4468:
4426:
2534:
4386:
4327:
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3095:
4261:
1938:
4347:
4176:
929:
4282:
3258:
4406:
4246:
4363:
4162:
147:
4305:
4169:
3789:
2529:{\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}}}
3296:
3425:
568:
287:
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2714:
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4058:
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4018:
4011:
4004:
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3950:
3943:
4230:
3997:
40:
4141:
4148:
924:{\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}}}
1193:
4155:
3220:
1699:
960:
3890:
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
1843:
3449:
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
3821:(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:
1357:
3886:
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.
2687:
1188:{\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}}}
1556:
3771:
3432:
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
1435:
2912:
1738:
3281:
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
3204:
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
1943:
965:
573:
4405:
3023:
4385:
2973:
2729:
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
3876:
3590:
3078:
2808:
1244:
3891:
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
3299:
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.
1694:{\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}}}
2572:
1521:
4467:
3461:. 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.
3598:
199:
1471:
209:
3511:
407:
204:
2564:
1930:
380:
480:
1903:
1883:
1236:
1216:
952:
560:
540:
520:
500:
352:
332:
308:
4281:
1857:. 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.
5018:
4448:
4362:
4674:
3270:
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
4326:
4425:
4245:
1368:
1838:{\displaystyle r={\frac {t}{2\tan {\mathord {\left({\frac {\pi }{5}}\right)}}}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.}
4346:
2824:
6196:
4533:
5631:
4779:
4304:
5012:
4805:
4763:
4129:
4260:
4750:
3912:
2720:
One method to construct a regular pentagon in a given circle is described by
Richmond and further discussed in Cromwell's
5066:
4835:
4458:
2981:
5661:
111:
4733:
4657:
4630:
3030:
2927:
3831:
3516:
1352:{\displaystyle t=R\ {\sqrt {\frac {5-{\sqrt {5}}}{2}}}=2R\sin 36^{\circ }=2R\sin {\frac {\pi }{5}}\approx 1.176~R,}
3362:
when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as
3035:
3025:, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos(54°), and CQ = 1 â 2cos(54°), which equals âcos(108°) by the cosine
6191:
19:
This article is about the geometric figure. For the headquarters of the United States
Department of Defense, see
4570:
3239:, either by inscribing one in a given circle or constructing one on a given edge. This process was described by
2779:
2737:
is marked halfway along its radius. This point is joined to the periphery vertically above the center at point
217:
3915:. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry.
3804:
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a
3224:
3121:
Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point
2682:{\displaystyle 3\left(\textstyle \sum _{i=1}^{5}d_{i}^{2}\right)^{2}=10\textstyle \sum _{i=1}^{5}d_{i}^{4}.}
3908:, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
5784:
5764:
4705:
4198:
4229:
3808:(one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that
1487:
6206:
5759:
5716:
5691:
4538:
4203:
3106:. This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:
191:
3766:{\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}}
3210:
8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.
24:
5819:
5089:
4902:
4507:
3236:
2698:
5744:
5059:
4900:
Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area",
4208:
2566:
are the distances from the vertices of a regular pentagon to any point on its circumcircle, then
2705:. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
1443:
5769:
5654:
4958:
3883:
3472:
3191:
The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
2773:
389:
4647:
4593:
3402:
is {5/2}. Its sides form the diagonals of a regular convex pentagon â in this arrangement the
6170:
6110:
5749:
5603:
5596:
5589:
4797:
4622:
4103:
3232:
3128:
Construct a vertical line through the center. Mark one intersection with the circle as point
440:
5128:
5106:
5094:
4723:
4556:"pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.
6054:
5824:
5754:
5696:
5260:
5207:
4925:
4124:
4111:
4095:
3419:
3369:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
3245:
3158:. Mark its intersection with the horizontal line (inside the original circle) as the point
3026:
2542:
1908:
1854:
4419:. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
8:
6201:
6160:
6135:
6105:
6100:
6059:
5774:
5615:
5514:
5264:
4940:
4119:
3437:
similarity, because it is equilateral and it is equiangular (its five angles are equal).
444:
359:
260:
3094:
462:
6165:
5706:
5484:
5434:
5384:
5341:
5311:
5271:
5234:
5052:
4966:
4881:
4697:
3350:. The dihedral symmetries are divided depending on whether they pass through vertices (
3339:
3283:
3103:
2815:
1888:
1868:
1717:
1221:
1201:
937:
545:
525:
505:
485:
337:
317:
293:
4808:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
3399:
422:
181:
6145:
5739:
5647:
5623:
4989:
4801:
4759:
4729:
4653:
4626:
4512:
4296:
4175:
3457:
There exist cyclic pentagons with rational sides and rational area; these are called
3308:
222:
171:
58:
1885:, whose distances to the centroid of the regular pentagon and its five vertices are
5674:
5627:
5192:
5181:
5170:
5159:
5150:
5141:
5080:
5076:
4911:
4873:
4844:
4818:
4689:
4616:
3793:
3458:
2722:
5035:
5006:
4946:
3403:
3257:
522:(distance between two farthest separated points, which equals the diagonal length
6140:
6120:
6115:
6085:
5804:
5779:
5711:
5217:
5202:
4921:
4251:
3825:
3783:
3451:
3392:
434:
416:
268:
264:
167:
160:
54:
4161:
6150:
6130:
6095:
6090:
5721:
5701:
5567:
4477:
4168:
3879:
3805:
3797:
3446:
3342:
labels these by a letter and group order. Full symmetry of the regular form is
3219:
3089:
452:
426:
256:
252:
238:
234:
104:
100:
78:
45:
5030:
4992:
4916:
4817:
Weisstein, Eric W. "Cyclic
Pentagon." From MathWorld--A Wolfram Web Resource.
146:
6185:
6125:
5976:
5869:
5789:
5731:
5584:
5472:
5465:
5458:
5422:
5415:
5408:
5372:
5365:
4522:
4490:
4369:
3374:
3267:
3102:
The
Carlyle circle was invented as a geometric method to find the roots of a
1728:
of the inscribed circle, of a regular pentagon is related to the side length
1473:
the regular pentagon fills approximately 0.7568 of its circumscribed circle.
4266:
3788:
1865:
For an arbitrary point in the plane of a regular pentagon with circumradius
1716:
Similar to every regular convex polygon, the regular convex pentagon has an
6155:
6025:
5981:
5945:
5935:
5930:
5524:
5024:
4784:
4502:
4496:
4454:
4412:
4396:
4392:
4188:
4183:
3407:
3395:
3324:
3316:
3195:
Steps 6â8 are equivalent to the following version, shown in the animation:
3188:. It intersects the original circle at two of the vertices of the pentagon.
3177:. It intersects the original circle at two of the vertices of the pentagon.
2702:
456:
311:
275:
120:
20:
6064:
5971:
5950:
5940:
5533:
5494:
5444:
5394:
5351:
5321:
5253:
5239:
4337:
3338:
These 4 symmetries can be seen in 4 distinct symmetries on the pentagon.
3295:
3266:
A regular pentagon may be created from just a strip of paper by tying an
4787:
and A.P. Rollett, second edition, 1961 (Oxford
University Press), p. 57.
1430:{\displaystyle A={\frac {5R^{2}}{4}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}};}
6069:
5925:
5915:
5799:
5519:
5503:
5453:
5403:
5360:
5330:
5244:
4885:
4849:
4830:
4701:
4473:
4353:
3428:
Equilateral pentagon built with four equal circles disposed in a chain.
