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2540:{\displaystyle {\begin{aligned}\textstyle \sum _{i=1}^{5}d_{i}^{2}&=5\left(R^{2}+L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{4}&=5\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{6}&=5\left(\left(R^{2}+L^{2}\right)^{3}+6R^{2}L^{2}\left(R^{2}+L^{2}\right)\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{8}&=5\left(\left(R^{2}+L^{2}\right)^{4}+12R^{2}L^{2}\left(R^{2}+L^{2}\right)^{2}+6R^{4}L^{4}\right).\end{aligned}}}
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4241:
935:{\displaystyle {\begin{aligned}H&={\frac {\sqrt {5+2{\sqrt {5}}}}{2}}~t\approx 1.539~t,\\W=D&={\frac {1+{\sqrt {5}}}{2}}~t\approx 1.618~t,\\W&={\sqrt {2-{\frac {2}{\sqrt {5}}}}}\cdot H\approx 1.051~H,\\R&={\sqrt {\frac {5+{\sqrt {5}}}{10}}}t\approx 0.8507~t,\\D&=R\ {\sqrt {\frac {5+{\sqrt {5}}}{2}}}=2R\cos 18^{\circ }=2R\cos {\frac {\pi }{10}}\approx 1.902~R.\end{aligned}}}
1204:
51:
4152:
4159:
4166:
1710:
971:
3901:
There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of
3231:
1854:
3460:
pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a
3832:(where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:
1368:
3897:
and WĂśden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has the optimal density among all packings of regular pentagons in the plane.
2698:
1199:{\displaystyle {\begin{aligned}A&={\frac {t^{2}{\sqrt {25+10{\sqrt {5}}}}}{4}}={\frac {5t^{2}\tan 54^{\circ }}{4}}\\&={\frac {{\sqrt {5(5+2{\sqrt {5}})}}\;t^{2}}{4}}={\frac {t^{2}{\sqrt {4\varphi ^{5}+3}}}{4}}\approx 1.720~t^{2}\end{aligned}}}
1567:
3782:
3443:
An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique
1446:
2923:
1749:
4478:
3292:
on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to make a
4416:
4396:
3215:
7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is the rightmost intersection of the horizontal line with the original
1954:
976:
584:
3034:
2984:
2740:
The top panel shows the construction used in
Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point
3887:
3601:
3089:
2819:
1255:
3902:
the pentagon must alternate around the pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all
3310:
Symmetries of a regular pentagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
1705:{\displaystyle A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{2}}={\frac {5t^{2}\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{4}}}
2583:
1532:
4292:
3472:. It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational.
3609:
4459:
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1941:
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963:
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531:
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343:
319:
4256:
1868:. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
5029:
4357:
4685:
3281:
into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a
6207:
5642:
4790:
4315:
1379:
4816:
4774:
4140:
1849:{\displaystyle r={\frac {t}{2\tan {\mathord {\left({\frac {\pi }{5}}\right)}}}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.}
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5672:
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4744:
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4641:
210:
6202:
220:
5023:
228:
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3235:
2731:
One method to construct a regular pentagon in a given circle is described by
Richmond and further discussed in Cromwell's
2992:
5795:
5775:
4209:
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202:
3842:
6217:
5770:
5727:
5702:
4549:
4214:
3527:
3041:
1363:{\displaystyle t=R\ {\sqrt {\frac {5-{\sqrt {5}}}{2}}}=2R\sin 36^{\circ }=2R\sin {\frac {\pi }{5}}\approx 1.176~R,}
4240:
3373:
when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as
3046:
3036:, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos(54°), and CQ = 1 â 2cos(54°), which equals âcos(108°) by the cosine
35:
30:
This article is about the geometric figure. For the headquarters of the United States
Department of Defense, see
4581:
3250:, either by inscribing one in a given circle or constructing one on a given edge. This process was described by
2790:
5830:
4913:
4518:
3247:
2748:
is marked halfway along its radius. This point is joined to the periphery vertically above the center at point
2709:
3926:. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry.
3815:
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a
3132:
Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point
5755:
5070:
4219:
2693:{\displaystyle 3\left(\textstyle \sum _{i=1}^{5}d_{i}^{2}\right)^{2}=10\textstyle \sum _{i=1}^{5}d_{i}^{4}.}
3919:, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
3105:
5780:
5665:
4969:
4544:
3894:
2784:
4716:
6181:
6121:
5760:
5614:
5607:
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4808:
4114:
3819:(one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that
3243:
1498:
451:
4186:
3117:. This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:
6065:
5835:
5765:
5707:
5271:
5218:
4135:
4122:
4106:
3777:{\displaystyle 3(a^{2}+b^{2}+c^{2}+d^{2}+e^{2})>d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}+d_{5}^{2}}
3430:
3256:
3221:
8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.
3037:
1865:
6212:
6171:
6146:
6116:
6111:
6070:
5785:
5626:
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5275:
4951:
4130:
3268:
455:
271:
6176:
5717:
5495:
5445:
5395:
5352:
5322:
5282:
5245:
5063:
4977:
4911:
Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area",
4892:
4708:
3350:
3294:
3114:
2826:
2577:
are the distances from the vertices of a regular pentagon to any point on its circumcircle, then
1728:
4172:
3410:
2716:. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
1454:
433:
192:
4179:
3483:
3230:
3202:
The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
400:
6156:
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5000:
4812:
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4664:
4658:
4637:
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3413:
is {5/2}. Its sides form the diagonals of a regular convex pentagon â in this arrangement the
3319:
233:
182:
157:
69:
4633:
3139:
Construct a vertical line through the center. Mark one intersection with the circle as point
5685:
5638:
5203:
5192:
5181:
5170:
5161:
5152:
5139:
5117:
5105:
5091:
5087:
4922:
4884:
4855:
4734:
4700:
4627:
4567:"pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.
3804:
3469:
2733:
4936:
3799:
3380:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
3169:. Mark its intersection with the horizontal line (inside the original circle) as the point
2553:
1919:
6151:
6131:
6126:
6096:
5815:
5790:
5722:
5228:
5213:
4932:
4430:. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
4262:
3836:
3794:
3462:
3403:
445:
427:
279:
275:
178:
171:
65:
3448:
similarity, because it is equilateral and it is equiangular (its five angles are equal).
370:
473:
6161:
6141:
6106:
6101:
5732:
5712:
5578:
4488:
3890:
3816:
3808:
3457:
3361:. The dihedral symmetries are divided depending on whether they pass through vertices (
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1899:
1879:
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1212:
948:
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496:
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304:
267:
263:
249:
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115:
111:
89:
56:
4819:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
6196:
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5742:
5595:
5483:
5476:
5469:
5433:
5426:
5419:
5383:
5376:
5100:
4533:
4501:
4380:
3468:
There exist cyclic pentagons with rational sides and rational area; these are called
3385:
3278:
3435:
1896:, whose distances to the centroid of the regular pentagon and its five vertices are
6166:
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5992:
5956:
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2713:
467:
322:
286:
131:
31:
5046:
5017:
4957:
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4021:
4014:
4007:
3981:
3974:
3967:
3960:
3953:
533:(distance between two farthest separated points, which equals the diagonal length
297:
6075:
5982:
5961:
5951:
5544:
5505:
5455:
5405:
5362:
5332:
5264:
5250:
4348:
3353:
labels these by a letter and group order. Full symmetry of the regular form is
6080:
5936:
5926:
5810:
5530:
5514:
5464:
5414:
5371:
5341:
5255:
5041:
5003:
4927:
4828:
Weisstein, Eric W. "Cyclic
Pentagon." From MathWorld--A Wolfram Web Resource.
4484:
4364:
50:
3113:
The
Carlyle circle was invented as a geometric method to find the roots of a
1739:
of the inscribed circle, of a regular pentagon is related to the side length
1484:
the regular pentagon fills approximately 0.7568 of its circumscribed circle.
