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5003:
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3120:
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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:
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:
1081:
3923:
4855:
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
4926:
1068:
2658:
2594:
2668:
2604:
1534:
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4013:
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130:
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102:
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590:
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125:
107:
2632:
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1931:
1361:
5339:
1334:
1314:
468:
5228:
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}}}
4910:
4895:
4824:
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
4980:
4948:
4751:
5369:
5303:
5163:
6642:
4735:
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
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).}
1216:
6072:
2753:
with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a
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
5250:
3663:
with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A
5507:
5399:
2327:
hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are
3881:
6102:
5337:
5157:
3428:
until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
5017:
2393:
572:
circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base),
4968:
6637:
4812:
4443:
4419:
4408:
4384:
5188:
4843:
690:
2645:
138:
3828:
5002:
4624:
3348:
418:
are hexagonal for this reason and because the shape makes efficient use of space and building materials. The
313:
4454:
4432:
3420:(also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any
6225:
6205:
4373:
4265:
3970:
3627:
is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the
1029:
5033:
4782:
6647:
6200:
6157:
6132:
5465:
4917:
4902:
94:
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),}
4591:
4397:
2964:
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
5453:
680:{\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}
6260:
5530:
5073:
4959:
1284:
that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long
437:
27:
20:
3557:
6185:
5500:
4243:
3966:
1933:
are the distances from the vertices of a regular hexagon to any point on its circumcircle, then
6210:
6095:
5056:
4333:
3517:
2925:
33:"Hexagonal" redirects here. For the FIFA World Cup qualifying tournament in North America, see
6611:
6551:
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366:
358:
270:
5569:
5547:
5535:
5147:
385:
with a vertex at the center of the regular hexagon and sharing one side with the hexagon is
6495:
6265:
6195:
6137:
5701:
5648:
5116:
5105:
4298:
4288:
4232:
4061:
4051:
3950:
3679:
3644:
3624:
3155:
3065:
2960:
2941:
2929:
2906:
2641:, are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
2617:
2365:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
1909:
1339:
1263:
495:
446:
412:
386:
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273:
34:
8:
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284:
280:
181:
3807:
of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex
2909:, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D
430:
6606:
6147:
5925:
5875:
5825:
5782:
5752:
5712:
5675:
5493:
5200:
5099:
4994:
3639:
3524:
3248:
3192:
2878:
2766:
2300:
1319:
1299:
453:
296:
5253:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
2892:
266:
84:
6586:
6180:
6088:
6064:
5420:
5246:
5153:
4862:
4834:
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3942:
3675:
2945:
2232:
The dihedral symmetries are divided depending on whether they pass through vertices (
393:
288:
143:
74:
4328:
4273:
There are other symmetry polyhedra with stretched or flattened hexagons, like these
2044:{\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}
480:
6115:
6068:
5633:
5622:
5611:
5600:
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2614:
2454:
346:
5397:
4438:
4414:
4403:
4379:
4317:
3091:
2851:
2799:
2729:-gon whose opposite sides are parallel and of equal length) can be dissected into
2709:
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.
6581:
6561:
6556:
6526:
6245:
6220:
6152:
5658:
5643:
5403:
5343:
5068:
5009:
4479:
3978:
3520:
states that the three main diagonals AD, BE, and CF intersect at a single point.
3445:
3437:
3385:
3328:
3324:
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2702:
2351:
2324:
2316:
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419:
260:
189:
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70:
63:
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3779:
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3501:
3441:
3305:
3174:
3105:
3016:
2865:
2813:
2754:
2652:
2621:
374:
354:
292:
247:
177:
173:
159:
155:
5459:
4368:
3697:(same as triangular antiprism) have regular skew hexagons as petrie polygons.
3023:
2974:
49:
6626:
6566:
6417:
6310:
6230:
6172:
6025:
5913:
5906:
5899:
5863:
5856:
5849:
5813:
5806:
5423:
4681:
4342:
3610:
3425:
3421:
2370:
2319:
hexagon constructed by three mirrors can alternate long and short edges, and
2228:
407:(three hexagons meeting at every vertex), and so are useful for constructing
216:
5214:
4674:
4392:
4238:
3002:
2988:
545:
6596:
6466:
6422:
6386:
6376:
6371:
5965:
5134:
4306:
3938:
3768:
3660:
3513:
3497:
3475:, then the three main diagonals intersect in a single point if and only if
3320:
3304:, hexagonal patterns are prevalent in nature due to their efficiency. In a
3188:
3143:
2995:
2825:
2762:
2389:
2359:
2331:
of each other and have half the symmetry order of the regular hexagon. The
2328:
2277:
2159:
1296:
For an arbitrary point in the plane of a regular hexagon with circumradius
1277:
557:
525:
499:
491:
471:
422:
of a regular triangular lattice is the honeycomb tessellation of hexagons.