3424:
1848:
4078:
4071:
4064:
4057:
4050:
4024:
2713:
6044:
6034:
6011:
6001:
5991:
5920:
5829:
5794:
5575:
5489:
5439:
5389:
5346:
5316:
5285:
4997:
4527:
4517:
4310:
4288:
4271:
4193:
4017:
4010:
4003:
3996:
3970:
3963:
3956:
3949:
3942:
3386:
3271:
2907:{\displaystyle \tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,}
1853:
Like every regular convex polygon, the regular convex pentagon has a
286:
126:
4877:
4693:
4499:, a polyhedron whose regular form is composed of 12 pentagonal faces
2745:
is bisected, and the bisector intersects the vertical axis at point
6049:
6039:
5996:
5955:
5884:
5864:
5683:
5549:
5304:
5300:
5031:
Renaissance artists' approximate constructions of regular pentagons
4971:
4432:
4333:
1535:
448:
70:
5639:
4864:
Robbins, D. P. (1995). "Areas of
Polygons Inscribed in a Circle".
39:
6006:
5986:
5899:
5894:
5889:
5879:
5854:
5809:
5670:
5558:
5528:
5295:
5290:
5281:
5222:
5009:
constructing an inscribed pentagon with compass and straightedge.
4675:"Carlyle circles and Lemoine simplicity of polygon constructions"
3278:
2776:
and two sides, the hypotenuse of the larger triangle is found as
1721:
1539:
96:
5814:
5498:
5448:
5398:
5355:
5325:
5276:
5212:
4416:
4147:
4140:
3454:
whose coefficients are functions of the sides of the pentagon.
3240:
3111:
3895:. To find the number of sides this polygon has, the result is
2764:
To determine the length of this side, the two right triangles
5859:
4292:
3434:
4154:
3114:
in which to inscribe the pentagon and mark the center point
5248:
4236:
3223:
Euclid's method for pentagon at a given circle, using the
5038:
How to calculate various dimensions of regular pentagons.
4594:"A Construction for a Regular Polygon of Seventeen Sides"
4571:"Cyclic Averages of Regular Polygons and Platonic Solids"
934:
The area of a convex regular pentagon with side length
502:(distance from one side to the opposite vertex), width
4534:
Pythagoras' theorem#Similar figures on the three sides
3373:
subgroup has no degrees of freedom but can be seen as
2783:
2638:
2584:
2329:
2165:
2034:
1946:
4752:
Euklid's
Elements of Geometry, Book 4, Proposition 11
4598:
The
Quarterly Journal of Pure and Applied Mathematics
3834:
3601:
3519:
3475:
3162:
and its intersection outside the circle as the point
3038:
2984:
2930:
2827:
2782:
2575:
2545:
1941:
1911:
1891:
1871:
1741:
1559:
1490:
1446:
1371:
1247:
1224:
1204:
963:
940:
571:
548:
528:
508:
488:
465:
392:
362:
340:
320:
296:
4721:
1849:
Chords from the circumscribed circle to the vertices
4254:, like many other flowers, have a pentagonal shape.
2921:are known from the larger triangle. The result is:
447:of order 5 (through 72°, 144°, 216° and 288°). The
4899:
4758:. Translated by Richard Fitzpatrick. p. 119.
3870:
3765:
3584:
3505:
3072:
3018:{\displaystyle m\angle \mathrm {CDP} =54^{\circ }}
3017:
2967:
2906:
2802:
2681:
2558:
2528:
1924:
1897:
1877:
1837:
1693:
1542:). Substituting the regular pentagon's values for
1515:
1465:
1429:
1351:
1230:
1210:
1187:
946:
923:
554:
534:
514:
494:
474:
401:
374:
346:
326:
302:
4987:
4645:
4614:
3796:of equal-sized regular pentagons on a plane is a
3319:there is one subgroup with dihedral symmetry: Dih
3199:6a. Construct point F as the midpoint of O and W.
1476:
6183:
3286:. The base of the pyramid is a regular pentagon.
3252:
2814:of the smaller triangle then is found using the
2761:is the required side of the inscribed pentagon.
1218:of a regular pentagon is given, its edge length
4544:
3882:packing shown. In a preprint released in 2016,
2978:If DP is truly the side of a regular pentagon,
2968:{\displaystyle h={\frac {{\sqrt {5}}-1}{4}}\ .}
4575:Communications in Mathematics and Applications
3871:{\displaystyle (5-{\sqrt {5}})/3\approx 0.921}
1440:since the area of the circumscribed circle is
5655:
5060:
3585:{\displaystyle d_{1},d_{2},d_{3},d_{4},d_{5}}
4568:
3800:structure which covers 92.131% of the plane.
3073:{\displaystyle \left({\sqrt {5}}-1\right)/4}
2692:
4957:
4748:
4372:, also echinoderms with a pentagonal shape.
3911:There are 15 classes of pentagons that can
3464:
2697:The regular pentagon is constructible with
5662:
5648:
5067:
5053:
4963:Packings of regular pentagons in the plane
4742:
2803:{\displaystyle \scriptstyle {\sqrt {5}}/2}
1100:
5025:Definition and properties of the pentagon
4970:
4915:
4848:
4831:"Areas of Polygons Inscribed in a Circle"
4087:
4672:
4652:(2nd ed.). CRC Press. p. 329.
4639:
4591:
4564:
4562:
4493:; A pentagon is an order-4 associahedron
4313:is another fruit with fivefold symmetry.
3787:
3423:
3413:
3294:
3256:
3218:
3093:
2712:
285:
5632:List of regular polytopes and compounds
4863:
4828:
4666:
4649:CRC concise encyclopedia of mathematics
4585:
4399:. The faces are true regular pentagons.
6184:
4539:Trigonometric constants for a pentagon
4216:
3777:
5643:
5015:with only a compass and straightedge.
4988:
4715:
4559:
4295:contains five carpels, arranged in a
2772:are depicted below the circle. Using
1530:is the perimeter of the polygon, and
4932:
3469:For all convex pentagons with sides
3380:
2708:
1481:The area of any regular polygon is:
459:to its sides. Given its side length
132:
5669:
5013:How to construct a regular pentagon
4836:Discrete and Computational Geometry
4800:, (2008) The Symmetries of Things,
4459:United States Department of Defense
3440:
3366:for their central gyration orders.
13:
4961:; Kusner, WĂśden (September 2016),
4615:Peter R. Cromwell (22 July 1999).
3592:, the following inequality holds:
3214:
3083:
2998:
2995:
2992:
2988:
1516:{\displaystyle A={\frac {1}{2}}Pr}
14:
6218:
4981:
4866:The American Mathematical Monthly
4682:The American Mathematical Monthly
4440:
4352:Another example of echinoderm, a
1860:
95: 'angle') is any five-sided
4466:
4447:
4435:of gold, half a centimeter tall.
4424:
4404:
4384:
4361:
4345:
4325:
4303:
4280:
4259:
4244:
4228:
4174:
4167:
4160:
4153:
4146:
4139:
4077:
4070:
4063:
4056:
4049:
4023:
4016:
4009:
4002:
3995:
3969:
3962:
3955:
3948:
3941:
207:
202:
197:
145:
38:
6197:Polygons by the number of sides
4951:
4892:
4857:
4822:
4811:
4796:John H. Conway, Heidi Burgiel,
4790:
4673:DeTemple, Duane W. (Feb 1991).
