:
3177:
3174:
3172:
3168:
3164:
3159:
3156:
3152:
3149:
3144:
3143:
3139:
3135:
3132:
3128:
3125:
3121:
3118:
3114:
3111:
3107:
3104:
3100:
3097:
3093:
3090:
3086:
3085:
3078:
3072:
3070:
3064:
3062:
3059:
3054:
3046:
3041:
3040:
3036:
3032:
3029:
3025:
3022:
3018:
3015:
3011:
3008:
3004:
3001:
2997:
2994:
2990:
2987:
2983:
2982:
2979:
2977:
2973:
2969:
2965:
2960:
2958:
2954:
2950:
2946:
2942:
2938:
2934:
2930:
2925:
2919:
2916:
2911:
2909:
2905:
2897:
2891:
2886:
2878:
2874:
2871:
2867:
2864:
2860:
2857:
2853:
2846:Square faces
2841:
2838:
2837:parallelogons
2831:
2826:
2822:
2819:
2815:
2812:
2808:
2805:
2801:
2793:
2789:
2784:
2781:
2779:
2775:
2774:parallelogons
2771:
2767:
2763:
2757:
2753:
2739:
2735:
2731:
2722:
2718:
2715:
2711:
2708:
2704:
2703:
2696:
2693:
2692:
2686:
2684:
2665:
2661:
2658:
2653:
2634:
2630:
2627:
2597:
2593:
2570:
2566:
2565:
2559:
2549:
2537:
2531:
2525:
2518:
2514:
2509:
2505:
2500:
2496:
2491:
2487:
2482:
2478:
2473:
2469:
2467:
2462:
2458:
2457:
2453:
2450:
2447:
2441:
2437:
2434:
2431:
2428:
2425:
2422:
2419:
2415:
2412:
2411:
2408:
2406:
2402:
2401:parallelogons
2398:
2394:
2390:
2385:
2383:
2379:
2374:
2372:
2371:parallelogons
2368:
2364:
2360:
2356:
2352:
2349:
2345:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2313:
2308:
2290:
2270:
2262:
2258:
2254:
2250:
2246:
2241:
2234:
2226:
2221:
2209:
2204:
2200:
2198:
2196:
2191:
2187:
2185:
2178:
2174:
2172:
2163:
2159:
2154:
2150:
2148:
2141:
2137:
2130:
2126:
2124:
2117:
2113:
2109:
2104:
2100:
2098:
2094:
2087:
2083:
2081:
2077:
2076:
2072:
2071:
2065:
2049:
2044:
2039:
2035:
2029:
2024:
2021:
2018:
2014:
2010:
2007:
2002:
1997:
1991:
1986:
1982:
1976:
1971:
1968:
1965:
1961:
1956:
1947:
1946:
1945:
1929:
1925:
1901:
1897:
1891:
1887:
1881:
1877:
1873:
1870:
1865:
1860:
1854:
1850:
1846:
1841:
1837:
1832:
1826:
1822:
1819:
1814:
1809:
1805:
1801:
1796:
1791:
1787:
1783:
1778:
1773:
1769:
1765:
1760:
1755:
1751:
1747:
1742:
1737:
1733:
1729:
1724:
1719:
1715:
1707:
1693:
1689:
1683:
1679:
1675:
1670:
1666:
1661:
1657:
1654:
1649:
1644:
1640:
1636:
1631:
1626:
1622:
1618:
1613:
1608:
1604:
1600:
1595:
1590:
1586:
1582:
1577:
1572:
1568:
1564:
1559:
1554:
1550:
1542:
1528:
1524:
1518:
1514:
1510:
1505:
1501:
1496:
1492:
1489:
1484:
1479:
1475:
1471:
1466:
1461:
1457:
1453:
1448:
1443:
1439:
1435:
1430:
1425:
1421:
1417:
1412:
1407:
1403:
1399:
1394:
1389:
1385:
1377:
1376:
1375:
1359:
1355:
1334:
1314:
1302:
1300:
1298:
1294:
1290:
1285:
1278:
1276:
1260:
1257:
1250:
1247:
1240:
1235:
1204:
1199:
1195:
1191:
1188:
1186:
1176:
1169:
1165:
1161:
1158:
1153:
1147:
1142:
1139:
1136:
1133:
1127:
1125:
1115:
1111:
1108:
1102:
1100:
1095:
1084:
1083:
1082:
1066:
1061:
1058:
1055:
1052:
1049:
1046:
1037:
1033:
1029:
1025:
1021:
1018:
1014:
991:
986:
982:
978:
975:
970:
966:
962:
959:
957:
947:
943:
939:
936:
931:
927:
923:
920:
