2384:
2368:
678:
42:
2338:
1080:
2314:
2302:
1061:
2284:
1140:
665:
860:
2435:
coins in the denominations of 2 Pula, 1 Pula, 50 Thebe and 5 Thebe are also shaped as equilateral-curve heptagons. Coins in the shape of
Reuleaux heptagons are also in circulation in Mauritius, U.A.E., Tanzania, Samoa, Papua New Guinea, São Tomé and Príncipe, Haiti, Jamaica, Liberia, Ghana, the
851:
2353:
packing of the
Euclidean plane of packing density approximately 0.89269. This has been conjectured to be the lowest density possible for the optimal double lattice packing density of any convex set, and more generally for the optimal packing density of any convex set.
889:
An approximation for practical use with an error of about 0.2% is to use half the side of an equilateral triangle inscribed in the same circle as the length of the side of a regular heptagon. It is unknown who first found this approximation, but it was mentioned by
2147:
367:
1869:
612:
996:
1040:
736:
494:
1395:
2330:. However, there is no tiling of the plane with only these polygons, because there is no way to fit one of them onto the third side of the triangle without leaving a gap or creating an overlap. In the
954:
1710:
1625:
1540:
1083:
Symmetries of a regular heptagon. Vertices are colored by their symmetry positions. Blue mirror lines are drawn through vertices and edges. Gyration orders are given in the center.
1801:
718:
2040:
2206:
1982:
1927:
1341:
1289:
1237:
94:
527:
104:
1450:
561:. It is also constructible with compass, straightedge and angle trisector. The impossibility of straightedge and compass construction follows from the observation that
2237:
413:
99:
2808:
2051:
2519:
294:
1820:
2803:
564:
2793:
962:
1001:
846:{\displaystyle 6\cos \left({\frac {2\pi }{7}}\right)=2{\sqrt {7}}\cos \left({\frac {1}{3}}\arctan \left(3{\sqrt {3}}\right)\right)-1.}
3368:
2798:
432:
1347:
2436:
Gambia, Jordan, Jersey, Guernsey, Isle of Man, Gibraltar, Guyana, Solomon
Islands, Falkland Islands and Saint Helena. The 1000
2638:
2345:
packing of the
Euclidean plane by regular heptagons, conjectured to have the lowest maximum packing density of any convex set
2160:
coinciding with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex) and angles
2835:
2783:
2452:
2334:, tilings by regular heptagons are possible. There are also concave heptagon tilings possible in the Euclidean plane.
2662:
2448:
913:
636:
1631:
1546:
1461:
1071:
There are other approximations of a heptagon using compass and straightedge, but they are time consuming to draw.
112:
2550:
898:
in the 1st century AD, was well known to medieval
Islamic mathematicians, and can be found in the work of
557:
and compass. It is the smallest regular polygon with this property. This type of construction is called a
2958:
2938:
2788:
2715:
Sycamore916, ed. "Heptagon." Polytope Wiki. Last modified
November 2023. Accessed January 20, 2024.
2408:
1745:
685:
3378:
2933:
2890:
2865:
2404:
2327:
86:
17:
1990:
2470:
2390:
2163:
1935:
1880:
2993:
1814:
The approximate lengths of the diagonals in terms of the side of the regular heptagon are given by
1295:
1243:
1191:
550:
232:-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle.
2427:; the sides are curved outwards to allow the coins to roll smoothly when they are inserted into a
2918:
2799:
Recently discovered and highly accurate approximation for the construction of a regular heptagon.
2424:
2420:
499:
372:
This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with
2943:
2828:
1411:
615:
2211:
3344:
3284:
2923:
2630:
2383:
1807:
with purely real terms exist for the solutions of this equation, because it is an example of
546:
387:
2259:, no convex polyhedron made entirely out of regular polygons contains a heptagon as a face.
3228:
2998:
2928:
2870:
2760:
2702:
2604:
2601:
2367:
2256:
1804:
1102:
654:
8:
3373:
3334:
3309:
3279:
3274:
3233:
2948:
2684:
2331:
2153:
1808:
1134:
891:
870:
729:
558:
155:
3339:
2880:
2738:
2689:
2587:
2542:
2142:{\displaystyle {\frac {b^{2}}{a^{2}}}+{\frac {c^{2}}{b^{2}}}+{\frac {a^{2}}{c^{2}}}=5.}
376:
at the center and at the heptagon's vertices, and then halving each triangle using the
2641:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
2272:
269:
76:
3319:
2913:
2821:
2658:
2634:
2459:
2157:
1092:
721:
373:
117:
66:
899:
2848:
2748:
2583:
2534:
2469:
In architecture, heptagonal floor plans are very rare. A remarkable example is the
2447:
25-cent coin has a heptagon inscribed in the coin's disk. Some old versions of the
2416:
2252:
998:
for the side of the heptagon inscribed in the unit circle while the exact value is
874:
725:
219:
677:
529:
thus the regular heptagon fills approximately 0.8710 of its circumscribed circle.