408:
307:
196:
5439:
3763:
3288:
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
6505:
6412:
6391:
6381:
5974:
5935:
5885:
5835:
5792:
5762:
5694:
5680:
5336:
Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter",
4986:
3958:
3615:
3077:
3046:
2487:
2478:
2460:
2449:
397:
4249:
4216:
3378:
3098:
2505:
2496:
2469:
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
6510:
6366:
6356:
6240:
5960:
5944:
5894:
5844:
5801:
5771:
5685:
4667:
4660:
4227:
3774:
3723:
3694:
3359:
3169:
2933:
2074:
1288:
of 1.0000000 will have a distance of 0.8660254 between parallel sides.
4485:
2208:
2191:
2178:
2165:
2150:
2141:
2128:
2117:
2104:
2091:
6485:
6475:
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6361:
6270:
6235:
6016:
5930:
5880:
5830:
5787:
5757:
5726:
5469:
5428:
5290:"Dao's theorem on six circumcenters associated with a cyclic hexagon"
5064:
5060:
4774:
3962:
3790:
3297:
3057:
2981:
2952:
2921:
2792:
569:
505:
four times on the circumscribed circle and connect the corner points.
415:
4889:; large masses must cool slowly to form a polygonal fracture pattern
4496:
4357:
3718:
517:
490:
is given, drawing a circular arc from point A and point B gives the
6490:
6480:
6437:
6396:
6325:
6315:
6305:
6124:
5990:
5745:
5741:
5668:
5476:
5205:
5093:
4955:
4789:
4742:
4465:
3628:
3272:
3263:
3254:
3243:
3234:
3225:
3049:
2937:
1285:
1281:
577:
549:
533:
400:
382:
208:
6080:
5358:"Equilateral triangles and Kiepert perspectors in complex numbers"
4474:
3126:
2944:
by adding a center point. This pattern repeats within the regular
6447:
6427:
6340:
6335:
6330:
6295:
6250:
6111:
5999:
5969:
5736:
5731:
5722:
5262:
5052:
4990:
4830:
3323:
and can also tile the plane by translation. In three dimensions,
2722:
2718:
2584:
2557:
2358:
hexagons, with opposite sides parallel are also called hexagonal
2339:
1005:
1001:
342:
243:
5265:, Mathematical recreations and Essays, Thirteenth edition, p.141
3744:, uniform and dual polyhedra and polytopes, shown in these skew
6255:
5939:
5889:
5839:
5796:
5766:
5717:
5653:
5024:
4878:
4806:, a hexagonal cloud pattern around the north pole of the planet
4718:
4711:
4704:
3785:
2956:
2844:
2695:
2683:
441:
4697:
3707:
6300:
4758:
4727:
350:
237:
231:
225:
219:
3119:
3112:
2775:
Dissection of hexagons into three rhombs and parallelograms
1255:{\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}
5689:
3712:
3690:
3319:
Irregular hexagons with parallel opposite edges are called
3159:
3009:
2873:
2765:
and projective directions of the cube are dissected within
2758:
5113:: 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
5189:"Cyclic Averages of Regular Polygons and Platonic Solids"
3631:
of opposite triangles form another equilateral triangle.
3309:
2392:
can tessellate the
Euclidean plane by translation. Other
3507:
2899:, {6,3}, with three hexagonal faces around each vertex.
584:. The maxima and minima are related by the same factor:
302:
The common length of the sides equals the radius of the
3412:
279:
A regular hexagon is defined as a hexagon that is both
3667:
has vertices alternating between two parallel planes.
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
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
5287:
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:
5227:: 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
6624:
5418:
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
5102:: single path, six-sided star, within a hexagon
2886:
2749:parallelograms. In particular this is true for
5193:Communications in Mathematics and Applications
2940:. A regular hexagon can be dissected into six
568:. The minimal diameter or the diameter of the
6096:
5501:
5355:
3798:
1015:. For the regular hexagon these are given by
276:, t{3}, which alternates two types of edges.