918:
908:
904:
898:
894:
888:
885:
882:
877:
874:
869:
864:
860:
854:
848:
843:
837:
835:
825:
821:
815:
810:
807:
804:
801:
798:
795:
790:
786:
780:
774:
769:
763:
761:
756:
745:
744:
743:
726:
723:
718:
714:
708:
705:
685:
680:
676:
670:
667:
662:
658:
652:
649:
641:
637:
630:
627:
624:
621:
618:
615:
610:
607:
598:
597:
596:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
551:= side length
550:
546:
542:
538:
534:
530:
523:
512:
508:
504:
493:
484:
483:Fermat primes
468:
460:
459:
454:
450:
443:
434:
432:
428:
425:
421:
420:tessellations
417:
413:
410:
406:
401:
399:
395:
387:
383:
379:
375:
371:
367:
363:
359:
355:
335:
331:
320:
316:
311:
309:
305:
301:
297:
293:
288:
286:
283:
279:
275:
273:
264:
262:
260:
256:
251:
239:
229:
225:
221:
211:
209:
205:
202:
198:
194:
190:
186:
183:
179:
175:
172:
168:
164:
156:
153:
151:
147:
109:
107:
103:
99:
97:
93:
89:
87:
83:
79:
76:
73:
69:
62:
57:
52:
47:
40:
33:
19:
6415:>20 sides
6350:Decagon (10)
6335:Heptagon (7)
6329:
6325:Pentagon (5)
6315:Triangle (3)
6210:Equidiagonal
6061:
6030:
6021:
6013:
6004:
5995:
5975:10-orthoplex
5711:Dodecahedron
5672:
5632:
5621:
5610:
5601:
5592:
5583:
5579:
5569:
5561:
5557:
5549:
5545:
5491:
5436:
5398:
5394:
5383:. Retrieved
5374:
5370:
5360:
5341:
5333:
5328:
5317:. Retrieved
5308:
5302:
5292:
5284:
5279:
5267:
5246:
5232:cite journal
5205:
5201:
5177:, retrieved
5157:
5150:
5144:Cube picture
5139:
4944:
4351:
4283:
4054:Hexagons in
3992:of the form
3944:There is no
3943:
3880:
3878:
3827:
3823:
3815:
3813:
3780:3-3 duoprism
3746:
3699:
3682:
3680:
3675:
3672:skew polygon
3668:skew hexagon
3667:
3665:
3658:, symmetry D
3646:Skew hexagon
3633:
3559:
3555:
3551:
3547:
3543:
3539:
3533:
3522:
3509:circumcircle
3506:
3498:
3492:
3488:
3483:
3479:
3475:
3471:
3467:
3463:
3461:
3446:
3427:
3415:
3329:
3306:
3288:Triple-tail
3279:Double-tail
3250:Center-flip
3197:
3167:skew hexagon
3155:star polygon
2961:
2926:
2912:
2901:
2832:Regular {6}
2755:
2751:
2737:
2728:
2654:
2623:
2451:
2445:
2439:
2435:
2429:
2423:
2417:
2413:
2396:
2392:
2388:
2386:
2377:
2375:
2366:
2358:
2354:
2343:
2331:
2323:
2319:
2315:
2309:
2268:
2266:
2260:
2256:
2252:
2248:
2244:
2232:
2171:parallelogon
1916:
1306:
1289:circumradius
1286:
1279:
1223:
1035:
1031:
1027:
1023:
1019:
1010:
741:
592:
584:
576:
572:
569:circumradius
564:
555:The maximal
554:
548:
540:
537:Circumradius
532:
511:line segment
503:intersection
456:
415:
402:
381:
373:
319:circumcircle
312:
289:
270:
268:
223:
217:
208:Dual polygon
161:), order 2×6
6611:Star-shaped
6586:Rectilinear
6556:Equilateral
6551:Equiangular
6515:Hendecagram
6359:11–20 sides
6340:Octagon (8)
6330:Hexagon (6)
6305:Monogon (1)
6147:Equilateral
5984:10-demicube
5945:9-orthoplex
5895:8-orthoplex
5845:7-orthoplex
5802:6-orthoplex
5772:5-orthoplex
5727:Pentachoron
5715:Icosahedron
5690:Tetrahedron
5377:: 105–114.