3919:15 monohedral pentagonal tiles
2753:intersects the circle at point
5019:How to fold a regular pentagon
4772:
4608:
4550:
4340:have fivefold radial symmetry.
3851:
3835:
3670:
3605:
3391:A pentagram or pentangle is a
3261:Overhand knot of a paper strip
3029:. This is the cosine of 72°,
2892:
2886:
2875:
2869:
2848:
2834:
1477:Derivation of the area formula
1095:
1076:
1:
4749:Fitzpatrick, Richard (2008).
4722:George Edward Martin (1998).
4592:Richmond, Herbert W. (1893).
4569:Meskhishvili, Mamuka (2020).
3253:Physical construction methods
5027:, with interactive animation
4545:In-line notes and references
4235:Pentagonal cross-section of
4181:
4091:
4047:
4030:
3993:
3976:
3939:
3098:Method using Carlyle circles
2749:. A horizontal line through
455:regular pentagon are in the
334:), inscribed circle radius (
110:A pentagon may be simple or
7:
5021:using only a strip of paper
4484:
4377:
4199:Pentagonal icositetrahedron
3913:monohedrally tile the plane
3346:and no symmetry is labeled
3290:
1711:
1238:is found by the expression
10:
6223:
5621:
5048:
4939:Inequalities proposed in â
4646:Eric W. Weisstein (2003).
4318:
4204:Pentagonal hexecontahedron
3781:
3417:
3404:sides of the two pentagons
3384:
3150:Draw a circle centered at
3087:
1466:{\displaystyle \pi R^{2},}
99:or 5-gon. The sum of the
18:
6078:
6024:
5964:
5908:
5847:
5838:
5730:
5682:
4917:10.1016/j.jnt.2007.05.005
4221:
3828:of a regular pentagon is
3506:{\displaystyle a,b,c,d,e}
3358:for perpendiculars), and
3315:, order 10. Since 5 is a
2917:where cosine and sine of
2693:Geometrical constructions
402:{\displaystyle \varphi t}
274:
248:
233:
216:
190:
180:
166:
156:
144:
139:
88: 'five' and
53:
37:
32:
25:Pentagon (disambiguation)
4903:Journal of Number Theory
4508:List of geometric shapes
3465:General convex pentagons
3354:for diagonal) or edges (
3237:compass and straightedge
3180:Draw a circle of radius
3169:Draw a circle of radius
2699:compass and straightedge
4829:Robbins, D. P. (1994).
4728:. Springer. p. 6.
4725:Geometric constructions
4395:formed as a pentagonal
4391:A Ho-Mg-Zn icosahedral
4209:Truncated trapezohedron
192:CoxeterâDynkin diagrams
6192:Constructible polygons
5007:Animated demonstration
4457:, headquarters of the
4088:Pentagons in polyhedra
3893:(360 â 108) / 2 = 126°
3872:
3801:
3767:
3586:
3507:
3429:
3300:
3262:
3231:A regular pentagon is
3228:
3227:, animation 1 min 39 s
3099:
3074:
3019:
2969:
2908:
2804:
2717:
2683:
2659:
2605:
2560:
2530:
2350:
2186:
2055:
1967:
1932:respectively, we have
1926:
1899:
1879:
1839:
1724:, which is the radius
1695:
1517:
1467:
1431:
1353:
1232:
1212:
1189:
948:
925:
556:
536:
516:
496:
476:
411:
403:
376:
348:
328:
304:
114:. A self-intersecting
23:. For other uses, see
4798:Chaim Goodman-Strauss
3897:360 / (180 â 126) = 6
3873:
3791:
3768:
3587:
3508:
3427:
3414:Equilateral pentagons
3298:
3260:
3222:
3097:
3075:
3020:
2970:
2909:
2805:
2716:
2684:
2639:
2585:
2561:
2559:{\displaystyle d_{i}}
2531:
2330:
2166:
2035:
1947:
1927:
1925:{\displaystyle d_{i}}
1900:
1880:
1840:
1696:
1518:
1468:
1432:
1354:
1233:
1213:
1190:
949:
926:
557:
537:
517:
497:
477:
441:reflectional symmetry
404:
377:
349:
329:
305:
289:
16:Shape with five sides
5895:Nonagon/Enneagon (9)
5825:Tangential trapezoid
3832:
3599:
3517:
3473:
3420:Equilateral pentagon
3277:Construct a regular
3135:Construct the point
3036:
3027:double angle formula
2982:
2928:
2825:
2780:
2573:
2543:
1939:
1909:
1889:
1869:
1855:circumscribed circle
1739:
1557:
1488:
1444:
1369:
1245:
1222:
1202:
1198:If the circumradius
961:
938:
569:
546:
526:
506:
486:
463:
390:
360:
338:
318:
294:
6007:Megagon (1,000,000)
5775:Isosceles trapezoid
5616:pentagonal polytope
5515:Uniform 10-polytope
5075:Fundamental convex
4941:Crux Mathematicorum
4780:Mathematical Models
4530:, the Chrysler logo
4368:An illustration of
4217:Pentagons in nature
3920:
3778:Pentagons in tiling
3762:
3744:
3726:
3708:
3690:
3139:as the midpoint of
2774:Pythagoras' theorem
2674:
2620:
2365:
2201:
2070:
1982:
542:) and circumradius
445:rotational symmetry
375:{\displaystyle R+r}
5977:Icositetragon (24)
5485:Uniform 9-polytope
5435:Uniform 8-polytope
5385:Uniform 7-polytope
5342:Uniform 6-polytope
5312:Uniform 5-polytope
5272:Uniform polychoron
5235:Uniform polyhedron
5083:in dimensions 2â10
4990:Weisstein, Eric W.
4850:10.1007/bf02574377
4513:Pentagonal numbers
3918:
3878:, achieved by the
3868:
3824:The maximum known
3802:
3794:best-known packing
3763:
3748:
3730:
3712:
3694:
3676:
3582:
3503:
3430:
3301:
3284:pentagonal pyramid
3263:
3229:
3154:through the point
3104:quadratic equation
3100:
3070:
3015:
2965:
2904:
2816:half-angle formula
2800:
2799:
2718:
2679:
2678:
2660:
2621:
2606:
2556:
2526:
2524:
2366:
2351:
2202:
2187:
2071:
2056:
1983:
1968:
1922:
1895:
1875:
1835:
1691:
1550:gives the formula
1538:(equivalently the
1513:
1463:
1427:
1349:
1228:
1208:
1185:
1183:
944:
921:
919:
552:
532:
512:
492:
475:{\displaystyle t,}
472:
439:has five lines of
412:
399:
372:
344:
324:
300:
151:A regular pentagon
107:pentagon is 540°.