5186:
4933:
4920:mirror is composed of 18 hexagonal segments.
3941:, not allowing the result to "fold up". The
3394:
2963:around it. This pattern repeats within the
2902:A regular hexagon can also be created as a
6103:
6089:
5508:
5494:
5444:construction with compass and straightedge
4728:Gallery of natural and artificial hexagons
3928:
3312:to construct and gain much strength under
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
5456:a website devoted to hexagon mathematics.
5204:
5152:, Cambridge University Press, p. 9,
5145:
5096:: six-sided star within a regular hexagon
3182:
250:(non-self-intersecting) hexagon is 720°.
5327:, Dover Publications, 2007 (orig. 1960).
5182:
5180:
3858:{\displaystyle {\frac {d_{1}}{a}}\leq 2}
3638:
3614:
3327:with parallel opposite faces are called
3287:
2227:
2207:
516:
6073:List of regular polytopes and compounds
3283:
1213:The regular hexagon fills the fraction
334:{\displaystyle {\tfrac {2}{\sqrt {3}}}}
6625:
5440:Definition and properties of a hexagon
5317:
3512:Let ABCDEF be a hexagon formed by six
6084:
5450:An Introduction to Hexagonal Geometry
5419:
5177:
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
5383:
5059: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
6110:
5274:Cartensen, Jens, "About hexagons",
5245:, (2008) The Symmetries of Things,
4741:The ideal crystalline structure of
3973:. These hexagons can be considered
2548:
1063:{\displaystyle {}=6R=4r{\sqrt {3}}}
13:
3945:with some hexagonal faces are the
3731:
2761:, with 3 of 6 square faces. Other
253:
14:
6659:
5412:
4942:for its vaguely hexagonal shape.
3496:If a hexagon has vertices on the
3431:
2394:hexagon shapes can tile the plane
1291:
576:, is twice the minimal radius or
556:, is twice the maximal radius or
295:(has a circumscribed circle) and
5032:
5016:
5001:
4979:
4967:
4947:
4925:
4909:
4894:
4870:
4854:
4842:
4823:
4811:
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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:
6643:Polygons by the number of sides
5442:with interactive animation and
5372:from the original on 2015-07-05
5349:
5306:from the original on 2014-12-05
5241:John H. Conway, Heidi Burgiel,
5166: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
3634:
3527:and that has consecutive sides
548:(which corresponds to the long
5330:
5281:
5268:
5256:
5235:
5139:
5128:
4512:Tilings with regular hexagons
3335:Hexagonal prism tessellations
2679:
2552:
2216:of a regular hexagon, with Dih
2058:
731:The area of a regular hexagon
635:
622:
1:
5187:Meskhishvili, Mamuka (2020).
5146:Wenninger, Magnus J. (1974),
5122:
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 (
2675:
2396:with different orientations.
2062:Example hexagons by symmetry
1276:It follows from the ratio of
512:
5288:Dergiades, Nikolaos (2014).
4455:triangular hebesphenorotunda
4433:triaugmented hexagonal prism
3600:{\displaystyle a+c+e=b+d+f.}
3072:
2969:
2887:Related polygons and tilings
357:. A regular hexagon has six
238:
226:
19:For the crystal system, see
7:
5325:Advanced Euclidean Geometry
5046:
4374:elongated triangular cupola
4266:truncated icosidodecahedron
3971:truncated icosidodecahedron
2248:and no symmetry is labeled
2054:
1271:PE + PF = PA + PB + PC + PD
299:(has an inscribed circle).
10:
6664:
6062:
5489:
5461:Hexagons are the Bestagons
5390: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:
6519:
6465:
6405:
6349:
6288:
6279:
6171:
6123:
5278:33(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:
195:
169:
154:
137:
93:
83:
69:
59:
47:
42:
5074:Hexagonal crystal system
4960: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
5215:10.26713/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
6638:Constructible polygons
5356:Dao Thanh Oai (2015).
5346:, Accessed 2012-04-17.
5217:(inactive 2024-09-12).