3314:
3294:
3289:
3259:
2978:
2953:
2885:
2756:
2652:
2428:
2412:
242:
163:
159:
62:
55:
3324:
3304:
3269:
3264:
2895:
2875:
2569:"Abu'l-Jūd's Answer to a Question of al-Bīrūnī Concerning the Regular Heptagon"
2463:
2437:
2350:
2342:
1740:
1398:
618:
538:
419:
265:
248:
229:
151:
147:
133:
129:
2568:
41:
3362:
3299:
3150:
3043:
2963:
2905:
2432:
415:
and the area of each of the 14 small triangles is one-fourth of the apothem.
362:{\displaystyle A={\frac {7}{4}}a^{2}\cot {\frac {\pi }{7}}\simeq 3.634a^{2}.}
2617:
2616:
raumannkidwai. "Heptagon." Chart. Geogebra. Accessed
January 20, 2024.
2520:"Angle trisection, the heptagon, and the triskaidecagon p. 186 (Fig.1) –187"
2240:
1163:
859:
3329:
3199:
3155:
3119:
3109:
3104:
2809:
A heptagon with a given side, an approximating construction as an animation
2777:
2752:
542:
170:
210:
3238:
3145:
3124:
3114:
2716:
1098:
1079:
224:
3243:
3099:
3089:
2973:
2682:
Abdilkadir
Altintas, "Some Collinearities in the Heptagonal Triangle",
2546:
2474:
2337:
1864:{\displaystyle b\approx 1.80193\cdot a,\qquad c\approx 2.24698\cdot a.}
1105:), order 28. The symmetry elements are: a 7-fold proper rotation axis C
3218:
3208:
3185:
3175:
3165:
3094:
3003:
2968:
2493:
2268:
607:{\displaystyle \scriptstyle {2\cos {\tfrac {2\pi }{7}}\approx 1.247}}
381:
2602:"The Polygons of Albrecht Dürer-1525, The Regular Heptagon", Fig. 11
2538:
2313:
682:
An animation from a neusis construction with radius of circumcircle
3223:
3213:
3170:
3129:
3048:
3038:
2857:
182:
2813:
2743:
2288:
Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
664:
3180:
3160:
3073:
3068:
3063:
3053:
3028:
2983:
2844:
2498:
2478:
2276:
991:{\displaystyle \scriptstyle {{\sqrt {3}} \over 2}\approx 0.86603}
377:
205:
194:
2301:
1060:
1035:{\displaystyle \scriptstyle 2\sin {\pi \over 7}\approx 0.86777}
2988:
2444:
1121:, in the plane of the heptagon and a horizontal mirror plane, σ
423:
252:
3033:
2484:
Many police badges in the US have a {7/2} heptagram outline.
2283:
1139:
554:
489:{\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},}
215:
2700:
Leon
Bankoff and Jack Garfunkel, "The heptagonal triangle",
2605:
the side of the
Heptagon (7) Fig. 15, image on the left side
1390:{\displaystyle {\frac {1}{a}}={\frac {1}{b}}+{\frac {1}{c}}}
2415:
are also heptagonal. Strictly, the shape of the coins is a
2400:
2374:
2239:
Thus its sides coincide with one side and two particular
2326:
A regular triangle, heptagon, and 42-gon can completely
2271:) can be constructed from regular heptagons, labeled by
2804:
Heptagon, an approximating construction as an animation
906:
lie on the circumference of the circumcircle. Draw arc
1005:
966:
917:
580:
568:
467:
437:
2214:
2166:
2054:
1993:
1938:
1883:
1823:
1748:
1634:
1549:
1464:
1414:
1350:
1298:
1246:
1194:
1004:
965:
956:
gives an approximation for the edge of the heptagon.
916:
873:
with marked ruler, according to David Johnson Leisk (
739:
688:
567:
502:
435:
390:
297:
2690:
http://forumgeom.fau.edu/FG2016volume16/FG201630.pdf
1128:
2399:The United Kingdom, since 1982, has two heptagonal
653:whereas the degree of the minimal polynomial for a
247:, in which all sides and all angles are equal, has
2373:Heptagon divided into triangles, clay tablet from
2231:
2200:
2141:
2034:
1976:
1921:
1863:
1795:
1704:
1619:
1534:
1444:
1389:
1335:
1283:
1231:
1034:
990:
948:
845:
712:
606:
521:
488:
407:
361:
2462:, have heptagonal symmetry in a shape called the
3360:
2729:Kallus, Yoav (2015). "Pessimal packing shapes".