6207:Elementary shapes
6179:
6178:
6020:
6019:
5997:Myriagon (10,000)
5982:Triacontagon (30)
5946:Heptadecagon (17)
5936:Pentadecagon (15)
5931:Tetradecagon (14)
5870:Quadrilateral (4)
5740:Antiparallelogram
5637:
5636:
5624:Polytope families
5081:uniform polytopes
4806:978-1-56881-220-5
4765:978-0-615-17984-1
4297:five-pointed star
4214:
4213:
4085:
4084:
3849:
3459:Robbins pentagons
3381:Regular pentagram
3049:
2961:
2957:
2945:
2900:
2896:
2789:
2709:Richmond's method
1898:{\displaystyle L}
1878:{\displaystyle R}
1818:
1815:
1813:
1786:
1777:
1704:with side length
1689:
1677:
1631:
1619:
1574:
1505:
1422:
1421:
1415:
1398:
1342:
1332:
1282:
1281:
1275:
1259:
1231:{\displaystyle t}
1211:{\displaystyle R}
1170:
1160:
1154:
1115:
1098:
1093:
1056:
1015:
1009:
1007:
947:{\displaystyle t}
910:
900:
850:
849:
843:
827:
801:
788:
787:
781:
745:
729:
727:
726:
688:
676:
672:
666:
625:
613:
609:
605:
603:
555:{\displaystyle R}
535:{\displaystyle D}
515:{\displaystyle W}
495:{\displaystyle H}
384:, width/diagonal
347:{\displaystyle r}
327:{\displaystyle R}
303:{\displaystyle t}
284:
283:
133:Regular pentagons
112:self-intersecting
67:
66:
6214:
5992:Chiliagon (1000)
5972:Icositrigon (23)
5951:Octadecagon (18)
5941:Hexadecagon (16)
5845:
5844:
5664:
5657:
5650:
5641:
5640:
5628:Regular polytope
5189:
5178:
5167:
5126:
5069:
5062:
5055:
5046:
5045:
5003:
5002:
4976:
4975:
4974:
4955:
4949:
4936:
4930:
4928:
4919:
4896:
4890:
4889:
4861:
4855:
4854:
4852:
4826:
4820:
4815:
4809:
4794:
4788:
4776:
4770:
4769:
4757:
4746:
4740:
4739:
4719:
4713:
4712:
4710:
4704:. Archived from
4679:
4670:
4664:
4663:
4643:
4637:
4636:
4612:
4606:
4605:
4589:
4583:
4582:
4566:
4557:
4554:
4470:
4451:
4428:
4408:
4388:
4365:
4349:
4329:
4307:
4284:
4263:
4248:
4232:
4178:
4171:
4164:
4157:
4150:
4143:
4092:
4081:
4074:
4067:
4060:
4053:
4027:
4020:
4013:
4006:
3999:
3973:
3966:
3959:
3952:
3945:
3921:
3917:
3907:
3906:
3905:
3901:
3894:
3877:
3875:
3874:
3869:
3858:
3850:
3845:
3820:
3819:
3818:
3814:
3772:
3770:
3769:
3764:
3761:
3756:
3743:
3738:
3725:
3720:
3707:
3702:
3689:
3684:
3669:
3668:
3656:
3655:
3643:
3642:
3630:
3629:
3617:
3616:
3591:
3589:
3588:
3583:
3581:
3580:
3568:
3567:
3555:
3554:
3542:
3541:
3529:
3528:
3512:
3510:
3509:
3504:
3441:Cyclic pentagons
3305:regular pentagon
3079:
3077:
3076:
3071:
3066:
3061:
3057:
3050:
3045:
3024:
3022:
3021:
3016:
3014:
3013:
3001:
2974:
2972:
2971:
2966:
2959:
2958:
2953:
2946:
2941:
2938:
2913:
2911:
2910:
2905:
2898:
2897:
2895:
2878:
2855:
2844:
2809:
2807:
2806:
2801:
2795:
2790:
2785:
2688:
2686:
2685:
2680:
2673:
2668:
2658:
2653:
2631:
2630:
2625:
2619:
2614:
2604:
2599:
2565:
2563:
2562:
2557:
2555:
2554:
2535:
2533:
2532:
2527:
2525:
2518:
2514:
2513:
2512:
2503:
2502:
2487:
2486:
2481:
2477:
2476:
2475:
2463:
2462:
2447:
2446:
2437:
2436:
2421:
2420:
2415:
2411:
2410:
2409:
2397:
2396:
2364:
2359:
2349:
2344:
2321:
2317:
2316:
2312:
2311:
2310:
2298:
2297:
2283:
2282:
2273:
2272:
2257:
2256:
2251:
2247:
2246:
2245:
2233:
2232:
2200:
2195:
2185:
2180:
2157:
2153:
2152:
2151:
2142:
2141:
2126:
2125:
2120:
2116:
2115:
2114:
2102:
2101:
2069:
2064:
2054:
2049:
2026:
2022:
2021:
2020:
2008:
2007:
1981:
1976:
1966:
1961:
1931:
1929:
1928:
1923:
1921:
1920:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1844:
1842:
1841:
1836:
1819:
1817:
1816:
1814:
1809:
1801:
1792:
1787:
1785:
1784:
1783:
1782:
1778:
1770:
1749:
1718:inscribed circle
1700:
1698:
1697:
1692:
1690:
1685:
1684:
1683:
1682:
1678:
1673:
1665:
1651:
1650:
1637:
1632:
1627:
1626:
1625:
1624:
1620:
1615:
1607:
1589:
1575:
1567:
1522:
1520:
1519:
1514:
1506:
1498:
1472:
1470:
1469:
1464:
1459:
1458:
1436:
1434:
1433:
1428:
1423:
1417:
1416:
1411:
1402:
1401:
1399:
1394:
1393:
1392:
1379:
1362:and its area is
1358:
1356:
1355:
1350:
1340:
1333:
1325:
1308:
1307:
1283:
1277:
1276:
1271:
1262:
1261:
1257:
1237:
1235:
1234:
1229:
1217:
1215:
1214:
1209:
1194:
1192:
1191:
1186:
1184:
1180:
1179:
1168:
1161:
1156:
1155:
1147:
1146:
1134:
1132:
1131:
1121:
1116:
1111:
1110:
1109:
1099:
1094:
1089:
1072:
1069:
1061:
1057:
1052:
1051:
1050:
1035:
1034:
1021:
1016:
1011:
1010:
1008:
1003:
992:
990:
989:
979:
953:
951:
950:
945:
930:
928:
927:
922:
920:
908:
901:
893:
876:
875:
851:
845:
844:
839:
830:
829:
825:
799:
789:
783:
782:
777:
768:
767:
743:
730:
728:
722:
718:
710:
686:
674:
673:
668:
667:
662:
653:
623:
611:
610:
604:
599:
588:
587:
561:
559:
558:
553:
541:
539:
538:
533:
521:
519:
518:
513:
501:
499:
498:
493:
481:
479:
478:
473:
410:
408:
406:
405:
400:
383:
381:
379:
378:
373:
353:
351:
350:
345:
333:
331:
330:
325:
309:
307:
306:
301:
212:
211:
210:
206:
205:
201:
200:
149:
140:Regular pentagon
137:
136:
116:regular pentagon
42:
30:
29:
6222:
6221:
6217:
6216:
6215:
6213:
6212:
6211:
6182:
6181:
6180:
6175:
6074:
6028:
6016:
5960:
5926:Tridecagon (13)
5916:Hendecagon (11)
5904:
5840:
5834:
5805:Right trapezoid
5726:
5678:
5668:
5638:
5607:
5600:
5593:
5476:
5469:
5462:
5426:
5419:
5412:
5376:
5369:
5203:Regular polygon