5311:: 243–246.
5208:: 335–355.
5072:facets, is
4996:The Hexagon
4943:In French,
4310:Icosahedral
4300:Tetrahedral
4073:Icosahedral
4063:Tetrahedral
3970:soccer ball
3820:equilateral
3325:compression
3307:From bees'
3058:Star figure
3049:t{3} = {6}
2697:projection
2312:John Conway
451:, given by
409:equilateral
398:equilateral
296:equiangular
292:equilateral
193:equilateral
6642:6 (number)
6636:Categories
6616:Tangential
6520:Dodecagram
6298:1–10 sides
6289:By number
6270:Tangential
6250:Right kite
5970:10-simplex
5954:9-demicube
5904:8-demicube
5854:7-demicube
5811:6-demicube
5781:5-demicube
5695:Octahedron
5385:2015-04-12
5319:2014-11-17
5215:2010.12340
5179:2015-11-06
5132:References
4963:Hexagonal
4947:refers to
4945:l'Hexagone
4784:A beehive
4768:Assembled
4531:1-uniform
4305:Octahedral
4068:Octahedral
3950:tessellate
3886:such that
3833:such that
3735:Octahedron
3706:octahedron
3502:concurrent
3309:honeycombs
3270:Fish-tail
3181:octahedron
3178:projection
3073:Alternated
2937:alternated
2924:symmetry.
2835:Hexagonal
2687:Dissection
2225:reflection
524:Parameters
376:) and six
352:times the
308:tangential
181:Properties
6596:Reinhardt
6505:Enneagram
6495:Heptagram
6485:Pentagram
6452:65537-gon
6310:Digon (2)
6280:Trapezoid
6245:Rectangle
6195:Bicentric
6157:Isosceles
6134:Triangles
6018:orthoplex
5940:9-simplex
5890:8-simplex
5840:7-simplex
5797:6-simplex
5767:5-simplex
5736:Tesseract
5438:MathWorld
5433:"Hexagon"
5074:self-dual
5070:orthoplex
4786:honeycomb
4477:Prismoids
3986:truncated
3974:fullerene
3861:≤
3802:5-simplex
3640:centroids
3260:Unicursal
3202:with the
3169:, within
3065:Truncated
3055:Stellated
3047:Truncated
2964:dodecagon
2945:stellated
2933:dodecagon
2929:truncated
2915:truncated
2432:2 (2222)
2348:elongated
2135:directed
2015:∑
1962:∑
1258:≈
1251:π
1189:≈
1137:⋅
976:≈
960:≈
937:≈
921:≈
642:∘
631:
581:inscribed
427:honeycomb
412:triangles
300:bicentric
282:truncated
100:{6}, t{3}
6571:Isotoxal
6566:Isogonal
6510:Decagram
6500:Octagram
6490:Hexagram
6291:of sides
6220:Harmonic
6121:Polygons
6072:Topics:
6035:demicube
6000:polytope
5994:Uniform
5755:600-cell
5751:120-cell
5704:Demicube
5678:Pentagon
5658:Triangle
5486:CGP Grey
5479:animated
5409:Archived
5379:Archived
5349:Archived
5313:Archived
5173:archived
5120:Havannah
5103:Hexagram
5056:See also
4971:minerals
4965:Hanksite
4801:carapace
4754:graphene
4528:Regular
4288:G(2,0):
3980:and the
3366:Regular
3160:Extended
3147:hexagon
2949:hexagram
2353:, while
2336:isotoxal
2328:isogonal
2147:isogonal
2123:isotoxal
2092:regular
2066:Symmetry
1297:diagonal
1293:inradius
589:inradius
561:diagonal
557:diameter
545:Inradius
458:Elements
394:triangle
364:are 120
298:. It is
220:geometry
201:isotoxal
197:isogonal
155:Dihedral
86:vertices
18:Hexagons
6591:Regular
6536:Concave
6529:Classes
6437:257-gon
6260:Rhombus
6200:Crossed
6009:simplex
5979:10-cube
5746:24-cell
5732:16-cell
5673:Hexagon
5527:regular
5475:YouTube
5272:Coxeter
5062:24-cell
5000:theatre
4842:Benzene