5090:of hexagons in a plane
4934:
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:
5276:Mathematical Spectrum
5243:Chaim Goodman-Strauss
4958: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
6336:Nonagon/Enneagon (9)
6266:Tangential trapezoid
5117:Central place theory
5106:Honeycomb conjecture
5008: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
6448:Megagon (1,000,000)
6216:Isosceles trapezoid
6057:pentagonal polytope
5956:Uniform 10-polytope
5516:Fundamental convex
5392:Crux Mathematicorum
5362:Forum Geometricorum
5323:Johnson, Roger A.,
5295:Forum Geometricorum
4940: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:
6418:Icositetragon (24)
5926:Uniform 9-polytope
5876:Uniform 8-polytope
5826:Uniform 7-polytope
5783:Uniform 6-polytope
5753:Uniform 5-polytope
5713:Uniform polychoron
5676:Uniform polyhedron
5524:in dimensions 2–10
5475:about hexagons by
5421:Weisstein, Eric W.
5402:2017-08-30 at the
5342:2012-05-11 at the
5100:Unicursal hexagram
4995: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:
6648:Elementary shapes
6620:
6619:
6461:
6460:
6438:Myriagon (10,000)
6423:Triacontagon (30)
6387:Heptadecagon (17)
6377:Pentadecagon (15)
6372:Tetradecagon (14)
6311:Quadrilateral (4)
6181:Antiparallelogram
6078:
6077:
6065:Polytope families
5522:uniform polytopes
5251:978-1-56881-220-5
5149:Polyhedron Models
5027: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
6655:
6433:Chiliagon (1000)
6413:Icositrigon (23)
6392:Octadecagon (18)
6382:Hexadecagon (16)
6286:
6285:
6105:
6098:
6091:
6082:
6081:
6069:Regular polytope
5630:
5619:
5608:
5567:
5510:
5503:
5496:
5487:
5486:
5462:
5434:
5433:
5407:
5387:
5381:
5380:
5378:
5377:
5353:
5347:
5334:
5328:
5321:
5315:
5314:
5312:
5311:
5285:
5279:
5272:
5266:
5260:
5254:
5239:
5233:
5232:
5226:
5218:
5208:
5184:
5175:
5173:
5172:
5171:
5143:
5137:
5132:
5084:Hexagonal tiling
5079:Hexagonal number
5067:and tessellates
5057:four-dimensional
5040:Hexagonal window
5036:
5023:Pavilion in the
5020:
5005:
4983:
4971:
4951:
4937:
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:
4508:
4499:
4488:
4477:
4462:
4461:
4452:
4441:
4430:
4417:
4406:
4395:
4382:
4371:
4360:
4348:
4347:
4331:
4320:
4309:
4280:
4279:
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:
4039:
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:
3750:
3721:
3710:
3703:
3699:
3606:
3604:
3603:
3598:
3484:
3418:Pascal's theorem
3407:Conway criterion
3401:Hexagonal tiling
3388:
3381:
3369:
3362:
3344:Hexagonal tiling
3337:
3333:
3329:parallelohedrons
3325:hexagonal prisms
3302:Giant's Causeway
3275:
3266:
3257:
3246:
3237:
3228:
3201:
3197:
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:
2771:
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:
2398:
2311:is no symmetry.