6055:
6045:
6022:
6012:
6002:
5931:
5840:
5805:
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5500:
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4528:
4321:
4299:
4282:
4277:
4204:
3414:
3397:
3282:
1876:
For an arbitrary point in the plane of a regular pentagon with circumradius
1727:
Similar to every regular convex polygon, the regular convex pentagon has an
137:
5035:
3206:
Steps 6â8 are equivalent to the following version, shown in the animation:
3199:. It intersects the original circle at two of the vertices of the pentagon.
3188:. It intersects the original circle at two of the vertices of the pentagon.
4158:
4151:
6060:
6050:
6007:
5966:
5895:
5875:
5694:
5560:
5315:
5311:
4443:
4344:
3349:
These 4 symmetries can be seen in 4 distinct symmetries on the pentagon.
3277:
A regular pentagon may be created from just a strip of paper by tying an
1546:
459:
81:
17:
4798:
and A.P. Rollett, second edition, 1961 (Oxford
University Press), p. 57.
1441:{\displaystyle A={\frac {5R^{2}}{4}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}};}
6017:
5997:
5910:
5905:
5900:
5890:
5865:
5820:
5681:
5569:
5539:
5306:
5301:
5292:
5233:
4896:
4860:
4841:
4712:
3439:
Equilateral pentagon built with four equal circles disposed in a chain.
3289:
1732:
1550:
107:
4165:
5825:
5509:
5459:
5409:
5366:
5336:
5287:
5223:
4427:
3251:
3122:
2918:{\displaystyle \tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,}
1864:
Like every regular convex polygon, the regular convex pentagon has a
4888:
4704:
4510:, a polyhedron whose regular form is composed of 12 pentagonal faces
2756:
is bisected, and the bisector intersects the vertical axis at point
5042:
Renaissance artists' approximate constructions of regular pentagons
4982:
5870:
5650:
4875:
Robbins, D. P. (1995). "Areas of
Polygons Inscribed in a Circle".
4303:
3798:
3445:
3434:
3267:
3104:
296:
5020:
constructing an inscribed pentagon with compass and straightedge.
4686:"Carlyle circles and Lemoine simplicity of polygon constructions"
2787:
and two sides, the hypotenuse of the larger triangle is found as
5259:
4247:
5654:
3465:
whose coefficients are functions of the sides of the pentagon.
3906:. To find the number of sides this polygon has, the result is
2775:
To determine the length of this side, the two right triangles
3125:
in which to inscribe the pentagon and mark the center point
3234:
Euclid's method for pentagon at a given circle, using the
5049:
How to calculate various dimensions of regular pentagons.
4605:"A Construction for a Regular Polygon of Seventeen Sides"
4582:"Cyclic Averages of Regular Polygons and Platonic Solids"
945:
The area of a convex regular pentagon with side length
513:(distance from one side to the opposite vertex), width
4545:
Pythagoras' theorem#Similar figures on the three sides
3384:
subgroup has no degrees of freedom but can be seen as
2794:
2649:
2595:
2340:
2176:
2045:
1957:
4763:
Euklid's
Elements of Geometry, Book 4, Proposition 11
4609:
The
Quarterly Journal of Pure and Applied Mathematics
3845:
3612:
3530:
3486:
3173:
and its intersection outside the circle as the point
3049:
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951:
582:
559:
539:
519:
499:
476:
403:
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1860:
Chords from the circumscribed circle to the vertices
6089:
6035:
5975:
5919:
5858:
5849:
5741:
5693:
4265:, like many other flowers, have a pentagonal shape.
2932:are known from the larger triangle. The result is:
458:of order 5 (through 72°, 144°, 216° and 288°). The
285:
259:
244:
227:
201:
191:
177:
167:
150:
64:
43:
4769:. Translated by Richard Fitzpatrick. p. 119.
3881:
3776:
3595:
3516:
3083:
3029:{\displaystyle m\angle \mathrm {CDP} =54^{\circ }}
3028:
2978:
2917:
2813:
2692:
2569:
2539:
1935:
1908:
1888:
1848:
1704:
1553:). Substituting the regular pentagon's values for
1526:
1476:
1440:
1362:
1241:
1221:
1198:
957:
934:
565:
545:
525:
505:
485:
412:
385:
357:
337:
313:
3807:of equal-sized regular pentagons on a plane is a
3330:there is one subgroup with dihedral symmetry: Dih
3210:6a. Construct point F as the midpoint of O and W.
3297:. The base of the pyramid is a regular pentagon.
2825:of the smaller triangle then is found using the
2772:is the required side of the inscribed pentagon.
1229:of a regular pentagon is given, its edge length
3893:packing shown. In a preprint released in 2016,
2989:If DP is truly the side of a regular pentagon,
2979:{\displaystyle h={\frac {{\sqrt {5}}-1}{4}}\ .}
4586:Communications in Mathematics and Applications
3882:{\displaystyle (5-{\sqrt {5}})/3\approx 0.921}
1451:since the area of the circumscribed circle is
5666:
5071:
3596:{\displaystyle d_{1},d_{2},d_{3},d_{4},d_{5}}
8:
3811:structure which covers 92.131% of the plane.
3084:{\displaystyle \left({\sqrt {5}}-1\right)/4}
4383:, also echinoderms with a pentagonal shape.
3922:There are 15 classes of pentagons that can
2708:The regular pentagon is constructible with
5855:
5673:
5659:
5651:
5078:
5064:
5056:
4974:Packings of regular pentagons in the plane
2814:{\displaystyle \scriptstyle {\sqrt {5}}/2}
1111:
5036:Definition and properties of the pentagon
4981:
4926:
4859:
4842:"Areas of Polygons Inscribed in a Circle"
3865:
3855:
3844:
3768:
3763:
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2014:
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306:
4663:(2nd ed.). CRC Press. p. 329.
4575:
4573:
4504:; A pentagon is an order-4 associahedron
4324:is another fruit with fivefold symmetry.
3928:
3305:
3229:
2723:
5643:List of regular polytopes and compounds
4660:CRC concise encyclopedia of mathematics
4560:
4455:
4410:. The faces are true regular pentagons.
4392:
4333:
4236:
4550:Trigonometric constants for a pentagon
147:
40:
5026:with only a compass and straightedge.
4306:contains five carpels, arranged in a
2783:are depicted below the circle. Using
1541:is the perimeter of the polygon, and
7:
3480:For all convex pentagons with sides
1492:The area of any regular polygon is:
470:to its sides. Given its side length
5024:How to construct a regular pentagon
4847:Discrete and Computational Geometry
4811:, (2008) The Symmetries of Things,
4470:United States Department of Defense
3377:for their central gyration orders.
4972:; Kusner, WĂśden (September 2016),
4626:Peter R. Cromwell (22 July 1999).
3603:, the following inequality holds:
3009:
3006:
3003:
2999:
1527:{\displaystyle A={\frac {1}{2}}Pr}
25:
4877:The American Mathematical Monthly
4693:The American Mathematical Monthly
4363:Another example of echinoderm, a
106: 'angle') is any five-sided
4477:
4458:
4446:of gold, half a centimeter tall.
4435:
4415:
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3980:
3973:
3966:
3959:
3952:
218:
213:
208:
156:
49:
6208:Polygons by the number of sides
4807:John H. Conway, Heidi Burgiel,
4684:DeTemple, Duane W. (Feb 1991).
3930:15 monohedral pentagonal tiles
2764:intersects the circle at point
5030:How to fold a regular pentagon
4351:have fivefold radial symmetry.
3862:
3846:
3681:
3616:
3402:A pentagram or pentangle is a
3272:Overhand knot of a paper strip
3040:. This is the cosine of 72°,
2903:
2897:
2886:
2880:
2859:
2845:
1488:Derivation of the area formula
1106:
1087:
1:
4760:Fitzpatrick, Richard (2008).
4733:George Edward Martin (1998).
4603:Richmond, Herbert W. (1893).
4580:Meskhishvili, Mamuka (2020).
3264:Physical construction methods
5038:, with interactive animation
4556:In-line notes and references
4246:Pentagonal cross-section of
4192:
4058:
4041:
4004:
3987:
3950:
3109:Method using Carlyle circles
2760:. A horizontal line through
466:regular pentagon are in the
345:), inscribed circle radius (
121:A pentagon may be simple or
5032:using only a strip of paper
4210:Pentagonal icositetrahedron
3924:monohedrally tile the plane
3357:and no symmetry is labeled
1249:is found by the expression
6234:
5632:
5059:
4950:Inequalities proposed in â
4657:Eric W. Weisstein (2003).