949:{\displaystyle \scriptstyle {BD={1 \over 2}BC}}
1053:, the absolute error of the 1st side would be
2829:
2654:Point group character tables and related data
2650:
1705:{\displaystyle a^{3}-2a^{2}b-ab^{2}+b^{3}=0,}
1620:{\displaystyle c^{3}-2c^{2}a-ca^{2}+a^{3}=0,}
1535:{\displaystyle b^{3}+2b^{2}c-bc^{2}-c^{3}=0,}
673:of the interior angle in a regular heptagon.
200:The heptagon is sometimes referred to as the
732:. This construction relies on the fact that
380:as the common side. The apothem is half the
2836:
2822:
2678:
2676:
2674:
2576:Annals of the New York Academy of Sciences
2742:
2657:. Cambridge: Cambridge University Press.
2618:https://www.geogebra.org/classic/CvsudDWr
2566:
2455:, used a {7/2} heptagram as an element.
2336:
1138:
1078:
858:
2789:Another approximate construction method
2778:Definition and properties of a heptagon
2671:
2517:
1066:
635:. Consequently, this polynomial is the
496:while the area of the circle itself is
284:) of a regular heptagon of side length
14:
3361:
2728:
2717:https://polytope.miraheze.org/Heptagon
2511:
2817:
2518:Gleason, Andrew Mattei (March 1988).
2357:
2307:Triangle, heptagon, and 42-gon vertex
2291:
1109:, a 7-fold improper rotation axis, S
2843:
2651:Salthouse, J.A; Ware, M.J. (1972).
2633:, (2008) The Symmetries of Things,
2440:coin of Zambia is a true heptagon.
1796:{\displaystyle t^{3}-2t^{2}-t+1=0.}
553:but is constructible with a marked
235:
24:
2588:10.1111/j.1749-6632.1987.tb37202.x
2279:being the interval of connection.
713:{\displaystyle {\overline {OA}}=6}
25:
3390:
2771:
2527:The American Mathematical Monthly
2458:A number of coins, including the
2262:
1129:Diagonals and heptagonal triangle
1125:, also in the heptagon's plane.
1049:At a circumscribed circle radius
2382:
2366:
2312:
2300:
2282:
2035:{\displaystyle a^{2}-c^{2}=-bc,}
1059:
1046:Example to illustrate the error:
884:
676:
663:
102:
97:
92:
40:
3369:Polygons by the number of sides
2722:
2629:John H. Conway, Heidi Burgiel,
2246:
2201:{\displaystyle \pi /7,2\pi /7,}
1977:{\displaystyle c^{2}-b^{2}=ab,}
1922:{\displaystyle b^{2}-a^{2}=ac,}
1842:
532:
418:The area of a regular heptagon
2709:
2694:
2644:
2623:
2610:
2607:, retrieved on 4 December 2015
2594:
2560:
1327:
1315:
1275:
1263:
1223:
1211:
545:, the regular heptagon is not
13:
1:
2504:
2267:Two kinds of star heptagons (
1336:{\displaystyle c^{2}=b(a+b),}
1284:{\displaystyle b^{2}=a(c+a),}
1232:{\displaystyle a^{2}=c(c-b),}
1113:, 7 vertical mirror planes, σ
1158:The regular heptagon's side
699:
659:
7:
2706:46 (1), January 1973, 7–19.
2487:
2349:The regular heptagon has a
2275:{7/2}, and {7/3}, with the
1117:, 7 2-fold rotation axes, C
1074:
10:
3395:
2784:Heptagon according Johnson
2780:With interactive animation
2567:Hogendijk, Jan P. (1987).
2319:Hyperbolic heptagon tiling
1132:
1055:approximately -1.7 mm
522:{\displaystyle \pi R^{2};}
3252:
3198:
3138:
3082:
3021:
3012:
2904:
2856:
2471:Mausoleum of Prince Ernst
2391:Mausoleum of Prince Ernst
2243:of the regular heptagon.
1445:{\displaystyle ab+ac=bc,}
169:
143:
128:
111:
85:
75:
61:
51:
39:
34:
2425:curves of constant width
2232:{\displaystyle 4\pi /7.}
959:This approximation uses
551:compass and straightedge
2731:Geometry & Topology
2449:coat of arms of Georgia
2389:Heptagonal dome of the
408:{\displaystyle \pi /7,}
275:
87:Coxeter–Dynkin diagrams
2753:10.2140/gt.2015.19.343
2346:
2233:
2202:
2143:
2036:
1978:
1923:
1865:
1797:
1706:
1621:
1536:
1446:
1391:
1337:
1285:
1233:
1169:, and longer diagonal
1155:
1084:
1036:
992:
950:
878:
847:
714:
657:must be a power of 2.