2194:
2181:
2168:
2153:
2144:
2131:
2120:
2107:
2094:
2077:
2068:
2067:
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:
39:
6663:
6662:
6658:
6657:
6656:
6654:
6653:
6652:
6623:
6622:
6621:
6616:
6515:
6469:
6457:
6401:
6367:Tridecagon (13)
6357:Hendecagon (11)
6345:
6281:
6275:
6246:Right trapezoid
6167:
6119:
6109:
6079:
6048:
6041:
6034:
5917:
5910:
5903:
5867:
5860:
5853:
5817:
5810:
5644:Regular polygon
5637:
5628:
5621:
5617:
5610:
5606:
5597:
5588:
5581:
5577:
5565:
5559:
5555:
5543:
5525:
5514:
5483:
5460:
5415:
5410:
5404:Wayback Machine
5388:
5384:
5375:
5373:
5354:
5350:
5344:Wayback Machine
5335:
5331:
5322:
5318:
5309:
5307:
5286:
5282:
5273:
5269:
5261:
5257:
5240:
5236:
5220:
5219:
5185:
5178:
5169:
5167:
5160:
5144:
5140:
5133:
5129:
5125:
5069:Euclidean space
5049:
5042:
5037:
5028:
5021:
5012:
5010:hexagonal chess
5006:
4997:
4984:
4975:
4972:
4963:
4952:
4943:
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:
3905:
3891:
3887:
3885:
3883:
3880:
3879:
3874:
3838:
3834:
3832:
3830:
3827:
3826:
3821:
3801:
3789:
3778:
3767:
3734:
3732:Petrie polygons
3722:
3711:
3685:
3682:with the same D
3650:
3637:
3613:
3559:
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:
2032:
2027:
2017:
2006:
1990:
1979:
1974:
1964:
1953:
1948:
1944:
1943:
1941:
1938:
1937:
1917:
1913:
1911:
1908:
1907:
1879:
1875:
1869:
1865:
1853:
1842:
1838:
1829:
1825:
1824:
1820:
1819:
1818:
1814:
1802:
1797:
1784:
1779:
1766:
1761:
1748:
1743:
1730:
1725:
1712:
1707:
1701:
1698:
1697:
1671:
1667:
1658:
1654:
1653:
1649:
1637:
1632:
1619:
1614:
1601:
1596:
1583:
1578:
1565:
1560:
1547:
1542:
1536:
1533:
1532:
1506:
1502:
1493:
1489:
1488:
1484:
1472:
1467:
1454:
1449:
1436:
1431:
1418:
1413:
1400:
1395:
1382:
1377:
1371:
1368:
1367:
1347:
1343:
1341:
1338:
1337:
1321:
1318:
1317:
1301:
1298:
1297:
1294:
1270:
1235:
1227:
1223:
1220:
1218:
1215:
1214:
1197:
1196:
1187:
1183:
1171:
1170:
1163:
1157:
1153:
1134:
1121:
1119:
1110:
1109:
1096:
1094:
1087:
1080:
1078:
1075:
1074:
1053:
1033:
1031:
1028:
1027:
984:
983:
974:
970:
958:
954:
942:
941:
935:
931:
919:
915:
903:
902:
896:
892:
880:
861:
852:
848:
835:
831:
829:
820:
819:
813:
809:
802:
778:
774:
761:
757:
755:
748:
741:
739:
736:
735:
700:
692:
689:
688:
662:
644:
629:
625:
594:
592:
589:
588:
515:
510:
509:
508:
507:
506:
502:
498:. Transfer the
487:
484:
476:
475:
455:
452:
451:
434:
420:Voronoi diagram
380:
345:(radius of the
317:
315:
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:
6661:
6651:
6650:
6645:
6640:
6635:
6618:
6617:
6615:
6614:
6609:
6604:
6599:
6594:
6589:
6584:
6579:
6574:
6572:Pseudotriangle
6569:
6564:
6559:
6554:
6549:
6544:
6539:
6534:
6529:
6523:
6521:
6517:
6516:
6514:
6513:
6508:
6503:
6498:
6493:
6488:
6483:
6478:
6472:
6470:
6463:
6462:
6459:
6458:
6456:
6455:
6450:
6445:
6440:
6435:
6430:
6425:
6420:
6415:
6409:
6407:
6403:
6402:
6400:
6399:
6394:
6389:
6384:
6379:
6374:
6369:
6364:
6362:Dodecagon (12)
6359:
6353:
6351:
6347:
6346:
6344:
6343:
6338:
6333:
6328:
6323:
6318:
6313:
6308:
6303:
6298:
6292:
6290:
6283:
6277:
6276:
6274:
6273:
6268:
6263:
6258:
6253:
6248:
6243:
6238:
6233:
6228:
6223:
6218:
6213:
6208:
6203:
6198:
6193:
6188:
6183:
6177:
6175:
6173:Quadrilaterals
6169:
6168:
6166:
6165:
6160:
6155:
6150:
6145:
6140:
6135:
6129:
6127:
6121:
6120:
6108:
6107:
6100:
6093:
6085:
6076:
6075:
6060:
6059:
6050:
6046:
6039:
6032:
6028:
6019:
6002:
5993:
5982:
5981:
5979:
5977:
5972:
5963:
5958:
5952:
5951:
5949:
5947:
5942:
5933:
5928:
5922:
5921:
5919:
5915:
5908:
5901:
5897:
5892:
5883:
5878:
5872:
5871:
5869:
5865:
5858:
5851:
5847:
5842:
5833:
5828:
5822:
5821:
5819:
5815:
5808:
5804:
5799:
5790:
5785:
5779:
5778:
5776:
5774:
5769:
5760:
5755:
5749:
5748:
5739:
5734:
5729:
5720:
5715:
5709:
5708:
5699:
5697:
5692:
5683:
5678:
5672:
5671:
5666:
5661:
5656:
5651:
5646:
5640:
5639:
5635:
5631:
5626:
5615:
5604:
5595:
5586:
5579:
5573:
5563:
5557:
5551:
5545:
5539:
5533:
5527:
5526:
5515:
5513:
5512:
5505:
5498:
5490:
5485:
5481:
5480:
5473:internet video
5457:
5447:
5436:
5435:
5414:
5413:External links
5411:
5409:
5408:
5382:
5348:
5329:
5316:
5280:
5267:
5255:
5234:
5176:
5158:
5138:
5126:
5124:
5121:
5120:
5119:
5114:
5108:
5103:
5097:
5091:
5088:regular tiling
5081:
5076:
5071:
5048:
5045:
5044:
5043:
5038:
5031:
5029:
5022:
5015:
5013:
5007:
5000:
4998:
4989:, a hexagonal
4985:
4978:
4976:
4974:Hexagonal barn
4973:
4966:
4964:
4953:
4946:
4944:
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:
9:
6:
4:
3:
2:
6660:
6649:
6646:
6644:
6641:
6639:
6636:
6634:
6631:
6630:
6628:
6613:
6612:Weakly simple
6610:
6608:
6605:
6603:
6600:
6598:
6595:
6593:
6590:
6588:
6585:
6583:
6580:
6578:
6575:
6573:
6570:
6568:
6565:
6563:
6560:
6558:
6555:
6553:
6552:Infinite skew
6550:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6524:
6522:
6518:
6512:
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6473:
6471:
6468:
6467:Star polygons
6464:
6454:
6453:Apeirogon (∞)
6451:
6449:
6446:
6444:
6441:
6439:
6436:
6434:
6431:
6429:
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6410:
6408:
6404:
6398:
6397:Icosagon (20)
6395:
6393:
6390:
6388:
6385:
6383:
6380:
6378:
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6354:
6352:
6348:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6317:
6314:
6312:
6309:
6307:
6304:
6302:
6299:
6297:
6294:
6293:
6291:
6287:
6284:
6278:
6272:
6269:
6267:
6264:
6262:
6259:
6257:
6254:
6252:
6249:
6247:
6244:
6242:
6239:
6237:
6234:
6232:
6231:Parallelogram
6229:
6227:
6226:Orthodiagonal
6224:
6222:
6219:
6217:
6214:
6212:
6209:
6207:
6206:Ex-tangential
6204:
6202:
6199:
6197:
6194:
6192:
6189:
6187:
6184:
6182:
6179:
6178:
6176:
6174:
6170:
6164:
6161:
6159:
6156:
6154:
6151:
6149:
6146:
6144:
6141:
6139:
6136:
6134:
6131:
6130:
6128:
6126:
6122:
6117:
6113:
6106:
6101:
6099:
6094:
6092:
6087:
6086:
6083:
6074:
6070:
6066:
6061:
6058:
6054:
6051:
6049:
6042:
6035:
6029:
6027:
6023:
6020:
6018:
6014:
6010:
6006:
6003:
6001:
5997:
5994:
5992:
5988:
5984:
5983:
5980:
5978:
5976:
5973:
5971:
5967:
5964:
5962:
5959:
5957:
5954:
5953:
5950:
5948:
5946:
5943:
5941:
5937:
5934:
5932:
5929:
5927:
5924:
5923:
5920:
5918:
5911:
5904:
5898:
5896:
5893:
5891:
5887:
5884:
5882:
5879:
5877:
5874:
5873:
5870:
5868:
5861:
5854:
5848:
5846:
5843:
5841:
5837:
5834:
5832:
5829:
5827:
5824:
5823:
5820:
5818:
5811:
5805:
5803:
5800:
5798:
5794:
5791:
5789:
5786:
5784:
5781:
5780:
5777:
5775:
5773:
5770:
5768:
5764:
5761:
5759:
5756:
5754:
5751:
5750:
5747:
5743:
5740:
5738:
5735:
5733:
5732:Demitesseract
5730:
5728:
5724:
5721:
5719:
5716:
5714:
5711:
5710:
5707:
5703:
5700:
5698:
5696:
5693:
5691:
5687:
5684:
5682:
5679:
5677:
5674:
5673:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5641:
5638:
5632:
5629:
5625:
5618:
5614:
5607:
5603:
5598:
5594:
5589:
5585:
5580:
5578:
5576:
5572:
5562:
5558:
5556:
5554:
5550:
5546:
5544:
5542:
5538:
5534:
5532:
5529:
5528:
5523:
5519:
5511:
5506:
5504:
5499:
5497:
5492:
5491:
5488:
5484:
5478:
5474:
5471:
5467:
5463:
5458:
5455:
5451:
5448:
5445:
5441:
5438:
5437:
5431:
5430:
5425:
5422:
5417:
5416:
5405:
5401:
5398:
5395:
5393:
5386:
5371:
5367:
5363:
5359:
5352:
5345:
5341:
5338:
5333:
5326:
5320:
5305:
5301:
5297:
5296:
5291:
5284:
5277:
5271:
5264:
5259:
5252:
5248:
5244:
5238:
5230:
5224:
5216:
5212:
5207:
5202:
5198:
5194:
5190:
5183:
5181:
5165:
5161:
5159:9780521098595
5155:
5151:
5150:
5142:
5136:
5131:
5127:
5118:
5115:
5112:
5109:
5107:
5104:
5101:
5098:
5095:
5092:
5089:
5085:
5082:
5080:
5077:
5075:
5072:
5070:
5066:
5062:
5058:
5054:
5051:
5050:
5041:
5035:
5030:
5026:
5019:
5014:
5011:
5004:
4999:
4996:
4992:
4988:
4982:
4977:
4970:
4965:
4961:
4957:
4950:
4945:
4941:
4936:
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:
4731:
4720:
4716:
4713:
4709:
4706:
4702:
4699:
4695:
4694:
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:
3912:
3907:
3902:
3897:
3892:
3888:
3878:
3877:
3876:
3871:
3852:
3849:
3844:
3839:
3835:
3825:
3824:
3823:
3818:
3814:
3810:
3806:
3792:
3787:
3783:
3781:
3776:
3772:
3770:
3765:
3761:
3756:
3752:
3749:
3747:
3743:
3739:
3725:
3720:
3716:
3714:
3709:
3705:
3704:
3698:
3696:
3692:
3687:
3681:
3677:
3673:
3668:
3666:
3662:
3658:
3646:
3641:
3632:
3630:
3626:
3617:
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
3505:
3503:
3499:
3494:
3492:
3486:
3483:
3479:
3474:
3470:
3466:
3462:
3458:
3454:
3449:
3447:
3443:
3439:
3429:
3427:
3423:
3422:conic section
3419:
3410:
3408:
3402:
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:
3274:
3270:
3265:
3261:
3256:
3252:
3250:
3245:
3241:
3236:
3232:
3230:Figure-eight
3227:
3223:
3215:
3202:
3196:
3194:
3190:
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:
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:
2673:
2654:
2650:
2647:
2642:
2623:
2619:
2616:
2586:
2582:
2559:
2555:
2554:
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:
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:
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:
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:
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:
6406:>20 sides
6341:Decagon (10)
6326:Heptagon (7)
6320:
6316:Pentagon (5)
6306:Triangle (3)
6201:Equidiagonal
6052:
6021:
6012:
6004:
5995:
5986:
5966:10-orthoplex
5702:Dodecahedron
5663:
5623:
5612:
5601:
5592:
5583:
5574:
5570:
5560:
5552:
5548:
5540:
5536:
5482:
5427:
5389:
5385:
5374:. Retrieved
5365:
5361:
5351:
5332:
5324:
5319:
5308:. Retrieved
5299:
5293:
5283:
5275:
5270:
5258:
5237:
5223:cite journal
5196:
5192:
5168:, retrieved
5148:
5141:
5135:Cube picture
5130:
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
6602:Star-shaped
6577:Rectilinear
6547:Equilateral
6542:Equiangular
6506:Hendecagram
6350:11–20 sides
6331:Octagon (8)
6321:Hexagon (6)
6296:Monogon (1)
6138:Equilateral
5975:10-demicube
5936:9-orthoplex
5886:8-orthoplex
5836:7-orthoplex
5793:6-orthoplex
5763:5-orthoplex
5718:Pentachoron
5706:Icosahedron
5681:Tetrahedron
5368:: 105–114.