4215:Pentagonal hexecontahedron
3792:
3428:
3415:sides of the two pentagons
3395:
3161:Draw a circle centered at
3098:
1477:{\displaystyle \pi R^{2},}
110:or 5-gon. The sum of the
29:
4928:10.1016/j.jnt.2007.05.005
3839:of a regular pentagon is
3517:{\displaystyle a,b,c,d,e}
3369:for perpendiculars), and
3326:, order 10. Since 5 is a
2928:where cosine and sine of
2704:Geometrical constructions
413:{\displaystyle \varphi t}
155:
99: 'five' and
48:
36:Pentagon (disambiguation)
4914:Journal of Number Theory
4519:List of geometric shapes
3476:General convex pentagons
3365:for diagonal) or edges (
3248:compass and straightedge
3191:Draw a circle of radius
3180:Draw a circle of radius
2710:compass and straightedge
4840:Robbins, D. P. (1994).
4739:. Springer. p. 6.
4736:Geometric constructions
4406:formed as a pentagonal
4402:A Ho-Mg-Zn icosahedral
4220:Truncated trapezohedron
203:CoxeterâDynkin diagrams
6203:Constructible polygons
5018:Animated demonstration
4468:, headquarters of the
4099:Pentagons in polyhedra
3904:(360 â 108) / 2 = 126°
3883:
3812:
3778:
3597:
3518:
3440:
3311:
3273:
3242:A regular pentagon is
3239:
3238:, animation 1 min 39 s
3225:
3110:
3085:
3030:
2980:
2919:
2815:
2728:
2719:
2694:
2670:
2616:
2571:
2541:
2361:
2197:
2066:
1978:
1943:respectively, we have
1937:
1910:
1890:
1850:
1735:, which is the radius
1706:
1528:
1478:
1442:
1364:
1243:
1223:
1200:
959:
936:
567:
547:
527:
507:
487:
422:
414:
387:
359:
339:
315:
125:. A self-intersecting
34:. For other uses, see
4809:Chaim Goodman-Strauss
3908:360 / (180 â 126) = 6
3884:
3802:
3779:
3598:
3519:
3438:
3425:Equilateral pentagons
3309:
3271:
3233:
3108:
3086:
3031:
2981:
2920:
2816:
2727:
2695:
2650:
2596:
2572:
2570:{\displaystyle d_{i}}
2542:
2341:
2177:
2046:
1958:
1938:
1936:{\displaystyle d_{i}}
1911:
1891:
1851:
1707:
1529:
1479:
1443:
1365:
1244:
1224:
1201:
960:
937:
568:
548:
528:
508:
488:
452:reflectional symmetry
415:
388:
360:
340:
316:
300:
27:Shape with five sides
5906:Nonagon/Enneagon (9)
5836:Tangential trapezoid
3843:
3610:
3528:
3484:
3431:Equilateral pentagon
3288:Construct a regular
3146:Construct the point
3047:
3038:double angle formula
2993:
2939:
2836:
2791:
2584:
2554:
1950:
1920:
1900:
1880:
1866:circumscribed circle
1750:
1568:
1499:
1455:
1380:
1256:
1233:
1213:
1209:If the circumradius
972:
949:
580:
557:
537:
517:
497:
474:
401:
371:
349:
329:
305:
6018:Megagon (1,000,000)
5786:Isosceles trapezoid
5627:pentagonal polytope
5526:Uniform 10-polytope
5086:Fundamental convex
4952:Crux Mathematicorum
4791:Mathematical Models
4541:, the Chrysler logo
4379:An illustration of
4228:Pentagons in nature
3931:
3789:Pentagons in tiling
3773:
3755:
3737:
3719:
3701:
3150:as the midpoint of
2785:Pythagoras' theorem
2685:
2631:
2376:
2212:
2081:
1993:
553:) and circumradius
456:rotational symmetry
386:{\displaystyle R+r}
5988:Icositetragon (24)
5496:Uniform 9-polytope
5446:Uniform 8-polytope
5396:Uniform 7-polytope
5353:Uniform 6-polytope
5323:Uniform 5-polytope
5283:Uniform polychoron
5246:Uniform polyhedron
5094:in dimensions 2â10
5001:Weisstein, Eric W.
4861:10.1007/bf02574377
4524:Pentagonal numbers
3929:
3889:, achieved by the
3879:
3835:The maximum known
3813:
3805:best-known packing
3774:
3759:
3741:
3723:
3705:
3687:
3593:
3514:
3441:
3312:
3295:pentagonal pyramid
3274:
3240:
3165:through the point
3115:quadratic equation
3111:
3081:
3026:
2976:
2915:
2827:half-angle formula
2811:
2810:
2729:
2690:
2689:
2671:
2632:
2617:
2567:
2537:
2535:
2377:
2362:
2213:
2198:
2082:
2067:
1994:
1979:
1933:
1906:
1886:
1846:
1702:
1561:gives the formula
1549:(equivalently the
1524:
1474:
1438:
1360:
1239:
1219:
1196:
1194:
955:
932:
930:
563:
543:
523:
503:
486:{\displaystyle t,}
483:
450:has five lines of
423:
410:
383:
355:
335:
311:
162:A regular pentagon
118:pentagon is 540°.
6218:Elementary shapes
6190:
6189:
6031:
6030:
6008:Myriagon (10,000)
5993:Triacontagon (30)
5957:Heptadecagon (17)
5947:Pentadecagon (15)
5942:Tetradecagon (14)
5881:Quadrilateral (4)
5751:Antiparallelogram
5648:
5647:
5635:Polytope families
5092:uniform polytopes
4817:978-1-56881-220-5
4776:978-0-615-17984-1
4308:five-pointed star
4225:
4224:
4096:
4095:
3860:
3470:Robbins pentagons
3392:Regular pentagram
3060:
2972:
2968:
2956:
2911:
2907:
2800:
2720:Richmond's method
1909:{\displaystyle L}
1889:{\displaystyle R}
1829:
1826:
1824:
1797:
1788:
1715:with side length
1700:
1688:
1642:
1630:
1585:
1516:
1433:
1432:
1426:
1409:
1353:
1343:
1293:
1292:
1286:
1270:
1242:{\displaystyle t}
1222:{\displaystyle R}
1181:
1171:
1165:
1126:
1109:
1104:
1067:
1026:
1020:
1018:
958:{\displaystyle t}
921:
911:
861:
860:
854:
838:
812:
799:
798:
792:
756:
740:
738:
737:
699:
687:
683:
677:
636:
624:
620:
616:
614:
566:{\displaystyle R}
546:{\displaystyle D}
526:{\displaystyle W}
506:{\displaystyle H}
395:, width/diagonal
358:{\displaystyle r}
338:{\displaystyle R}
314:{\displaystyle t}
295:
294:
144:Regular pentagons
123:self-intersecting
78:
77:
16:(Redirected from
6225:
6003:Chiliagon (1000)
5983:Icositrigon (23)
5962:Octadecagon (18)
5952:Hexadecagon (16)
5856:
5675:
5668:
5661:
5652:
5639:Regular polytope
5200:
5189:
5178:
5137:
5080:
5073:
5066:
5057:
5014:
5013:
4987:
4986:
4985:
4966:
4960:
4947:
4941:
4939:
4930:
4907:
4901:
4900:
4872:
4866:
4865:
4863:
4837:
4831:
4826:
4820:
4805:
4799:
4787:
4781:
4780:
4768:
4757:
4751:
4750:
4730:
4724:
4723:
4721:
4715:. Archived from
4690:
4681:
4675:
4674:
4654:
4648:
4647:
4623:
4617:
4616:
4600:
4594:
4593:
4577:
4568:
4565:
4481:
4462:
4439:
4419:
4399:
4376:
4360:
4340:
4318:
4295:
4274:
4259:
4243:
4189:
4182:
4175:
4168:
4161:
4154:
4103:
4092:
4085:
4078:
4071:
4064:
4038:
4031:
4024:
4017:
4010:
3984:
3977:
3970:
3963:
3956:
3932:
3918:
3917:
3916:
3912:
3905:
3888:
3886:
3885:
3880:
3869:
3861:
3856:
3831:
3830:
3829:
3825:
3783:
3781:
3780:
3775:
3772:
3767:
3754:
3749:
3736:
3731:
3718:
3713:
3700:
3695:
3680:
3679:
3667:
3666:
3654:
3653:
3641:
3640:
3628:
3627:
3602:
3600:
3599:
3594:
3592:
3591:
3579:
3578:
3566:
3565:
3553:
3552:
3540:
3539:
3523:
3521:
3520:
3515:
3452:Cyclic pentagons
3316:regular pentagon
3090:
3088:
3087:
3082:
3077:
3072:
3068:
3061:
3056:
3035:
3033:
3032:
3027:
3025:
3024:
3012:
2985:
2983:
2982:
2977:
2970:
2969:
2964:
2957:
2952:
2949:
2924:
2922:
2921:
2916:
2909:
2908:
2906:
2889:
2866:
2855:
2820:
2818:
2817:
2812:
2806:
2801:
2796:
2699:
2697:
2696:
2691:
2684:
2679:
2669:
2664:
2642:
2641:
2636:
2630:
2625:
2615:
2610:
2576:
2574:
2573:
2568:
2566:
2565:
2546:
2544:
2543:
2538:
2536:
2529:
2525:
2524:
2523:
2514:
2513:
2498:
2497:
2492:
2488:
2487:
2486:
2474:
2473:
2458:
2457:
2448:
2447:
2432:
2431:
2426:
2422:
2421:
2420:
2408:
2407:
2375:
2370:
2360:
2355:
2332:
2328:
2327:
2323:
2322:
2321:
2309:
2308:
2294:
2293:
2284:
2283:
2268:
2267:
2262:
2258:
2257:
2256:
2244:
2243:
2211:
2206:
2196:
2191:
2168:
2164:
2163:
2162:
2153:
2152:
2137:
2136:
2131:
2127:
2126:
2125:
2113:
2112:
2080:
2075:
2065:
2060:
2037:
2033:
2032:
2031:
2019:
2018:
1992:
1987:
1977:
1972:
1942:
1940:
1939:
1934:
1932:
1931:
1915:
1913:
1912:
1907:
1895:
1893:
1892:
1887:
1855:
1853:
1852:
1847:
1830:
1828:
1827:
1825:
1820:
1812:
1803:
1798:
1796:
1795:
1794:
1793:
1789:
1781:
1760:
1729:inscribed circle
1711:
1709:
1708:
1703:
1701:
1696:
1695:
1694:
1693:
1689:
1684:
1676:
1662:
1661:
1648:
1643:
1638:
1637:
1636:
1635:
1631:
1626:
1618:
1600:
1586:
1578:
1533:
1531:
1530:
1525:
1517:
1509:
1483:
1481:
1480:
1475:
1470:
1469:
1447:
1445:
1444:
1439:
1434:
1428:
1427:
1422:
1413:
1412:
1410:
1405:
1404:
1403:
1390:
1373:and its area is
1369:
1367:
1366:
1361:
1351:
1344:
1336:
1319:
1318:
1294:
1288:
1287:
1282:
1273:
1272:
1268:
1248:
1246:
1245:
1240:
1228:
1226:
1225:
1220:
1205:
1203:
1202:
1197:
1195:
1191:
1190:
1179:
1172:
1167:
1166:
1158:
1157:
1145:
1143:
1142:
1132:
1127:
1122:
1121:
1120:
1110:
1105:
1100:
1083:
1080:
1072:
1068:
1063:
1062:
1061:
1046:
1045:
1032:
1027:
1022:
1021:
1019:
1014:
1003:
1001:
1000:
990:
964:
962:
961:
956:
941:
939:
938:
933:
931:
919:
912:
904:
887:
886:
862:
856:
855:
850:
841:
840:
836:
810:
800:
794:
793:
788:
779:
778:
754:
741:
739:
733:
729:
721:
697:
685:
684:
679:
678:
673:
664:
634:
622:
621:
615:
610:
599:
598:
572:
570:
569:
564:
552:
550:
549:
544:
532:
530:
529:
524:
512:
510:
509:
504:
492:
490:
489:
484:
421:
419:
417:
416:
411:
394:
392:
390:
389:
384:
364:
362:
361:
356:
344:
342:
341:
336:
320:
318:
317:
312:
223:
222:
221:
217:
216:
212:
211:
160:
151:Regular pentagon
148:
127:regular pentagon
53:
41:
21:
6233:
6232:
6228:
6227:
6226:
6224:
6223:
6222:
6193:
6192:
6191:
6186:
6085:
6039:
6027:
5971:
5937:Tridecagon (13)
5927:Hendecagon (11)
5915:
5851:
5845:
5816:Right trapezoid
5737:
5689:
5679:
5649:
5618:
5611:
5604:
5487:
5480:
5473:
5437:
5430:
5423:
5387:
5380:
5214:Regular polygon
5207:
5198:
5191:
5187:
5180:
5176:
5167:
5158:
5151:
5147:
5135:
5129:
5125:
5113:
5095:
5084:
5053:
4999:
4998:
4995:
4990:
4968:
4967:
4963:
4948:
4944:
4910:
4908:
4904:
4889:10.2307/2974766
4874:
4873:
4869:
4839:
4838:
4834:
4827:
4823:
4806:
4802:
4796:H. Martyn Cundy
4788:
4784:
4777:
4766:
4759:
4758:
4754:
4747:
4732:
4731:
4727:
4719:
4705:10.2307/2323939
4688:
4683:
4682:
4678:
4671:
4656:
4655:
4651:
4644:
4625:
4624:
4620:
4602:
4601:
4597:
4579:
4578:
4571:
4566:
4562:
4558:
4498:
4491:
4482:
4473:
4463:
4454:
4447:
4440:
4431:
4420:
4411:
4400:
4391:
4384:
4377:
4368:
4361:
4352:
4341:
4332:
4325:
4319:
4310:
4296:
4287:
4275:
4266:
4263:Morning glories
4260:
4251:
4244:
4235:
4230:
4144:
4126:
4118:
4110:
4101:
3914:
3910:
3909:
3907:
3903:
3841:
3840:
3837:packing density
3827:
3823:
3822:
3821:360° / 108° = 3
3820:
3797:
3795:Pentagon tiling
3791:
3671:
3658:
3645:
3632:
3619:
3608:
3607:
3583:
3570:
3557:
3544:
3531:
3526:
3525:
3482:
3481:
3478:
3463:septic equation
3454:
3433:
3427:
3411:Schläfli symbol
3400:
3394:
3345:
3341:
3333:
3323:
3304:
3266:
3236:golden triangle
3228:
3226:Euclid's method