608:
523:
490:
409:
363:
27:Shape with seven sides
2631:Chaim Goodman-Strauss
2340:
2234:
2203:
2144:
2037:
1979:
1924:
1866:
1805:algebraic expressions
1798:
1707:
1622:
1537:
1447:
1392:
1338:
1286:
1234:
1142:
1082:
1037:
993:
951:
862:
848:
715:
609:
524:
491:
410:
364:
3069:Nonagon/Enneagon (9)
2999:Tangential trapezoid
2794:Polygons – Heptagons
2703:Mathematics Magazine
2556:on 19 December 2015.
2377:, 2nd millennium BCE
2257:heptagonal antiprism
2212:
2164:
2052:
1991:
1936:
1881:
1821:
1746:
1632:
1547:
1462:
1412:
1348:
1296:
1244:
1192:
1103:Schoenflies notation
1067:Other approximations
1002:
963:
914:
869:An animation from a
737:
686:
655:constructible number
565:
500:
433:
388:
295:
204:, using "sept-" (an
3181:Megagon (1,000,000)
2949:Isosceles trapezoid
2685:Forum Geometricorum
2423:heptagon which has
2328:fill a plane vertex
2154:heptagonal triangle
1809:casus irreducibilis
1135:Heptagonal triangle
892:Heron of Alexandria
871:neusis construction
671:neusis construction
559:neusis construction
3151:Icositetragon (24)
2688:16, 2016, 249–256.
2358:Empirical examples
2347:
2292:Tiling and packing
2229:
2198:
2139:
2032:
1974:
1919:
1861:
1793:
1702:
1617:
1532:
1442:
1387:
1333:
1281:
1229:
1156:
1085:
1032:
1031:
988:
987:
946:
945:
879:
843:
710:
637:minimal polynomial
604:
603:
594:
519:
486:
481:
458:
405:
359:
46:A regular heptagon
3379:Elementary shapes
3353:
3352:
3194:
3193:
3171:Myriagon (10,000)
3156:Triacontagon (30)
3120:Heptadecagon (17)
3110:Pentadecagon (15)
3105:Tetradecagon (14)
3044:Quadrilateral (4)
2914:Antiparallelogram
2639:978-1-56881-220-5
2460:20 euro cent coin
2417:Reuleaux heptagon
2131:
2104:
2077:
1385:
1372:
1359:
1023:
979:
974:
936:
865:given side length
857:
856:
825:
804:
783:
766:
722:Andrew M. Gleason
702:
614:is a zero of the
593:
480:
457:
338:
312:
193:is a seven-sided
179:
178:
16:(Redirected from
3386:
3166:Chiliagon (1000)
3146:Icositrigon (23)
3125:Octadecagon (18)
3115:Hexadecagon (16)
3019:
3018:
2838:
2831:
2824:
2815:
2814:
2765:
2764:
2746:
2726:
2720:
2713:
2707:
2698:
2692:
2680:
2669:
2668:
2648:
2642:
2627:
2621:
2614:
2608:
2598:
2592:
2591:
2573:
2564:
2558:
2557:
2555:
2549:. Archived from
2524:
2515:
2386:
2370:
2332:hyperbolic plane
2316:
2304:
2286:
2273:Schläfli symbols
2253:heptagonal prism
2238:
2236:
2235:
2230:
2225:
2207:
2205:
2204:
2199:
2191:
2174:
2148:
2146:
2145:
2140:
2132:
2130:
2129:
2120:
2119:
2110:
2105:
2103:
2102:
2093:
2092:
2083:
2078:
2076:
2075:
2066:
2065:
2056:
2041:
2039:
2038:
2033:
2016:
2015:
2003:
2002:
1983:
1981:
1980:
1975:
1961:
1960:
1948:
1947:
1928:
1926:
1925:
1920:
1906:
1905:
1893:
1892:
1870:
1868:
1867:
1862:
1802:
1800:
1799:
1794:
1774:
1773:
1758:
1757:
1739:all satisfy the
1711:
1709:
1708:
1703:
1692:
1691:
1679:
1678:
1660:
1659:
1644:
1643:
1626:
1624:
1623:
1618:
1607:
1606:
1594:
1593:
1575:
1574:
1559:
1558:
1541:
1539:
1538:
1533:
1522:
1521:
1509:
1508:
1490:
1489:
1474:
1473:
1451:
1449:
1448:
1443:
1396:
1394:
1393:
1388:
1386:
1378:
1373:
1365:
1360:
1352:
1342:
1340:
1339:
1334:
1308:
1307:
1290:
1288:
1287:
1282:
1256:
1255:
1238:
1236:
1235:
1230:
1204:
1203:
1089:regular heptagon
1063:
1041:
1039:
1038:
1033:
1024:
1016:
997:
995:
994:
989:
980:
975:
970:
968:
955:
953:
952:
947:
944:
937:
929:
875:Crockett Johnson
852:
850:
849:
844:
836:
832:
831:
827:
826:
821:
805:
797:
784:
779:
771:
767:
762:
754:
728:by means of the
726:angle trisection
719:
717:
716:
711:
703:
698:
690:
680:
667:
660:
652:
650:
649:
645:
634:
613:
611:
610:
605:
602:
595:
589:
581:
528:
526:
525:
520:
515:
514:
495:
493:
492:
487:
482:
476:
468:
459:
453:
452:
451:
438:
414:
412:
411:
406:
398:
368:
366:
365:
360:
355:
354:
339:
331:
323:
322:
313:
305:
264:
263:
259:
236:Regular heptagon
220:numerical prefix
107:
106:
105:
101:
100:
96:
95:
44:
35:Regular heptagon
32:
31:
21:
3394:
3393:
3389:
3388:
3387:
3385:
3384:
3383:
3359:
3358:
3354:
3349:
3248:
3202:
3190:
3134:
3100:Tridecagon (13)
3090:Hendecagon (11)
3078:
3014:
3008:
2979:Right trapezoid
2900:
2852:
2842:
2774:
2769:
2768:
2727:
2723:
2714:
2710:
2699:
2695:
2681:
2672:
2665:
2649:
2645:
2628:
2624:
2615:
2611:
2599:
2595:
2571:
2565:
2561:
2553:
2539:10.2307/2323624
2522:
2516:
2512:
2507:
2490:
2451:, including in
2429:vending machine
2413:Barbados Dollar
2397:
2396:
2395:
2394:
2393:
2387:
2379:
2378:
2371:
2360:
2324:
2323:
2322:
2321:
2320:
2317:
2309:
2308:
2305:
2294:
2287:
2265:
2251:Apart from the
2249:
2221:
2213:
2210:
2209:
2187:
2170:
2165:
2162:
2161:
2125:
2121:
2115:
2111:
2109:
2098:
2094:
2088:
2084:
2082:
2071:
2067:
2061:
2057:
2055:
2053:
2050:
2049:
2011:
2007:
1998:
1994:
1992:
1989:
1988:
1956:
1952:
1943:
1939:
1937:
1934:
1933:
1901:
1897:
1888:
1884:
1882:
1879:
1878:
1822:
1819:
1818:
1769:
1765:
1753:
1749:
1747:
1744:
1743:
1687:
1683:
1674:
1670:
1655:
1651:
1639:
1635:
1633:
1630:
1629:
1602:
1598:
1589:
1585:
1570:
1566:
1554:
1550:
1548:
1545:
1544:
1517:
1513:
1504:
1500:
1485:
1481:
1469:
1465:
1463:
1460:
1459:
1413:
1410:
1409:
1377:
1364:
1351:
1349:
1346:
1345:
1303:
1299:
1297:
1294:
1293:
1251:
1247:
1245:
1242:
1241:
1199:
1195:
1193:
1190:
1189:
1137:
1131:
1124:
1120:
1116:
1112:
1108:
1096:
1091:belongs to the
1077:
1069:
1047:
1015:
1003:
1000:
999:
969:
967:
964:
961:
960:
928:
918:
915:
912:
911:
887:
881:
868:
820:
816:
812:
796:
795:
791:
778:
755:
753:
749:
738:
735:
734:
720:, according to
691:
689:
687:
684:
683:
681:
668:
647:
643:
642:
640:
621:
582:
579:
569:
566:
563:
562:
535:
510:
506:
501:
498:
497:
469:
466:
447:
443:
439:
436:
434:
431:
430:
422:in a circle of
394:
389:
386:
385:
350:
346:
330:
318:
314:
304:
296:
293:
292:
278:
270:Schläfli symbol
261:
257:
256:
249:internal angles
238:
123:
103:
98:
93:
91:
77:Schläfli symbol
56:Regular polygon
47:
28:
23:
22:
15:
12:
11:
5:
3392:
3382:
3381:
3376:
3371:
3351:
3350:
3348:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3305:Pseudotriangle
3302:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3256:
3254:
3250:
3249:
3247:
3246:
3241:
3236:
3231:
3226:
3221:
3216:
3211:
3205:
3203:
3196:
3195:
3192:
3191:
3189:
3188:
3183:
3178:
3173:
3168:
3163:
3158:
3153:
3148:
3142:
3140:
3136:
3135:
3133:
3132:
3127:
3122:
3117:
3112:
3107:
3102:
3097:
3095:Dodecagon (12)
3092:
3086:
3084:
3080:
3079:
3077:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3025:
3023:
3016:
3010:
3009:
3007:
3006:
3001:
2996:
2991:
2986:
2981:
2976:
2971:
2966:
2961:
2956:
2951:
2946:
2941:
2936:
2931:
2926:
2921:
2916:
2910:
2908:
2906:Quadrilaterals
2902:
2901:
2899:
2898:
2893:
2888:
2883:
2878:
2873:
2868:
2862:
2860:
2854:
2853:
2841:
2840:
2833:
2826:
2818:
2812:
2811:
2806:
2801:
2796:
2791:
2786:
2781:
2773:
2772:External links
2770:
2767:
2766:
2737:(1): 343–363.