5302:: 243–246.
5199:: 335–355.
5063:facets, is
4987: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
6633:6 (number)
6627:Categories
6607:Tangential
6511:Dodecagram
6289:1–10 sides
6280:By number
6261:Tangential
6241:Right kite
5961:10-simplex
5945:9-demicube
5895:8-demicube
5845:7-demicube
5802:6-demicube
5772:5-demicube
5686:Octahedron
5376:2015-04-12
5310:2014-11-17
5206:2010.12340
5170:2015-11-06
5123:References
4954:Hexagonal
4938:refers to
4935: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
6587:Reinhardt
6496:Enneagram
6486:Heptagram
6476:Pentagram
6443:65537-gon
6301:Digon (2)
6271:Trapezoid
6236:Rectangle
6186:Bicentric
6148:Isosceles
6125:Triangles
6009:orthoplex
5931:9-simplex
5881:8-simplex
5831:7-simplex
5788:6-simplex
5758:5-simplex
5727:Tesseract
5429:MathWorld
5424:"Hexagon"
5065:self-dual
5061: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}
6562:Isotoxal
6557:Isogonal
6501:Decagram
6491:Octagram
6481:Hexagram
6282:of sides
6211:Harmonic
6112:Polygons
6063:Topics:
6026:demicube
5991:polytope
5985:Uniform
5746:600-cell
5742:120-cell
5695:Demicube
5669:Pentagon
5649:Triangle
5477:CGP Grey
5470:animated
5400:Archived
5370:Archived
5340:Archived
5304:Archived
5164:archived
5111:Havannah
5094:Hexagram
5047:See also
4962:minerals
4956: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
6582:Regular
6527:Concave
6520:Classes
6428:257-gon
6251:Rhombus
6191:Crossed
6000:simplex
5970:10-cube
5737:24-cell
5723:16-cell
5664:Hexagon
5518:regular
5466:YouTube
5263:Coxeter
5053:24-cell
4991: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
6592:Simple
6537:Cyclic
6532:Convex
6256:Square
6196:Cyclic
6158:Obtuse
6153:Kepler
5940:9-cube
5890:8-cube
5840:7-cube
5797:6-cube
5767:5-cube
5654:Square
5531:Family
5454:Hexnet
5249:
5156:
5025: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
6567:Magic
6163:Right
6143:Ideal
6133:Acute
5659:p-gon
5468:– an
5201: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
6597:Skew
6221:Kite
6116:List
6017:cube
5690:Cube
5520:and
5247:ISBN
5229:link
5154:ISBN
5086:: a
5055:: 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
5566:(p)
5464:on
5452:on
5211:doi
4993: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
6629::
6071:•
6067:•
6047:21
6043:•
6040:k1
6036:•
6033:k2
6011:•
5968:•
5938:•
5916:21
5912:•
5909:41
5905:•
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5888:•
5866:21
5862:•
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5855:•
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5838:•
5816:21
5812:•
5809:22
5795:•
5765:•
5744:•
5725:•
5704:•
5688:•
5620:/
5609:/
5599:/
5590:/
5568:/
5426:.
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5364:.
5360:.
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5298:.
5292:.
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5221:{{
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3649:3d
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3493:.
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3467:,
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1273:.
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1019:=
627:30
580:,
560:,
536:;
532:=
528:;
524:=
503:AB
488:AB
258:A
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224:,
221:ἕξ
188:,
184:,
180:,
176:,
146:(D
6118:)
6114:(
6104:e
6097:t
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6053:n
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5987:n
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3586:d
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3580:b
3577:=
3574:e
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3568:c
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2535:1
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2523:2
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2429:m
2425:p
2419:p
2407:m
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2294:1
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2252:.
2242:g
2238:p
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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:+
1855:2
1850:)
1844:2
1840:L
1836:+
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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
1155:r
1151:2
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
933:r
921:2
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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:=
746:A
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
582:r
574:d
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562:R
554:D
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474:.
458:=
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377:D
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