4636:tr{6,3}
4603:rr{6,3}
3976:fame),
3753:regular
3311:to the
3145:Crossed
3075:h{6} =
3067:t{6} =
3042:Regular
2968:squares
2794:Rhombs
2746:⁄
2734:zonogon
2730:Coxeter
2426:(2*22)
2420:(*632)
2351:rhombus
2168:general
1273:of its
1017:apothem
1013:polygon
424:beehive
405:squares
366:degrees
354:apothem
274:hexagon
272:regular
255:polygon
224:hexagon
171:degrees
6601:Simple
6546:Cyclic
6541:Convex
6265:Square
6205:Cyclic
6167:Obtuse
6162:Kepler
5949:9-cube
5899:8-cube
5849:7-cube
5806:6-cube
5776:5-cube
5663:Square
5540:Family
5463:Hexnet
5258:
5165:
5034:Taiwan
4890:basalt
4570:r{6,3}
3634:If an
3558:, and
3511:of an
3453:cyclic
2695:6-cube
2448:(22*)
2442:(3*3)
2395:, and
2289:cyclic
1261:0.8270
1034:, and
963:0.6495
453:Euclid
362:angles
304:cyclic
259:simple
226:(from
189:cyclic
185:Convex
6576:Magic
6172:Right
6152:Ideal
6142:Acute
5668:p-gon
5477:– an
5210:arXiv
4770:E-ELT
4537:{6,3}
3670:is a
3451:is a
3351:Form
2454:(××)
2399:, as
2363:kites
2340:duals
2334:, an
2326:, an
2287:), 4
2271:has D
1192:3.464
1081:, so
979:0.866
940:3.464
924:2.598
403:Like
250:gonía
244:γωνία
228:Greek
82:Edges
6606:Skew
6230:Kite
6125:List
6026:cube
5699:Cube
5529:and
5256:ISBN
5238:link
5163:ISBN
5095:: a
5064:: a
4927:The
4020:and
3972:and
3968:(of
3914:>
3724:Cube
3704:and
3702:cube
3700:The
3447:The
3171:cube
3069:{12}
3061:2{3}
3044:{6}
2970:and
2885:Cube
2770:cube
2758:− 1)
2736:(a 2
2357:and
2291:: (Z
2267:The
1347:and
407:and
294:and
276:has
222:, a
212:Self
176:120°
84:and
71:Type
5575:(p)
5473:on
5461:on
5220:doi
5002:in
4896:in
3768:5D
3765:4D
3685:is
3493:bdf
3489:ace
3321:wax
3227:Dih
3221:Dih
3215:Dih
3077:{3}
2843:3D
2791:2D
2544:Dih
2532:Dih
2526:Dih
2521:a1
2512:p2
2503:d2
2494:d2
2485:g2
2476:i4
2466:r12
2446:pmg
2424:cmm
2397:r12
2316:r12
2303:, Z
2299:, Z
2295:, Z
2257:r12
2233:r12
2231:or
2207:a1
2194:g3
2181:p2
2157:d2
2107:i4
2090:r12
1917:If
1291:to
628:cos
455:'s
317:or
238:hex
218:In
6638::
6080:•
6076:•
6056:21
6052:•
6049:k1
6045:•
6042:k2
6020:•
5977:•
5947:•
5925:21
5921:•
5918:41
5914:•
5911:42
5897:•
5875:21
5871:•
5868:31
5864:•
5861:32
5847:•
5825:21
5821:•
5818:22
5804:•
5774:•
5753:•
5734:•
5713:•
5697:•
5629:/
5618:/
5608:/
5599:/
5577:/
5435:.
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5375:15
5373:.
5369:.
5309:14
5307:.
5301:.
5234:}}
5230:{{
5218:.
5206:11
5204:.
5200:.
5188:^
5171:,
4048:.
3964:,
3960:,
3814:A
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3695:3d
3681:A
3666:A
3660:3d
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3554:,
3550:,
3546:,
3542:,
3504:.
3496:.
3491:=
3482:,
3478:,
3474:,
3470:,
3466:,
3459:.
3327:.
3165:A
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2978:.
2959:.
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2660:G2
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2452:pg
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2324:p6
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2261:a1
2166:g2
2144:p6
2133:g6
2120:d6
1284:.
1277:.