5196:
5187:
5180:
5176:
5169:
5165:
5156:
5147:
5140:
5136:
5124:
5118:
5114:
5102:
5084:
5073:
5042:
4984:
4979:
4956:
4952:
4937:
4933:
4897:
4893:
4878:10.2307/2974766
4862:
4858:
4827:
4823:
4816:
4812:
4795:
4791:
4785:H. Martyn Cundy
4777:
4773:
4766:
4755:
4747:
4743:
4736:
4720:
4716:
4708:
4694:10.2307/2323939
4677:
4671:
4667:
4660:
4644:
4640:
4633:
4613:
4609:
4590:
4586:
4567:
4560:
4555:
4551:
4547:
4487:
4480:
4471:
4462:
4452:
4443:
4436:
4429:
4420:
4409:
4400:
4389:
4380:
4373:
4366:
4357:
4350:
4341:
4330:
4321:
4314:
4308:
4299:
4285:
4276:
4264:
4255:
4252:Morning glories
4249:
4240:
4233:
4224:
4219:
4133:
4115:
4107:
4099:
4090:
3903:
3899:
3898:
3896:
3892:
3854:
3844:
3833:
3830:
3829:
3826:packing density
3816:
3812:
3811:
3810:360° / 108° = 3
3809:
3786:
3784:Pentagon tiling
3780:
3757:
3752:
3739:
3734:
3721:
3716:
3703:
3698:
3685:
3680:
3664:
3660:
3651:
3647:
3638:
3634:
3625:
3621:
3612:
3608:
3600:
3597:
3596:
3576:
3572:
3563:
3559:
3550:
3546:
3537:
3533:
3524:
3520:
3518:
3515:
3514:
3474:
3471:
3470:
3467:
3452:septic equation
3443:
3422:
3416:
3400:Schläfli symbol
3389:
3383:
3334:
3330:
3322:
3312:
3293:
3255:
3225:golden triangle
3217:
3215:Euclid's method
3092:
3086:
3084:Carlyle circles
3062:
3044:
3043:
3039:
3037:
3034:
3033:
3009:
3005:
2991:
2983:
2980:
2979:
2940:
2939:
2937:
2929:
2926:
2925:
2879:
2856:
2854:
2840:
2826:
2823:
2822:
2791:
2784:
2781:
2778:
2777:
2733:and a midpoint
2711:
2695:
2669:
2664:
2654:
2643:
2626:
2615:
2610:
2600:
2589:
2580:
2579:
2574:
2571:
2570:
2550:
2546:
2544:
2541:
2540:
2523:
2522:
2508:
2504:
2498:
2494:
2482:
2471:
2467:
2458:
2454:
2453:
2449:
2448:
2442:
2438:
2432:
2428:
2416:
2405:
2401:
2392:
2388:
2387:
2383:
2382:
2381:
2377:
2367:
2360:
2355:
2345:
2334:
2326:
2325:
2306:
2302:
2293:
2289:
2288:
2284:
2278:
2274:
2268:
2264:
2252:
2241:
2237:
2228:
2224:
2223:
2219:
2218:
2217:
2213:
2203:
2196:
2191:
2181:
2170:
2162:
2161:
2147:
2143:
2137:
2133:
2121:
2110:
2106:
2097:
2093:
2092:
2088:
2087:
2086:
2082:
2072:
2065:
2060:
2050:
2039:
2031:
2030:
2016:
2012:
2003:
1999:
1998:
1994:
1984:
1977:
1972:
1962:
1951:
1942:
1940:
1937:
1936:
1916:
1912:
1910:
1907:
1906:
1890:
1887:
1886:
1870:
1867:
1866:
1863:
1851:
1808:
1800:
1796:
1791:
1769:
1765:
1764:
1763:
1753:
1748:
1740:
1737:
1736:
1714:
1666:
1664:
1660:
1659:
1658:
1646:
1642:
1638:
1636:
1608:
1606:
1602:
1601:
1600:
1590:
1588:
1566:
1558:
1555:
1554:
1497:
1489:
1486:
1485:
1479:
1454:
1450:
1445:
1442:
1441:
1410:
1403:
1400:
1388:
1384:
1380:
1378:
1370:
1367:
1366:
1324:
1303:
1299:
1270:
1263:
1260:
1246:
1243:
1242:
1223:
1220:
1219:
1203:
1200:
1199:
1182:
1181:
1175:
1171:
1142:
1138:
1133:
1127:
1123:
1122:
1120:
1105:
1101:
1088:
1071:
1070:
1068:
1059:
1058:
1046:
1042:
1030:
1026:
1022:
1020:
1002:
991:
985:
981:
980:
978:
971:
964:
962:
959:
958:
939:
936:
935:
918:
917:
892:
871:
867:
838:
831:
828:
815:
809:
808:
776:
769:
766:
759:
753:
752:
717:
709:
702:
696:
695:
661:
654:
652:
645:
633:
632:
598:
586:
579:
572:
570:
567:
566:
547:
544:
543:
527:
524:
523:
507:
504:
503:
487:
484:
483:
464:
461:
460:
427:interior angles
423:Schläfli symbol
391:
388:
387:
385:
361:
358:
357:
355:
339:
336:
335:
319:
316:
315:
295:
292:
291:
228:
208:
203:
198:
196:
182:Schläfli symbol
161:Regular polygon
152:
135:
101:internal angles
49:
28:
17:
12:
11:
5:
6220:
6210:
6209:
6204:
6199:
6194:
6177:
6176:
6174:
6173:
6168:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6131:Pseudotriangle
6128:
6123:
6118:
6113:
6108:
6103:
6098:
6093:
6088:
6082:
6080:
6076:
6075:
6073:
6072:
6067:
6062:
6057:
6052:
6047:
6042:
6037:
6031:
6029:
6022:
6021:
6018:
6017:
6015:
6014:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5968:
5966:
5962:
5961:
5959:
5958:
5953:
5948:
5943:
5938:
5933:
5928:
5923:
5921:Dodecagon (12)
5918:
5912:
5910:
5906:
5905:
5903:
5902:
5897:
5892:
5887:
5882:
5877:
5872:
5867:
5862:
5857:
5851:
5849:
5842:
5836:
5835:
5833:
5832:
5827:
5822:
5817:
5812:
5807:
5802:
5797:
5792:
5787:
5782:
5777:
5772:
5767:
5762:
5757:
5752:
5747:
5742:
5736:
5734:
5732:Quadrilaterals
5728:
5727:
5725:
5724:
5719:
5714:
5709:
5704:
5699:
5694:
5688:
5686:
5680:
5679:
5667:
5666:
5659:
5652:
5644:
5635:
5634:
5619:
5618:
5609:
5605:
5598:
5591:
5587:
5578:
5561:
5552:
5541:
5540:
5538:
5536:
5531:
5522:
5517:
5511:
5510:
5508:
5506:
5501:
5492:
5487:
5481:
5480:
5478:
5474:
5467:
5460:
5456:
5451:
5442:
5437:
5431:
5430:
5428:
5424:
5417:
5410:
5406:
5401:
5392:
5387:
5381:
5380:
5378:
5374:
5367:
5363:
5358:
5349:
5344:
5338:
5337:
5335:
5333:
5328:
5319:
5314:
5308:
5307:
5298:
5293:
5288:
5279:
5274:
5268:
5267:
5258:
5256:
5251:
5242:
5237:
5231:
5230:
5225:
5220:
5215:
5210:
5205:
5199:
5198:
5194:
5190:
5185:
5174:
5163:
5154:
5145:
5138:
5132:
5122:
5116:
5110:
5104:
5098:
5092:
5086:
5085:
5074:
5072:
5071:
5064:
5057:
5049:
5044:
5040:
5039:
5033:
5028:
5022:
5016:
5010:
5004:
4983:
4982:External links
4980:
4978:
4977:
4950:
4931:
4891:
4872:(6): 523â530.