3103:
3097:
3095:Carlyle circles
3054:
3050:
3045:
3044:
3016:
2991:
2990:
2950:
2937:
2936:
2890:
2867:
2834:
2833:
2789:
2788:
2744:and a midpoint
2722:
2706:
2591:
2590:
2582:
2581:
2557:
2552:
2551:
2534:
2533:
2515:
2505:
2478:
2465:
2464:
2460:
2459:
2449:
2439:
2412:
2399:
2398:
2394:
2393:
2392:
2388:
2378:
2337:
2336:
2313:
2300:
2299:
2295:
2285:
2275:
2248:
2235:
2234:
2230:
2229:
2228:
2224:
2214:
2173:
2172:
2154:
2144:
2117:
2104:
2103:
2099:
2098:
2097:
2093:
2083:
2042:
2041:
2023:
2010:
2009:
2005:
1995:
1948:
1947:
1923:
1918:
1917:
1898:
1897:
1878:
1877:
1874:
1862:
1807:
1776:
1764:
1748:
1747:
1725:
1677:
1671:
1653:
1649:
1619:
1613:
1601:
1566:
1565:
1497:
1496:
1490:
1461:
1453:
1452:
1414:
1395:
1391:
1378:
1377:
1310:
1274:
1254:
1253:
1231:
1230:
1211:
1210:
1193:
1192:
1182:
1149:
1134:
1133:
1112:
1081:
1070:
1069:
1053:
1037:
1033:
992:
991:
982:
970:
969:
947:
946:
929:
928:
878:
842:
826:
820:
819:
780:
770:
764:
763:
713:
707:
706:
665:
656:
644:
643:
590:
578:
577:
555:
554:
535:
534:
515:
514:
495:
494:
472:
471:
438:interior angles
434:Schläfli symbol
399:
398:
396:
369:
368:
366:
347:
346:
327:
326:
303:
302:
239:
219:
214:
209:
207:
193:Schläfli symbol
172:Regular polygon
163:
146:
112:internal angles
60:
39:
28:
23:
22:
15:
12:
11:
5:
6231:
6229:
6221:
6220:
6215:
6210:
6205:
6195:
6194:
6188:
6187:
6185:
6184:
6179:
6174:
6169:
6164:
6159:
6154:
6149:
6144:
6142:Pseudotriangle
6139:
6134:
6129:
6124:
6119:
6114:
6109:
6104:
6099:
6093:
6091:
6087:
6086:
6084:
6083:
6078:
6073:
6068:
6063:
6058:
6053:
6048:
6042:
6040:
6033:
6032:
6029:
6028:
6026:
6025:
6020:
6015:
6010:
6005:
6000:
5995:
5990:
5985:
5979:
5977:
5973:
5972:
5970:
5969:
5964:
5959:
5954:
5949:
5944:
5939:
5934:
5932:Dodecagon (12)
5929:
5923:
5921:
5917:
5916:
5914:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5878:
5873:
5868:
5862:
5860:
5853:
5847:
5846:
5844:
5843:
5838:
5833:
5828:
5823:
5818:
5813:
5808:
5803:
5798:
5793:
5788:
5783:
5778:
5773:
5768:
5763:
5758:
5753:
5747:
5745:
5743:Quadrilaterals
5739:
5738:
5736:
5735:
5730:
5725:
5720:
5715:
5710:
5705:
5699:
5697:
5691:
5690:
5680:
5678:
5677:
5670:
5663:
5655:
5646:
5645:
5630:
5629:
5620:
5616:
5609:
5602:
5598:
5589:
5572:
5563:
5552:
5551:
5549:
5547:
5542:
5533:
5528:
5522:
5521:
5519:
5517:
5512:
5503:
5498:
5492:
5491:
5489:
5485:
5478:
5471:
5467:
5462:
5453:
5448:
5442:
5441:
5439:
5435:
5428:
5421:
5417:
5412:
5403:
5398:
5392:
5391:
5389:
5385:
5378:
5374:
5369:
5360:
5355:
5349:
5348:
5346:
5344:
5339:
5330:
5325:
5319:
5318:
5309:
5304:
5299:
5290:
5285:
5279:
5278:
5269:
5267:
5262:
5253:
5248:
5242:
5241:
5236:
5231:
5226:
5221:
5216:
5210:
5209:
5205:
5201:
5196:
5185:
5174:
5165:
5156:
5149:
5143:
5133:
5127:
5121:
5115:
5109:
5103:
5097:
5096:
5085:
5083:
5082:
5075:
5068:
5060:
5055:
5051:
5050:
5044:
5039:
5033:
5027:
5021:
5015:
4994:
4993:External links
4991:
4989:
4988:
4961:
4942:
4902:
4883:(6): 523â530.
4867:
4854:(2): 223â236.
4832:
4821:
4800:
4782:
4775:
4752:
4745:
4725:
4722:on 2015-12-21.
4676:
4669:
4649:
4642:
4618:
4595:
4569:
4559:
4557:
4554:
4553:
4552:
4547:
4542:
4536:
4531:
4526:
4521:
4516:
4511:
4505:
4497:
4494:
4493:
4492:
4489:baseball field
4483:
4476:
4474:
4464:
4457:
4453:
4452:Other examples
4450:
4449:
4448:
4441:
4434:
4432:
4421:
4414:
4412:
4401:
4394:
4390:
4387:
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4385:
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4371:
4369:
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4355:
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4328:
4327:
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4320:
4313:
4311:
4297:
4290:
4288:
4276:
4269:
4267:
4261:
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4245:
4238:
4234:
4231:
4229:
4226:
4223:
4222:
4217:
4212:
4207:
4202:
4197:
4191:
4190:
4183:
4176:
4169:
4162:
4155:
4147:
4146:
4142:
4138:
4133:
4128:
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4120:
4116:
4112:
4108:
4100:
4097:
4094:
4093:
4086:
4079:
4072:
4065:
4057:
4056:
4053:
4050:
4047:
4044:
4040:
4039:
4032:
4025:
4018:
4011:
4003:
4002:
3999:
3996:
3993:
3990:
3986:
3985:
3978:
3971:
3964:
3957:
3949:
3948:
3945:
3942:
3939:
3936:
3891:double lattice
3878:
3875:
3872:
3868:
3864:
3859:
3854:
3851:
3848:
3817:regular tiling
3809:double lattice
3793:Main article:
3790:
3787:
3786:
3785:
3771:
3766:
3762:
3758:
3753:
3748:
3744:
3740:
3735:
3730:
3726:
3722:
3717:
3712:
3708:
3704:
3699:
3694:
3690:
3686:
3683:
3678:
3674:
3670:
3665:
3661:
3657:
3652:
3648:
3644:
3639:
3635:
3631:
3626:
3622:
3618:
3615:
3590:
3586:
3582:
3577:
3573:
3569:
3564:
3560:
3556:
3551:
3547:
3543:
3538:
3534:
3524:and diagonals
3513:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3477:
3474:
3453:
3450:
3429:Main article:
3426:
3423:
3409:pentagon. Its
3396:Main article:
3393:
3390:
3386:directed edges
3343:
3339:
3331:
3321:
3303:
3300:
3299:
3298:
3286:
3265:
3262:
3260:circa 300 BC.