1030:=
638:30
591:,
571:,
547:;
543:=
539:;
535:=
514:AB
499:AB
269:A
247:,
235:,
232:ἕξ
199:,
195:,
191:,
187:,
157:(D
6127:)
6123:(
6113:e
6106:t
6099:v
6064:-
6062:n
6054:k
6047:2
6040:1
6033:-
6031:n
6024:-
6022:n
6016:-
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6005:n
5998:-
5996:n
5923:4
5916:2
5909:1
5873:3
5866:2
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5816:1
5645:n
5643:H
5636:2
5633:G
5625:4
5622:F
5614:8
5611:E
5605:7
5602:E
5596:6
5593:E
5584:n
5580:D
5573:2
5570:I
5562:n
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5488:.
5455:.
5441:.
5415:.
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5388:.
5322:.
5240:)
5222::
5212::
5183:.
3924:.
3919:3
3909:a
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3864:2
3856:a
3851:1
3847:d
3831:1
3828:d
3824:a
3606:.
3603:f
3600:+
3597:d
3594:+
3591:b
3588:=
3585:e
3582:+
3579:c
3576:+
3573:a
3560:f
3556:e
3552:d
3548:c
3544:b
3540:a
3484:f
3480:e
3476:d
3472:c
3468:b
3464:a
3229:3
3223:1
3217:2
2922:3
2756:m
2754:(
2752:m
2748:2
2744:1
2738:m
2552:1
2550:Z
2546:1
2540:2
2538:Z
2534:2
2528:6
2440:m
2436:p
2430:p
2418:m
2416:6
2414:p
2305:1
2301:2
2297:3
2293:6
2285:2
2283:D
2277:6
2273:6
2263:.
2253:g
2249:p
2245:d
2229:6
2050:.
2045:4
2040:i
2036:d
2030:6
2025:1
2022:=
2019:i
2011:4
2008:=
2003:2
1998:)
1992:2
1987:i
1983:d
1977:6
1972:1
1969:=
1966:i
1957:(
1930:i
1926:d
1902:.
1898:)
1892:2
1888:L
1882:2
1878:R
1874:2
1871:+
1866:2
1861:)
1855:2
1851:L
1847:+
1842:2
1838:R
1833:(
1827:(
1823:3
1820:=
1815:4
1810:6
1806:d
1802:+
1797:4
1792:4
1788:d
1784:+
1779:4
1774:2
1770:d
1766:=
1761:4
1756:5
1752:d
1748:+
1743:4
1738:3
1734:d
1730:+
1725:4
1720:1
1716:d
1694:,
1690:)
1684:2
1680:L
1676:+
1671:2
1667:R
1662:(
1658:3
1655:=
1650:2
1645:6
1641:d
1637:+
1632:2
1627:4
1623:d
1619:+
1614:2
1609:2
1605:d
1601:=
1596:2
1591:5
1587:d
1583:+
1578:2
1573:3
1569:d
1565:+
1560:2
1555:1
1551:d
1529:,
1525:)
1519:2
1515:L
1511:+
1506:2
1502:R
1497:(
1493:2
1490:=
1485:2
1480:6
1476:d
1472:+
1467:2
1462:3
1458:d
1454:=
1449:2
1444:5
1440:d
1436:+
1431:2
1426:2
1422:d
1418:=
1413:2
1408:4
1404:d
1400:+
1395:2
1390:1
1386:d
1360:i
1356:d
1335:L
1315:R
1248:2
1241:3
1236:3
1205:.
1200:2
1196:r
1177:3
1170:2
1166:r
1162:2
1159:=
1154:2
1148:3
1143:r
1140:4
1134:r
1128:=
1116:2
1112:p
1109:a
1103:=
1096:A
1067:3
1062:r
1059:4
1056:=
1053:R
1050:6
1047:=
1036:p
1032:r
1028:a
1024:p
1020:a
992:.
987:2
983:d
971:2
967:D
948:2
944:r
932:2
928:R
909:2
905:d
899:2
895:3
889:=
886:d
883:D
878:4
875:3
870:=
865:2
861:D
855:8
849:3
844:3
838:=
826:2
822:r
816:3
811:2
808:=
805:r
802:R
799:3
796:=
791:2
787:R
781:2
775:3
770:3
764:=
757:A
727:.
724:D
719:2
715:3
709:=
706:d
686:t
681:2
677:3
671:=
668:R
663:2
659:3
653:=
650:R
647:)
634:(
625:=
622:r
619:=
616:d
611:2
608:1
593:r
585:d
577:t
573:R
565:D
549:t
541:r
533:R
485:.
469:=
390:6
388:D
380:(
372:(
336:3
332:2
173:)
169:(
159:6
90:6
48:.
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