4856:
4843:(2): 223â236.
4821:
4810:
4789:
4771:
4764:
4741:
4734:
4714:
4711:on 2015-12-21.
4665:
4658:
4638:
4631:
4607:
4584:
4558:
4548:
4546:
4543:
4542:
4541:
4536:
4531:
4525:
4520:
4515:
4510:
4505:
4500:
4494:
4486:
4483:
4482:
4481:
4478:baseball field
4472:
4465:
4463:
4453:
4446:
4442:
4441:Other examples
4439:
4438:
4437:
4430:
4423:
4421:
4410:
4403:
4401:
4390:
4383:
4379:
4376:
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3974:
3967:
3960:
3953:
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3938:
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3934:
3931:
3928:
3925:
3880:double lattice
3867:
3864:
3861:
3857:
3853:
3848:
3843:
3840:
3837:
3806:regular tiling
3798:double lattice
3782:Main article:
3779:
3776:
3775:
3774:
3760:
3755:
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3607:
3604:
3579:
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3571:
3566:
3562:
3558:
3553:
3549:
3545:
3540:
3536:
3532:
3527:
3523:
3513:and diagonals
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3466:
3463:
3442:
3439:
3418:Main article:
3415:
3412:
3398:pentagon. Its
3385:Main article:
3382:
3379:
3375:directed edges
3332:
3328:
3320:
3310:
3292:
3289:
3288:
3287:
3275:
3254:
3251:
3249:circa 300 BC.
3216:
3213:
3212:
3211:
3207:
3206:
3201:
3200:
3193:
3192:
3189:
3178:
3167:
3148:
3133:
3126:
3119:
3090:Carlyle circle
3088:Main article:
3085:
3082:
3069:
3065:
3060:
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3012:
3008:
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2063:
2059:
2053:
2048:
2045:
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2038:
2033:
2032:
2029:
2025:
2019:
2015:
2011:
2006:
2002:
1997:
1993:
1990:
1987:
1985:
1980:
1975:
1971:
1965:
1960:
1957:
1954:
1950:
1945:
1944:
1919:
1915:
1894:
1874:
1862:
1861:Point in plane
1859:
1850:
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622:
619:
616:
608:
602:
597:
594:
591:
585:
582:
580:
578:
575:
574:
562:are given by:
551:
531:
511:
491:
471:
468:
398:
395:
371:
368:
365:
343:
323:
299:
282:
281:
278:
272:
271:
250:
246:
245:
242:
235:Internal angle
231:
230:
226:
220:
218:Symmetry group
214:
213:
194:
188:
187:
184:
178:
177:
174:
164:
163:
158:
154:
153:
150:
142:
141:
134:
131:
125:) is called a
65:
64:
61:
51:
50:
43:
35:
34:
15:
9:
6:
4:
3:
2:
6219:
6208:
6205:
6203:
6200:
6198:
6195:
6193:
6190:
6189:
6187:
6172:
6171:Weakly simple
6169:
6167:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6111:Infinite skew
6109:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6087:
6084:
6083:
6081:
6077:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6051:
6048:
6046:
6043:
6041:
6038:
6036:
6033:
6032:
6030:
6027:
6026:Star polygons
6023:
6013:
6012:Apeirogon (â)
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5980:
5978:
5975:
5973:
5970:
5969:
5967:
5963:
5957:
5956:Icosagon (20)
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5913:
5911:
5907:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5853:
5852:
5850:
5846:
5843:
5837:
5831:
5828:
5826:
5823:
5821:
5818:
5816:
5813:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5790:Parallelogram
5788:
5786:
5785:Orthodiagonal
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5765:Ex-tangential
5763:
5761:
5758:
5756:
5753:
5751:
5748:
5746:
5743:
5741:
5738:
5737:
5735:
5733:
5729:
5723:
5720:
5718:
5715:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5689:
5687:
5685:
5681:
5676:
5672:
5665:
5660:
5658:
5653:
5651:
5646:
5645:
5642:
5633:
5629:
5625:
5620:
5617:
5613:
5610:
5608:
5601:
5594:
5588:
5586:
5582:
5579:
5577:
5573:
5569:
5565:
5562:
5560:
5556:
5553:
5551:
5547:
5543:
5542:
5539:
5537:
5535:
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5530:
5526:
5523:
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5516:
5513:
5512:
5509:
5507:
5505:
5502:
5500:
5496:
5493:
5491:
5488:
5486:
5483:
5482:
5479:
5477:
5470:
5463:
5457:
5455:
5452:
5450:
5446:
5443:
5441:
5438:
5436:
5433:
5432:
5429:
5427:
5420:
5413:
5407:
5405:
5402:
5400:
5396:
5393:
5391:
5388:
5386:
5383:
5382:
5379:
5377:
5370:
5364:
5362:
5359:
5357:
5353:
5350:
5348:
5345:
5343:
5340:
5339:
5336:
5334:
5332:
5329:
5327:
5323:
5320:
5318:
5315:
5313:
5310:
5309:
5306:
5302:
5299:
5297:
5294:
5292:
5291:Demitesseract
5289:
5287:
5283:
5280:
5278:
5275:
5273:
5270:
5269:
5266:
5262:
5259:
5257:
5255:
5252:
5250:
5246:
5243:
5241:
5238:
5236:
5233:
5232:
5229:
5226:
5224:
5221:
5219:
5216:
5214:
5211:
5209:
5206:
5204:
5201:
5200:
5197:
5191:
5188:
5184:
5177:
5173:
5166:
5162:
5157:
5153:
5148:
5144:
5139:
5137:
5135:
5131:
5121:
5117:
5115:
5113:
5109:
5105:
5103:
5101:
5097:
5093:
5091:
5088:
5087:
5082:
5078:
5070:
5065:
5063:
5058:
5056:
5051:
5050:
5047:
5043:
5037:
5034:
5032:
5029:
5026:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
5000:
4999:
4994:
4991:
4986:
4985:
4973:
4968:
4964:
4960:
4959:Hales, Thomas
4954:
4947:
4944:
4942:
4935:
4927:
4923:
4918:
4913:
4909:
4905:
4904:
4895:
4887:
4883:
4879:
4875:
4871:
4867:
4860:
4851:
4846:
4842:
4838:
4837:
4832:
4825:
4819:
4814:
4807:
4803:
4799:
4793:
4786:
4782:
4781:
4775:
4767:
4761:
4754:
4753:
4745:
4737:
4735:0-387-98276-0
4731:
4727:
4726:
4718:
4707:
4703:
4699:
4695:
4691:
4688:(2): 97â108.
4687:
4683:
4676:
4669:
4661:
4659:1-58488-347-2
4655:
4651:
4650:
4642:
4634:
4632:0-521-66405-5
4628:
4624:
4620:
4619:
4611:
4603:
4599:
4595:
4588:
4580:
4576:
4572:
4565:
4563:
4553:
4549:
4540:
4537:
4535:
4532:
4529:
4526:
4524:
4523:Pentagram map
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4504:
4501:
4498:
4495:
4492:
4491:Associahedron
4489:
4488:
4479:
4475:
4469:
4464:
4460:
4456:
4450:
4445:
4444:
4434:
4427:
4422:
4418:
4414:
4407:
4402:
4398:
4394:
4387:
4382:
4381:
4371:
4370:brittle stars
4364:
4359:
4356:endoskeleton.