3227:
3224:
3223:
3222:
3218:
3217:
3212:
3211:
3204:
3203:
3200:
3189:
3178:
3159:
3144:
3137:
3130:
3101:Carlyle circle
3099:Main article:
3096:
3093:
3080:
3076:
3071:
3067:
3064:
3059:
3053:
3023:
3019:
3015:
3011:
3008:
3005:
3001:
2998:
2987:
2986:
2975:
2967:
2963:
2960:
2955:
2947:
2944:
2926:
2925:
2914:
2905:
2902:
2899:
2896:
2893:
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2882:
2879:
2876:
2873:
2870:
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2799:
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2688:
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2648:
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2629:
2624:
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2614:
2609:
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2603:
2599:
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2532:
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2522:
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2512:
2508:
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2326:
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2312:
2307:
2303:
2298:
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2288:
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2255:
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2247:
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2205:
2201:
2195:
2190:
2187:
2184:
2180:
2175:
2174:
2171:
2167:
2161:
2157:
2151:
2147:
2143:
2140:
2135:
2130:
2124:
2120:
2116:
2111:
2107:
2102:
2096:
2092:
2089:
2086:
2084:
2079:
2074:
2070:
2064:
2059:
2056:
2053:
2049:
2044:
2043:
2040:
2036:
2030:
2026:
2022:
2017:
2013:
2008:
2004:
2001:
1998:
1996:
1991:
1986:
1982:
1976:
1971:
1968:
1965:
1961:
1956:
1955:
1930:
1926:
1905:
1885:
1873:
1872:Point in plane
1870:
1861:
1858:
1857:
1856:
1845:
1842:
1839:
1836:
1833:
1823:
1818:
1815:
1810:
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1801:
1792:
1787:
1784:
1779:
1773:
1770:
1767:
1763:
1758:
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1724:
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1656:
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1629:
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1610:
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1592:
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1584:
1581:
1576:
1573:
1535:
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1523:
1520:
1515:
1512:
1507:
1504:
1489:
1486:
1473:
1468:
1464:
1460:
1449:
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1437:
1431:
1425:
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1417:
1408:
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1398:
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1388:
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1371:
1370:
1359:
1356:
1350:
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1342:
1339:
1334:
1331:
1328:
1325:
1322:
1317:
1313:
1309:
1306:
1303:
1300:
1297:
1291:
1285:
1280:
1277:
1267:
1264:
1261:
1238:
1218:
1207:
1206:
1189:
1185:
1178:
1175:
1170:
1164:
1161:
1156:
1152:
1148:
1141:
1137:
1130:
1125:
1119:
1115:
1108:
1103:
1098:
1095:
1092:
1089:
1086:
1078:
1075:
1073:
1071:
1066:
1060:
1056:
1052:
1049:
1044:
1040:
1036:
1030:
1025:
1017:
1012:
1009:
1006:
999:
995:
988:
985:
983:
981:
978:
977:
954:
943:
942:
927:
924:
918:
915:
910:
907:
902:
899:
896:
893:
890:
885:
881:
877:
874:
871:
868:
865:
859:
853:
848:
845:
835:
832:
829:
827:
825:
822:
821:
818:
815:
809:
806:
803:
797:
791:
786:
783:
776:
773:
771:
769:
766:
765:
762:
759:
753:
750:
747:
744:
736:
732:
727:
724:
719:
716:
714:
712:
709:
708:
705:
702:
696:
693:
690:
682:
676:
671:
668:
662:
659:
657:
655:
652:
649:
646:
645:
642:
639:
633:
630:
627:
619:
613:
608:
605:
602:
596:
593:
591:
589:
586:
585:
573:are given by:
562:
542:
522:
502:
482:
479:
409:
406:
382:
379:
376:
354:
334:
310:
293:
292:
289:
283:
282:
261:
257:
256:
253:
246:Internal angle
242:
241:
237:
231:
229:Symmetry group
225:
224:
205:
199:
198:
195:
189:
188:
185:
175:
174:
169:
165:
164:
161:
153:
152:
145:
142:
136:) is called a
76:
75:
72:
62:
61:
54:
46:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6230:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6201:
6200:
6198:
6183:
6182:Weakly simple
6180:
6178:
6175:
6173:
6170:
6168:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6133:
6130:
6128:
6125:
6123:
6122:Infinite skew
6120:
6118:
6115:
6113:
6110:
6108:
6105:
6103:
6100:
6098:
6095:
6094:
6092:
6088:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6057:
6054:
6052:
6049:
6047:
6044:
6043:
6041:
6038:
6037:Star polygons
6034:
6024:
6023:Apeirogon (â)
6021:
6019:
6016:
6014:
6011:
6009:
6006:
6004:
6001:
5999:
5996:
5994:
5991:
5989:
5986:
5984:
5981:
5980:
5978:
5974:
5968:
5967:Icosagon (20)
5965:
5963:
5960:
5958:
5955:
5953:
5950:
5948:
5945:
5943:
5940:
5938:
5935:
5933:
5930:
5928:
5925:
5924:
5922:
5918:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5882:
5879:
5877:
5874:
5872:
5869:
5867:
5864:
5863:
5861:
5857:
5854:
5848:
5842:
5839:
5837:
5834:
5832:
5829:
5827:
5824:
5822:
5819:
5817:
5814:
5812:
5809:
5807:
5804:
5802:
5801:Parallelogram
5799:
5797:
5796:Orthodiagonal
5794:
5792:
5789:
5787:
5784:
5782:
5779:
5777:
5776:Ex-tangential
5774:
5772:
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5748:
5746:
5744:
5740:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5700:
5698:
5696:
5692:
5687:
5683:
5676:
5671:
5669:
5664:
5662:
5657:
5656:
5653:
5644:
5640:
5636:
5631:
5628:
5624:
5621:
5619:
5612:
5605:
5599:
5597:
5593:
5590:
5588:
5584:
5580:
5576:
5573:
5571:
5567:
5564:
5562:
5558:
5554:
5553:
5550:
5548:
5546:
5543:
5541:
5537:
5534:
5532:
5529:
5527:
5524:
5523:
5520:
5518:
5516:
5513:
5511:
5507:
5504:
5502:
5499:
5497:
5494:
5493:
5490:
5488:
5481:
5474:
5468:
5466:
5463:
5461:
5457:
5454:
5452:
5449:
5447:
5444:
5443:
5440:
5438:
5431:
5424:
5418:
5416:
5413:
5411:
5407:
5404:
5402:
5399:
5397:
5394:
5393:
5390:
5388:
5381:
5375:
5373:
5370:
5368:
5364:
5361:
5359:
5356:
5354:
5351:
5350:
5347:
5345:
5343:
5340:
5338:
5334:
5331:
5329:
5326:
5324:
5321:
5320:
5317:
5313:
5310:
5308:
5305:
5303:
5302:Demitesseract
5300:
5298:
5294:
5291:
5289:
5286:
5284:
5281:
5280:
5277:
5273:
5270:
5268:
5266:
5263:
5261:
5257:
5254:
5252:
5249:
5247:
5244:
5243:
5240:
5237:
5235:
5232:
5230:
5227:
5225:
5222:
5220:
5217:
5215:
5212:
5211:
5208:
5202:
5199:
5195:
5188:
5184:
5177:
5173:
5168:
5164:
5159:
5155:
5150:
5148:
5146:
5142:
5132:
5128:
5126:
5124:
5120:
5116:
5114:
5112:
5108:
5104:
5102:
5099:
5098:
5093:
5089:
5081:
5076:
5074:
5069:
5067:
5062:
5061:
5058:
5054:
5048:
5045:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5011:
5010:
5005:
5002:
4997:
4996:
4992:
4984:
4979:
4975:
4971:
4970:Hales, Thomas
4965:
4962:
4958:
4955:
4953:
4946:
4943:
4938:
4934:
4929:
4924:
4920:
4916:
4915:
4906:
4903:
4898:
4894:
4890:
4886:
4882:
4878:
4871:
4868:
4862:
4857:
4853:
4849:
4848:
4843:
4836:
4833:
4830:
4825:
4822:
4818:
4814:
4810:
4804:
4801:
4797:
4793:
4792:
4786:
4783:
4778:
4772:
4765:
4764:
4756:
4753:
4748:
4746:0-387-98276-0
4742:
4738:
4737:
4729:
4726:
4718:
4714:
4710:
4706:
4702:
4699:(2): 97â108.
4698:
4694:
4687:
4680:
4677:
4672:
4670:1-58488-347-2
4666:
4662:
4661:
4653:
4650:
4645:
4643:0-521-66405-5
4639:
4635:
4631:
4630:
4622:
4619:
4614:
4610:
4606:
4599:
4596:
4591:
4587:
4583:
4576:
4574:
4570:
4564:
4561:
4555:
4551:
4548:
4546:
4543:
4540:
4537:
4535:
4534:Pentagram map
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4509:
4506:
4503:
4502:Associahedron
4500:
4499:
4495:
4490:
4486:
4480:
4475:
4471:
4467:
4461:
4456:
4451:
4445:
4438:
4433:
4429:
4425:
4418:
4413:
4409:
4405:
4398:
4393:
4388:
4382:
4381:brittle stars
4375:
4370:
4367:endoskeleton.