2721:
2708:
2693:
2670:
2663:
2643:
2622:
2609:
2593:
2582:(1): 175–183.
2559:
2533:(3): 185–194.
2509:
2508:
2506:
2503:
2502:
2501:
2496:
2489:
2486:
2464:Spanish flower
2388:
2381:
2380:
2372:
2365:
2364:
2363:
2362:
2361:
2359:
2356:
2351:double lattice
2343:double lattice
2318:
2311:
2310:
2306:
2299:
2298:
2297:
2296:
2295:
2293:
2290:
2264:
2263:Star heptagons
2261:
2248:
2245:
2228:
2224:
2220:
2217:
2197:
2194:
2190:
2186:
2183:
2180:
2177:
2173:
2169:
2150:
2149:
2138:
2135:
2128:
2124:
2118:
2114:
2108:
2101:
2097:
2091:
2087:
2081:
2074:
2070:
2064:
2060:
2043:
2042:
2031:
2028:
2025:
2022:
2019:
2014:
2010:
2006:
2001:
1997:
1985:
1984:
1973:
1970:
1967:
1964:
1959:
1955:
1951:
1946:
1942:
1930:
1929:
1918:
1915:
1912:
1909:
1904:
1900:
1896:
1891:
1887:
1872:
1871:
1860:
1857:
1854:
1851:
1848:
1845:
1841:
1838:
1835:
1832:
1829:
1826:
1792:
1789:
1786:
1783:
1780:
1777:
1772:
1768:
1764:
1761:
1756:
1752:
1741:cubic equation
1713:
1712:
1701:
1698:
1695:
1690:
1686:
1682:
1677:
1673:
1669:
1666:
1663:
1658:
1654:
1650:
1647:
1642:
1638:
1627:
1616:
1613:
1610:
1605:
1601:
1597:
1592:
1588:
1584:
1581:
1578:
1573:
1569:
1565:
1562:
1557:
1553:
1542:
1531:
1528:
1525:
1520:
1516:
1512:
1507:
1503:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1472:
1468:
1453:
1452:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1403:
1402:
1399:optic equation
1384:
1381:
1376:
1371:
1368:
1363:
1358:
1355:
1343:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1306:
1302:
1291:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1254:
1250:
1239:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1202:
1198:
1133:Main article:
1130:
1127:
1122:
1118:
1114:
1110:
1106:
1094:
1076:
1073:
1068:
1065:
1030:
1027:
1022:
1019:
1014:
1011:
1008:
986:
983:
978:
973:
943:
940:
935:
932:
927:
924:
921:
900:Albrecht Dürer
886:
883:
863:Heptagon with
855:
854:
842:
839:
835:
830:
824:
819:
815:
811:
808:
803:
800:
794:
790:
787:
782:
777:
774:
770:
765:
761:
758:
752:
748:
745:
742:
709:
706:
701:
697:
694:
674:
601:
598:
592:
588:
585:
578:
575:
572:
539:Pierpont prime
534:
531:
518:
513:
509:
505:
485:
479:
475:
472:
465:
462:
456:
450:
446:
442:
404:
401:
397:
393:
370:
369:
358:
353:
349:
345:
342:
337:
334:
329:
326:
321:
317:
311:
308:
303:
300:
277:
274:
237:
234:
222:, rather than
177:
176:
173:
167:
166:
145:
141:
140:
137:
130:Internal angle
126:
125:
121:
115:
113:Symmetry group
109:
108:
89:
83:
82:
79:
73:
72:
69:
59:
58:
53:
49:
48:
45:
37:
36:
26:
9:
6:
4:
3:
2:
3391:
3380:
3377:
3375:
3372:
3370:
3367:
3366:
3364:
3357:
3346:
3345:Weakly simple
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3285:Infinite skew
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3257:
3255:
3251:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3206:
3204:
3201:
3200:Star polygons
3197:
3187:
3186:Apeirogon (∞)
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3143:
3141:
3137:
3131:
3130:Icosagon (20)
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3087:
3085:
3081:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3026:
3024:
3020:
3017:
3011:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2964:Parallelogram
2962:
2960:
2959:Orthodiagonal
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2939:Ex-tangential