4355:
4348:
4343:
4339:
4335:
4328:
4323:
4322:
4312:
4306:
4301:
4298:
4294:
4290:
4283:
4278:
4274:
4273:
4268:
4262:
4257:
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4238:
4231:
4226:
4225:
4210:
4207:
4205:
4202:
4200:
4197:
4195:
4192:
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4187:
4185:
4182:
4177:
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4170:
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4159:
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4149:
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4142:
4138:
4137:
4134:
4128:
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4094:
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4080:
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3998:
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3411:
3409:
3405:
3401:
3397:
3394:
3388:
3378:
3376:
3372:
3367:
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3336:
3327:symmetries: Z
3326:
3318:
3314:
3306:
3297:
3285:
3280:
3276:
3274:when backlit.
3273:
3269:
3268:overhand knot
3265:
3264:
3259:
3250:
3248:
3247:
3242:
3238:
3234:
3233:constructible
3226:
3221:
3209:
3208:
3203:
3202:
3198:
3197:
3196:
3190:
3187:
3183:
3179:
3176:
3172:
3168:
3165:
3161:
3157:
3153:
3149:
3146:
3142:
3138:
3134:
3131:
3127:
3124:
3120:
3117:
3113:
3109:
3108:
3107:
3105:
3096:
3091:
3081:
3067:
3063:
3058:
3054:
3051:
3046:
3040:
3032:
3028:
3010:
3006:
3002:
2985:
2962:
2954:
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2947:
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2934:
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2924:
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2922:
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2901:
2889:
2883:
2880:
2872:
2866:
2863:
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2851:
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2837:
2831:
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2821:
2820:
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2817:
2813:
2796:
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2767:
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2760:
2756:
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2748:
2744:
2740:
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2727:
2725:
2724:
2715:
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2704:
2700:
2675:
2670:
2665:
2661:
2655:
2650:
2647:
2644:
2640:
2635:
2632:
2627:
2622:
2616:
2611:
2607:
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2596:
2593:
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2567:
2551:
2547:
2519:
2515:
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2322:
2318:
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2307:
2303:
2299:
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2290:
2285:
2279:
2275:
2269:
2265:
2261:
2258:
2253:
2248:
2242:
2238:
2234:
2229:
2225:
2220:
2214:
2210:
2207:
2205:
2197:
2192:
2188:
2182:
2177:
2174:
2171:
2167:
2158:
2154:
2148:
2144:
2138:
2134:
2130:
2127:
2122:
2117:
2111:
2107:
2103:
2098:
2094:
2089:
2083:
2079:
2076:
2074:
2066:
2061:
2057:
2051:
2046:
2043:
2040:
2036:
2027:
2023:
2017:
2013:
2009:
2004:
2000:
1995:
1991:
1988:
1986:
1978:
1973:
1969:
1963:
1958:
1955:
1952:
1948:
1935:
1934:
1933:
1917:
1913:
1892:
1872:
1858:
1856:
1832:
1829:
1826:
1823:
1820:
1810:
1805:
1802:
1797:
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1788:
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1766:
1760:
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1727:
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1709:
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1686:
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1597:
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1579:
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1533:
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1510:
1507:
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1499:
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1474:
1460:
1455:
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1424:
1418:
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1407:
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1395:
1389:
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1375:
1372:
1365:
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1346:
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1334:
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1326:
1321:
1318:
1315:
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1309:
1304:
1300:
1296:
1293:
1290:
1287:
1284:
1278:
1272:
1267:
1264:
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1248:
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1239:
1225:
1205:
1176:
1172:
1165:
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1157:
1151:
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1135:
1128:
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1117:
1112:
1106:
1102:
1090:
1085:
1082:
1079:
1073:
1065:
1063:
1053:
1047:
1043:
1039:
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1031:
1027:
1023:
1017:
1012:
1004:
999:
996:
993:
986:
982:
975:
973:
968:
957:
956:
955:
941:
914:
911:
905:
902:
897:
894:
889:
886:
883:
880:
877:
872:
868:
864:
861:
858:
855:
852:
846:
840:
835:
832:
822:
819:
817:
812:
805:
802:
796:
793:
790:
784:
778:
773:
770:
763:
761:
756:
749:
746:
740:
737:
734:
731:
723:
719:
714:
711:
706:
704:
699:
692:
689:
683:
680:
677:
669:
663:
658:
655:
649:
647:
642:
639:
636:
629:
626:
620:
617:
614:
606:
600:
595:
592:
589:
583:
581:
576:
565:
564:
563:
549:
529:
509:
489:
469:
466:
458:
454:
450:
446:
442:
438:
436:
430:
428:
424:
420:
418:
396:
393:
369:
366:
363:
341:
321:
313:
297:
288:
279:
277:
273:
270:
266:
262:
258:
254:
251:
247:
243:
240:
236:
232:
224:
221:
219:
215:
195:
193:
189:
185:
183:
179:
175:
173:
169:
165:
162:
159:
155:
148:
143:
138:
130:
128:
124:
122:
117:
113:
108:
106:
102:
98:
94:
90:
87:
83:
80:
76:
72:
62:
60:
56:
52:
47:
41:
36:
31:
26:
22:
5965:>20 sides
5900:Decagon (10)
5885:Heptagon (7)
5875:Pentagon (5)
5874:
5865:Triangle (3)
5760:Equidiagonal
5611:
5580:
5571:
5563:
5554:
5545:
5525:10-orthoplex
5261:Dodecahedron
5227:
5182:
5171:
5160:
5151:
5142:
5133:
5129:
5119:
5111:
5107:
5099:
5095:
5041:
4996:
4962:
4953:
4938:
4934:
4910:(1): 17â48,
4907:
4901:
4894:
4869:
4865:
4859:
4840:
4834:
4824:
4813:
4792:
4778:
4774:
4751:
4744:
4724:
4717:
4706:the original
4685:
4681:
4668:
4648:
4641:
4617:
4610:
4601:
4597:
4587:
4578:
4574:
4552:
4503:Golden ratio
4497:Dodecahedron
4455:The Pentagon
4413:pyritohedral
4397:dodecahedron
4393:quasicrystal
4270:
4189:Pyritohedron
4184:Dodecahedron
3910:
3889:
3884:Thomas Hales
3823:
3803:
3468:
3456:
3444:
3431:
3408:golden ratio
3390:
3370:
3368:
3363:
3359:
3355:
3351:
3347:
3343:
3337:
3325:cyclic group
3317:prime number
3304:
3302:
3244:
3230:
3194:
3185:
3181:
3174:
3170:
3163:
3159:
3155:
3151:
3144:
3140:
3136:
3129:
3122:
3115:
3101:
3080:as desired.