4366:
4359:
4354:
4350:
4346:
4339:
4334:
4329:
4323:
4317:
4312:
4309:
4305:
4301:
4294:
4289:
4285:
4284:
4279:
4273:
4268:
4264:
4258:
4253:
4249:
4242:
4237:
4232:
4227:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4188:
4184:
4181:
4177:
4174:
4170:
4167:
4163:
4160:
4156:
4153:
4149:
4148:
4145:
4139:
4137:
4134:
4132:
4129:
4127:
4121:
4119:
4113:
4111:
4105:
4104:
4098:
4091:
4087:
4084:
4080:
4077:
4073:
4070:
4066:
4063:
4059:
4054:
4051:
4048:
4045:
4042:
4037:
4033:
4030:
4026:
4023:
4019:
4016:
4012:
4009:
4005:
4000:
3997:
3994:
3991:
3988:
3983:
3979:
3976:
3972:
3969:
3965:
3962:
3958:
3955:
3951:
3946:
3943:
3940:
3937:
3934:
3933:
3927:
3925:
3920:
3899:
3896:
3892:
3876:
3873:
3870:
3866:
3857:
3852:
3849:
3838:
3833:
3818:
3810:
3806:
3801:
3796:
3788:
3769:
3764:
3760:
3756:
3751:
3746:
3742:
3738:
3733:
3728:
3724:
3720:
3715:
3710:
3706:
3702:
3697:
3692:
3688:
3684:
3676:
3672:
3668:
3663:
3659:
3655:
3650:
3646:
3642:
3637:
3633:
3629:
3624:
3620:
3613:
3606:
3605:
3604:
3588:
3584:
3580:
3575:
3571:
3567:
3562:
3558:
3554:
3549:
3545:
3541:
3536:
3532:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3475:
3473:
3471:
3466:
3464:
3459:
3451:
3449:
3447:
3437:
3432:
3424:
3422:
3420:
3416:
3412:
3408:
3405:
3399:
3391:
3389:
3387:
3383:
3378:
3376:
3372:
3368:
3364:
3360:
3356:
3352:
3347:
3338:symmetries: Z
3337:
3329:
3325:
3317:
3308:
3301:
3296:
3291:
3287:
3285:when backlit.
3284:
3280:
3279:overhand knot
3276:
3275:
3270:
3263:
3261:
3259:
3258:
3253:
3249:
3245:
3244:constructible
3237:
3232:
3220:
3219:
3214:
3213:
3209:
3208:
3207:
3201:
3198:
3194:
3190:
3187:
3183:
3179:
3176:
3172:
3168:
3164:
3160:
3157:
3153:
3149:
3145:
3142:
3138:
3135:
3131:
3128:
3124:
3120:
3119:
3118:
3116:
3107:
3102:
3094:
3092:
3078:
3074:
3069:
3065:
3062:
3057:
3051:
3043:
3039:
3021:
3017:
3013:
2996:
2973:
2965:
2961:
2958:
2953:
2945:
2942:
2935:
2934:
2933:
2931:
2912:
2900:
2894:
2891:
2883:
2877:
2874:
2871:
2868:
2862:
2856:
2852:
2848:
2842:
2839:
2832:
2831:
2830:
2828:
2824:
2807:
2803:
2797:
2786:
2782:
2778:
2773:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2743:
2738:
2736:
2735:
2726:
2717:
2715:
2711:
2703:
2686:
2681:
2676:
2672:
2666:
2661:
2658:
2655:
2651:
2646:
2643:
2638:
2633:
2627:
2622:
2618:
2612:
2607:
2604:
2601:
2597:
2592:
2587:
2580:
2579:
2578:
2562:
2558:
2530:
2526:
2520:
2516:
2510:
2506:
2502:
2499:
2494:
2489:
2483:
2479:
2475:
2470:
2466:
2461:
2454:
2450:
2444:
2440:
2436:
2433:
2428:
2423:
2417:
2413:
2409:
2404:
2400:
2395:
2389:
2385:
2382:
2380:
2372:
2367:
2363:
2357:
2352:
2349:
2346:
2342:
2333:
2329:
2324:
2318:
2314:
2310:
2305:
2301:
2296:
2290:
2286:
2280:
2276:
2272:
2269:
2264:
2259:
2253:
2249:
2245:
2240:
2236:
2231:
2225:
2221:
2218:
2216:
2208:
2203:
2199:
2193:
2188:
2185:
2182:
2178:
2169:
2165:
2159:
2155:
2149:
2145:
2141:
2138:
2133:
2128:
2122:
2118:
2114:
2109:
2105:
2100:
2094:
2090:
2087:
2085:
2077:
2072:
2068:
2062:
2057:
2054:
2051:
2047:
2038:
2034:
2028:
2024:
2020:
2015:
2011:
2006:
2002:
1999:
1997:
1989:
1984:
1980:
1974:
1969:
1966:
1963:
1959:
1946:
1945:
1944:
1928:
1924:
1903:
1883:
1871:
1869:
1867:
1859:
1843:
1840:
1837:
1834:
1831:
1821:
1816:
1813:
1808:
1804:
1799:
1790:
1785:
1782:
1777:
1771:
1768:
1765:
1761:
1756:
1753:
1746:
1745:
1744:
1742:
1738:
1734:
1730:
1722:
1720:
1718:
1697:
1690:
1685:
1681:
1678:
1672:
1666:
1663:
1658:
1654:
1650:
1644:
1639:
1632:
1627:
1623:
1620:
1614:
1608:
1605:
1602:
1596:
1593:
1590:
1587:
1582:
1579:
1574:
1571:
1564:
1563:
1562:
1560:
1556:
1552:
1548:
1544:
1540:
1521:
1518:
1513:
1510:
1505:
1502:
1495:
1494:
1493:
1487:
1485:
1471:
1466:
1462:
1458:
1435:
1429:
1423:
1418:
1415:
1406:
1400:
1396:
1392:
1386:
1383:
1376:
1375:
1374:
1357:
1354:
1348:
1345:
1340:
1337:
1332:
1329:
1326:
1323:
1320:
1315:
1311:
1307:
1304:
1301:
1298:
1295:
1289:
1283:
1278:
1275:
1265:
1262:
1259:
1252:
1251:
1250:
1236:
1216:
1187:
1183:
1176:
1173:
1168:
1162:
1159:
1154:
1150:
1146:
1139:
1135:
1128:
1123:
1117:
1113:
1101:
1096:
1093:
1090:
1084:
1076:
1074:
1064:
1058:
1054:
1050:
1047:
1042:
1038:
1034:
1028:
1023:
1015:
1010:
1007:
1004:
997:
993:
986:
984:
979:
968:
967:
966:
952:
925:
922:
916:
913:
908:
905:
900:
897:
894:
891:
888:
883:
879:
875:
872:
869:
866:
863:
857:
851:
846:
843:
833:
830:
828:
823:
816:
813:
807:
804:
801:
795:
789:
784:
781:
774:
772:
767:
760:
757:
751:
748:
745:
742:
734:
730:
725:
722:
717:
715:
710:
703:
700:
694:
691:
688:
680:
674:
669:
666:
660:
658:
653:
650:
647:
640:
637:
631:
628:
625:
617:
611:
606:
603:
600:
594:
592:
587:
576:
575:
574:
560:
540:
520:
500:
480:
477:
469:
465:
461:
457:
453:
449:
447:
441:
439:
435:
431:
429:
407:
404:
380:
377:
374:
352:
332:
324:
308:
299:
290:
288:
284:
281:
277:
273:
269:
265:
262:
258:
254:
251:
247:
243:
235:
232:
230:
226:
206:
204:
200:
196:
194:
190:
186:
184:
180:
176:
173:
170:
166:
159:
154:
149:
143:
141:
139:
135:
133:
128:
124:
119:
117:
113:
109:
105:
101:
98:
94:
91:
87:
83:
73:
71:
67:
63:
58:
52:
47:
42:
37:
33:
19:
5976:>20 sides
5911:Decagon (10)
5896:Heptagon (7)
5886:Pentagon (5)
5885:
5876:Triangle (3)
5771:Equidiagonal
5622:
5591:
5582:
5574:
5565:
5556:
5536:10-orthoplex
5272:Dodecahedron
5238:
5193:
5182:
5171:
5162:
5153:
5144:
5140:
5130:
5122:
5118:
5110:
5106:
5052:
5007:
4973:
4964:
4949:
4945:
4921:(1): 17â48,
4918:
4912:
4905:
4880:
4876:
4870:
4851:
4845:
4835:
4824:
4803:
4789:
4785:
4762:
4755:
4735:
4728:
4717:the original
4696:
4692:
4679:
4659:
4652:
4628:
4621:
4612:
4608:
4598:
4589:
4585:
4563:
4514:Golden ratio
4508:Dodecahedron
4466:The Pentagon
4424:pyritohedral
4408:dodecahedron
4404:quasicrystal
4281:
4200:Pyritohedron
4195:Dodecahedron
3921:
3900:
3895:Thomas Hales
3834:
3814:
3479:
3467:
3455:
3442:
3419:golden ratio
3401:
3381:
3379:
3374:
3370:
3366:
3362:
3358:
3354:
3348:
3336:cyclic group
3328:prime number
3315:
3313:
3255:
3241:
3205:
3196:
3192:
3185:
3181:
3174:
3170:
3166:
3162:
3155:
3151:
3147:
3140:
3133:
3126:
3112:
3091:as desired.
3042:which equals
2988:
2929:
2927:
2822:
2780:
2776:
2774:
2769:
2768:, and chord
2765:
2761:
2757:
2753:
2749:
2745:
2741:
2739:
2732:
2730:
2714:Fermat prime
2712:, as 5 is a
2707:
2549:
1875:
1863:
1740:
1736:
1726:
1716:
1714:
1558:
1554:
1542:
1538:
1536:
1491:
1450:
1372:
1208:
965:is given by
944:
468:golden ratio
444:
442:
426:
424:
323:circumradius
287:Dual polygon
240:), order 2Ă5
130:
126:
120:
103:
100:
96:
93:
85:
79:
32:The Pentagon
6172:Star-shaped
6147:Rectilinear
6117:Equilateral
6112:Equiangular
6076:Hendecagram
5920:11â20 sides
5901:Octagon (8)
5891:Hexagon (6)
5866:Monogon (1)
5708:Equilateral
5545:10-demicube
5506:9-orthoplex
5456:8-orthoplex
5406:7-orthoplex
5363:6-orthoplex
5333:5-orthoplex
5288:Pentachoron
5276:Icosahedron
5251:Tetrahedron
4426:crystal of
4349:echinoderms
3417:are in the
3351:John Conway
3195:and center
3184:and center
493:its height
272:equilateral
6213:5 (number)
6197:Categories
6177:Tangential
6081:Dodecagram
5859:1â10 sides
5850:By number
5831:Tangential
5811:Right kite
5531:10-simplex
5515:9-demicube
5465:8-demicube
5415:7-demicube
5372:6-demicube
5342:5-demicube
5256:Octahedron
5004:"Pentagon"
4983:1602.07220
4615:: 206â207.
4592:: 335â355.
4485:Home plate
4365:sea urchin
365:), height
260:Properties
88:(from
6157:Reinhardt
6066:Enneagram
6056:Heptagram
6046:Pentagram
6013:65537-gon
5871:Digon (2)
5841:Trapezoid
5806:Rectangle
5756:Bicentric
5718:Isosceles
5695:Triangles
5579:orthoplex
5501:9-simplex
5451:8-simplex
5401:7-simplex
5358:6-simplex
5328:5-simplex
5297:Tesseract
5047:Pentagon.
5009:MathWorld
4629:Polyhedra
4539:Pentastar
4529:Pentagram
4322:Starfruit
4300:gynoecium
4283:Rafflesia
4205:Tetartoid
3874:≈
3853:−
3398:Pentagram
3283:pentagram
3063:−
3022:∘
3000:∠
2959:−
2901:ϕ
2895:
2884:ϕ
2878:
2872:−
2849:ϕ
2843:
2734:Polyhedra
2652:∑
2598:∑
2343:∑
2179:∑
2048:∑
1960:∑
1838:⋅
1832:≈
1817:−
1783:π
1772:
1682:π
1667:
1624:π
1609:
1597:⋅
1588:⋅
1459:π
1346:≈
1338:π
1333:
1316:∘
1308:
1279:−
1174:≈
1151:φ
1059:∘
1051:
914:≈
906:π
901:
884:∘
876:
805:≈
749:≈
743:⋅
726:−
692:≈
629:≈
460:diagonals
440:of 108°.
405:φ
138:pentagram
18:Pentagons
6132:Isotoxal
6127:Isogonal
6071:Decagram
6061:Octagram
6051:Hexagram
5852:of sides
5781:Harmonic
5682:Polygons
5633:Topics:
5596:demicube
5561:polytope
5555:Uniform
5316:600-cell
5312:120-cell
5265:Demicube
5239:Pentagon
5219:Triangle
4496:See also
4444:Fiveling
4389:Minerals
4345:sea star
4280:tube of
4278:Perigone
3334:, and 2
3324:symmetry
3302:Symmetry
3257:Elements
3246:using a
2752:. Angle
1723:Inradius
1547:inradius
448:pentagon
436:{5} and
430:pentagon
280:isotoxal
276:isogonal
234:Dihedral
183:vertices
134:pentagon
86:pentagon
82:geometry
70:vertices
59:pentagon
44:Pentagon
6152:Regular
6097:Concave
6090:Classes
5998:257-gon
5821:Rhombus
5761:Crossed
5570:simplex
5540:10-cube
5307:24-cell
5293:16-cell
5234:Hexagon
5088:regular
4937:2382768
4897:2974766
4713:2323939
4347:. Many
4330:Animals
4286:flower.
3913:⁄
3826:⁄
3404:regular
3342:, and Z
3290:hexagon
3254:in his
3216:circle.
3121:Draw a
2821:. Side
1733:apothem
1551:apothem
1545:is the
446:regular
428:regular
250:degrees
108:polygon
104:(gonia)
97:(pente)
6162:Simple
6107:Cyclic
6102:Convex
5826:Square
5766:Cyclic
5728:Obtuse
5723:Kepler
5510:9-cube
5460:8-cube
5410:7-cube
5367:6-cube
5337:5-cube
5224:Square
5101:Family
4935:
4895:
4815:
4773:
4743:
4711:
4667:
4640:
4428:pyrite
4302:of an
4233:Plants
3458:cyclic
3252:Euclid
3123:circle
2971:
2910:
1835:0.6882
1731:. The
1537:where
1352:
1269:
1180:
920:
837:
811:
808:0.8507
755:
698:
686:
635:
623:
464:convex
454:, and
301:Side (
268:cyclic
264:Convex
116:simple
57:cyclic
6137:Magic
5733:Right
5713:Ideal
5703:Acute
5229:p-gon
4978:arXiv
4893:JSTOR
4767:(PDF)
4720:(PDF)
4709:JSTOR
4689:(PDF)
4634:p. 63
4487:of a
4304:apple
3877:0.921
3446:up to
1349:1.176
1177:1.720
917:1.902
752:1.051
695:1.618
632:1.539
462:of a
179:Edges
114:in a
102:ÎłĎνίι
95:ĎÎνĎÎľ
92:
90:Greek
66:Edges
6167:Skew
5791:Kite
5686:List
5587:cube
5260:Cube
5090:and
4813:ISBN
4771:ISBN
4741:ISBN
4665:ISBN
4638:ISBN
4298:The
4248:okra
3803:The
3685:>
3407:star
3318:has
3314:The
3154:and
2779:and
1916:and
1557:and
432:has
291:Self
255:108°
181:and
168:Type
132:star
129:(or
84:, a
68:and
5136:(p)
4923:doi
4919:128
4885:doi
4881:102
4856:doi
4794:by
4701:doi
4055:15
4001:10
3355:r10
3320:Dih
2892:sin
2875:cos
2840:tan
2781:QCM
2777:DCM
2754:CMD
2550:If
1769:tan
1743:by
1664:tan
1606:tan
1330:sin
1305:sin
1048:tan
898:cos
873:cos
321:),
197:{5}
80:In
6199::
5641:â˘
5637:â˘
5617:21
5613:â˘
5610:k1
5606:â˘
5603:k2
5581:â˘
5538:â˘
5508:â˘
5486:21
5482:â˘
5479:41
5475:â˘
5472:42
5458:â˘
5436:21
5432:â˘
5429:31
5425:â˘
5422:32
5408:â˘
5386:21
5382:â˘
5379:22
5365:â˘
5335:â˘
5314:â˘
5295:â˘
5274:â˘
5258:â˘
5190:/
5179:/
5169:/
5160:/
5138:/
5006:.
4976:,
4956:,
4933:MR
4931:,
4917:,
4891:.
4879:.
4852:12
4850:.
4844:.
4707:.
4697:98
4695:.
4691:.
4636:.
4632:.
4613:26
4611:.
4607:.
4590:11
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