3031:which equals
2977:
2918:
2916:
2811:
2769:
2765:
2763:
2758:
2757:, and chord
2754:
2750:
2746:
2742:
2738:
2734:
2730:
2728:
2721:
2719:
2703:Fermat prime
2701:, as 5 is a
2696:
2538:
1864:
1852:
1729:
1725:
1715:
1705:
1703:
1547:
1543:
1531:
1527:
1525:
1480:
1439:
1361:
1197:
954:is given by
933:
457:golden ratio
433:
431:
415:
413:
312:circumradius
276:Dual polygon
229:), order 2Ă5
119:
115:
109:
92:
89:
85:
82:
74:
68:
21:The Pentagon
6161:Star-shaped
6136:Rectilinear
6106:Equilateral
6101:Equiangular
6065:Hendecagram
5909:11â20 sides
5890:Octagon (8)
5880:Hexagon (6)
5855:Monogon (1)
5697:Equilateral
5534:10-demicube
5495:9-orthoplex
5445:8-orthoplex
5395:7-orthoplex
5352:6-orthoplex
5322:5-orthoplex
5277:Pentachoron
5265:Icosahedron
5240:Tetrahedron
4415:crystal of
4338:echinoderms
3406:are in the
3340:John Conway
3184:and center
3173:and center
482:its height
261:equilateral
6202:5 (number)
6186:Categories
6166:Tangential
6070:Dodecagram
5848:1â10 sides
5839:By number
5820:Tangential
5800:Right kite
5520:10-simplex
5504:9-demicube
5454:8-demicube
5404:7-demicube
5361:6-demicube
5331:5-demicube
5245:Octahedron
4993:"Pentagon"
4972:1602.07220
4604:: 206â207.
4581:: 335â355.
4474:Home plate
4354:sea urchin
354:), height
249:Properties
77:(from
6146:Reinhardt
6055:Enneagram
6045:Heptagram
6035:Pentagram
6002:65537-gon
5860:Digon (2)
5830:Trapezoid
5795:Rectangle
5745:Bicentric
5707:Isosceles
5684:Triangles
5568:orthoplex
5490:9-simplex
5440:8-simplex
5390:7-simplex
5347:6-simplex
5317:5-simplex
5286:Tesseract
5036:Pentagon.
4998:MathWorld
4618:Polyhedra
4528:Pentastar
4518:Pentagram
4311:Starfruit
4289:gynoecium
4272:Rafflesia
4194:Tetartoid
3863:≈
3842:−
3387:Pentagram
3272:pentagram
3052:−
3011:∘
2989:∠
2948:−
2890:ϕ
2884:
2873:ϕ
2867:
2861:−
2838:ϕ
2832:
2723:Polyhedra
2641:∑
2587:∑
2332:∑
2168:∑
2037:∑
1949:∑
1827:⋅
1821:≈
1806:−
1772:π
1761:
1671:π
1656:
1613:π
1598:
1586:⋅
1577:⋅
1448:π
1335:≈
1327:π
1322:
1305:∘
1297:
1268:−
1163:≈
1140:φ
1048:∘
1040:
903:≈
895:π
890:
873:∘
865:
794:≈
738:≈
732:⋅
715:−
681:≈
618:≈
449:diagonals
429:of 108°.
394:φ
127:pentagram
6121:Isotoxal
6116:Isogonal
6060:Decagram
6050:Octagram
6040:Hexagram
5841:of sides
5770:Harmonic
5671:Polygons
5622:Topics:
5585:demicube
5550:polytope
5544:Uniform
5305:600-cell
5301:120-cell
5254:Demicube
5228:Pentagon
5208:Triangle
4485:See also
4433:Fiveling
4378:Minerals
4334:sea star
4269:tube of
4267:Perigone
3323:, and 2
3313:symmetry
3291:Symmetry
3246:Elements
3235:using a
2741:. Angle
1712:Inradius
1536:inradius
437:pentagon
425:{5} and
419:pentagon
269:isotoxal
265:isogonal
223:Dihedral
172:vertices
123:pentagon
75:pentagon
71:geometry
59:vertices
48:pentagon
33:Pentagon
6141:Regular
6086:Concave
6079:Classes
5987:257-gon
5810:Rhombus
5750:Crossed
5559:simplex
5529:10-cube
5296:24-cell
5282:16-cell
5223:Hexagon
5077:regular
4926:2382768
4886:2974766
4702:2323939
4336:. Many
4319:Animals
4275:flower.
3902:⁄
3815:⁄
3393:regular
3331:, and Z
3279:hexagon
3243:in his
3205:circle.
3110:Draw a
2810:. Side
1722:apothem
1540:apothem
1534:is the
435:regular
417:regular
239:degrees
97:polygon
93:(gonia)
86:(pente)
6151:Simple
6096:Cyclic
6091:Convex
5815:Square
5755:Cyclic
5717:Obtuse
5712:Kepler
5499:9-cube
5449:8-cube
5399:7-cube
5356:6-cube
5326:5-cube
5213:Square
5090:Family
4924:
4884:
4804:
4762:
4732:
4700:
4656:
4629:
4417:pyrite
4291:of an
4222:Plants
3447:cyclic
3241:Euclid
3112:circle
2960:
2899:
1824:0.6882
1720:. The
1526:where
1341:
1258:
1169:
909:
826:
800:
797:0.8507
744:
687:
675:
624:
612:
453:convex
443:, and
290:Side (
257:cyclic
253:Convex
105:simple
46:cyclic
6126:Magic
5722:Right
5702:Ideal
5692:Acute
5218:p-gon
4967:arXiv
4882:JSTOR
4756:(PDF)
4709:(PDF)
4698:JSTOR
4678:(PDF)
4623:p. 63
4476:of a
4293:apple
3866:0.921
3435:up to
1338:1.176
1166:1.720
906:1.902
741:1.051
684:1.618
621:1.539
451:of a
168:Edges
103:in a
91:ÎłĎνίι
84:ĎÎνĎÎľ
81:
79:Greek
55:Edges
6156:Skew
5780:Kite
5675:List
5576:cube
5249:Cube
5079:and
4802:ISBN
4760:ISBN
4730:ISBN
4654:ISBN
4627:ISBN
4287:The
4237:okra
3792:The
3674:>
3396:star
3307:has
3303:The
3143:and
2768:and
1905:and
1546:and
421:has
280:Self
244:108°
170:and
157:Type
121:star
118:(or
73:, a
57:and
5125:(p)
4912:doi
4908:128
4874:doi
4870:102
4845:doi
4783:by
4690:doi
4044:15
3990:10
3344:r10
3309:Dih
2881:sin
2864:cos
2829:tan
2770:QCM
2766:DCM
2743:CMD
2539:If
1758:tan
1732:by
1653:tan
1595:tan
1319:sin
1294:sin
1037:tan
887:cos
862:cos
310:),
186:{5}
69:In
6188::
5630:â˘
5626:â˘
5606:21
5602:â˘
5599:k1
5595:â˘
5592:k2
5570:â˘
5527:â˘
5497:â˘
5475:21
5471:â˘
5468:41
5464:â˘
5461:42
5447:â˘
5425:21
5421:â˘
5418:31
5414:â˘
5411:32
5397:â˘
5375:21
5371:â˘
5368:22
5354:â˘
5324:â˘
5303:â˘
5284:â˘
5263:â˘
5247:â˘
5179:/
5168:/
5158:/
5149:/
5127:/
4995:.
4965:,
4945:,
4922:MR
4920:,
4906:,
4880:.
4868:.
4841:12
4839:.
4833:.
4696:.
4686:98
4684:.
4680:.
4625:.
4621:.
4602:26
4600:.
4596:.
4579:11
4577:.
4573:.
4561:^
4431:A
4411:A
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