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2909:
2907:
2903:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2863:
2861:
2859:
2855:
2850:
2846:
2839:
2834:
2832:
2827:
2825:
2820:
2819:
2816:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2779:
2776:
2775:
2762:
2758:
2754:
2750:
2745:
2740:
2736:
2732:
2725:
2718:
2712:
2705:
2704:
2697:
2691:
2687:
2686:
2679:
2677:
2675:
2666:
2664:0-521-08139-4
2660:
2656:
2655:
2647:
2640:
2636:
2632:
2626:
2619:
2613:
2606:
2603:
2600:G.H. Hughes,
2597:
2589:
2585:
2581:
2577:
2570:
2563:
2552:
2548:
2544:
2540:
2536:
2532:
2528:
2521:
2514:
2510:
2500:
2497:
2495:
2492:
2491:
2485:
2482:
2480:
2476:
2472:
2467:
2465:
2461:
2456:
2454:
2450:
2446:
2441:
2439:
2434:
2433:Botswana pula
2430:
2426:
2422:
2418:
2414:
2410:
2406:
2402:
2392:
2385:
2376:
2369:
2355:
2352:
2344:
2339:
2335:
2333:
2329:
2315:
2303:
2289:
2285:
2280:
2278:
2274:
2270:
2260:
2258:
2254:
2244:
2242:
2226:
2222:
2218:
2215:
2195:
2192:
2188:
2184:
2181:
2178:
2175:
2171:
2167:
2159:
2155:
2136:
2133:
2126:
2122:
2116:
2112:
2106:
2099:
2095:
2089:
2085:
2079:
2072:
2068:
2062:
2058:
2048:
2047:
2046:
2029:
2026:
2023:
2020:
2017:
2012:
2008:
2004:
1999:
1995:
1987:
1986:
1971:
1968:
1965:
1962:
1957:
1953:
1949:
1944:
1940:
1932:
1931:
1916:
1913:
1910:
1907:
1902:
1898:
1894:
1889:
1885:
1877:
1876:
1875:
1874:We also have
1858:
1855:
1852:
1849:
1846:
1843:
1839:
1836:
1833:
1830:
1827:
1824:
1817:
1816:
1815:
1812:
1810:
1806:
1790:
1787:
1784:
1781:
1778:
1775:
1770:
1766:
1762:
1759:
1754:
1750:
1742:
1738:
1734:
1730:
1726:
1722:
1718:
1699:
1696:
1693:
1688:
1684:
1680:
1675:
1671:
1667:
1664:
1661:
1656:
1652:
1648:
1645:
1640:
1636:
1628:
1614:
1611:
1608:
1603:
1599:
1595:
1590:
1586:
1582:
1579:
1576:
1571:
1567:
1563:
1560:
1555:
1551:
1543:
1529:
1526:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1494:
1491:
1486:
1482:
1478:
1475:
1470:
1466:
1458:
1457:
1456:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1408:
1407:
1406:
1400:
1382:
1379:
1374:
1369:
1366:
1361:
1356:
1353:
1344:
1330:
1324:
1321:
1318:
1312:
1309:
1304:
1300:
1292:
1278:
1272:
1269:
1266:
1260:
1257:
1252:
1248:
1240:
1226:
1220:
1217:
1214:
1208:
1205:
1200:
1196:
1188:
1187:
1186:
1184:
1180:
1176:
1172:
1168:
1165:
1161:
1153:
1149:
1145:
1141:
1136:
1126:
1104:
1100:
1097:
1090:
1081:
1072:
1064:
1062:
1057:
1056:
1052:
1048:
1043:
1028:
1025:
1020:
1017:
1012:
1009:
1006:
984:
981:
976:
971:
957:
941:
938:
933:
930:
925:
922:
919:
909:
905:
901:
897:
893:
885:Approximation
882:
876:
872:
866:
861:
853:
840:
837:
833:
828:
822:
817:
813:
809:
806:
801:
798:
792:
788:
785:
780:
775:
772:
768:
763:
759:
756:
750:
746:
743:
740:
731:
727:
724:based on the
723:
707:
704:
695:
692:
679:
675:
672:
666:
662:
661:
658:
656:
638:
632:
628:
624:
620:
617:
599:
596:
590:
586:
583:
576:
573:
570:
560:
556:
552:
548:
547:constructible
544:
540:
530:
516:
511:
507:
503:
483:
477:
473:
470:
463:
460:
454:
448:
444:
440:
428:
425:
421:
416:
402:
399:
395:
391:
383:
379:
375:
356:
351:
347:
343:
340:
335:
332:
327:
324:
319:
315:
309:
306:
301:
298:
291:
290:
289:
288:is given by:
287:
283:
273:
271:
267:
254:
250:
246:
244:
233:
231:
227:
226:
221:
217:
213:
212:
207:
203:
198:
196:
192:
188:
184:
174:
172:
168:
165:
161:
157:
153:
149:
146:
142:
138:
135:
131:
127:
119:
116:
114:
110:
90:
88:
84:
80:
78:
74:
70:
68:
64:
60:
57:
54:
50:
43:
38:
33:
30:
19:
3355:
3139:>20 sides
3074:Decagon (10)
3059:Heptagon (7)
3058:
3049:Pentagon (5)
3039:Triangle (3)
2934:Equidiagonal
2734:
2730:
2724:
2711:
2701:
2696:
2683:
2653:
2646:
2625:
2612:
2596:
2579:
2575:
2562:
2551:the original
2530:
2526:
2513:
2483:
2468:
2457:
2442:
2411:pieces. The
2398:
2348:
2341:The densest
2325:
2281:
2266:
2250:
2247:In polyhedra
2151:
2044:
1873:
1813:
1803:However, no
1736:
1732:
1728:
1724:
1720:
1716:
1714:
1454:
1404:
1182:
1178:
1174:
1170:
1166:
1159:
1157:
1154:=green lines
1151:
1147:
1143:
1088:
1086:
1070:
1058:
1054:
1051:r = 1 m
1050:
1045:
1044:
958:
907:
903:
895:
888:
880:
864:
733:
670:
630:
626:
622:
543:Fermat prime
536:
533:Construction
426:
417:
371:
285:
281:
279:
241:
239:
223:
209:
201:
199:
190:
186:
180:
171:Dual polygon
124:), order 2×7
29:
3335:Star-shaped
3310:Rectilinear
3280:Equilateral
3275:Equiangular
3239:Hendecagram
3083:11–20 sides
3064:Octagon (8)
3054:Hexagon (6)
3029:Monogon (1)
2871:Equilateral
2453:Soviet days
2421:curvilinear
1099:point group
616:irreducible
156:equilateral
3374:7 (number)
3363:Categories
3340:Tangential
3244:Dodecagram
3022:1–10 sides
3013:By number
2994:Tangential
2974:Right kite
2505:References
2475:Stadthagen
2269:heptagrams
1405:and hence
1185:, satisfy
1162:, shorter
541:but not a
537:As 7 is a
280:The area (
197:or 7-gon.
144:Properties
18:Heptagonal
3356:Heptagon
3320:Reinhardt
3229:Enneagram
3219:Heptagram
3209:Pentagram
3176:65537-gon
3034:Digon (2)
3004:Trapezoid
2969:Rectangle
2919:Bicentric
2881:Isosceles
2858:Triangles
2744:1305.0289
2494:Heptagram
2445:Brazilian
2241:diagonals
2219:π
2185:π
2168:π
2021:−
2005:−
1950:−
1895:−
1853:⋅
1847:≈
1834:⋅
1828:≈
1776:−
1760:−
1665:−
1646:−
1580:−
1561:−
1511:−
1495:−
1218:−
1026:≈
1018:π
1013:
982:≈
838:−
810:
789:
760:π
747:
700:¯
597:≈
587:π
577:
504:π
474:π
464:
420:inscribed
392:π
382:cotangent
341:≃
333:π
328:
218:-derived
139:≈128.571°
3295:Isotoxal
3290:Isogonal
3234:Decagram
3224:Octagram
3214:Hexagram
3015:of sides
2944:Harmonic
2845:Polygons
2488:See also
2158:vertices
1164:diagonal
1075:Symmetry
730:tomahawk
374:vertices
272:is {7}.
251:of 5π/7
245:heptagon
202:septagon
191:septagon
187:heptagon
183:geometry
164:isotoxal
160:isogonal
118:Dihedral
67:vertices
3315:Regular
3260:Concave
3253:Classes
3161:257-gon
2984:Rhombus
2924:Crossed
2761:3318753
2547:2323624
2499:Polygon
2479:Germany
2277:divisor
1850:2.24698
1831:1.80193
1173:, with
1150:=blue,
1029:0.86777
985:0.86603
910:. Then
896:Metrica
646:⁄
378:apothem
268:). Its
266:degrees
260:⁄
253:radians
243:regular
211:septua-
206:elision
195:polygon
134:degrees
3325:Simple
3270:Cyclic
3265:Convex
2989:Square
2929:Cyclic
2891:Obtuse
2886:Kepler
2759:
2661:
2637:
2545:
2438:Kwacha
2403:, the
1731:, and
1715:Thus –
1146:=red,
902:. Let
807:arctan
424:radius
225:hepta-
152:cyclic
148:Convex
3300:Magic
2896:Right
2876:Ideal
2866:Acute
2739:arXiv
2572:(PDF)
2554:(PDF)
2543:JSTOR
2523:(PDF)
2401:coins
1397:(the
641:2cos(
619:cubic
600:1.247
555:ruler
549:with
344:3.634
230:Greek
216:Latin
63:Edges
3330:Skew
2954:Kite
2849:List
2659:ISBN
2635:ISBN
2443:The
2419:, a
2407:and
2375:Susa
2255:and
2208:and
2156:has
2045:and
1455:and
1181:<
1177:<
1087:The
276:Area
255:(128
228:, a
214:, a
185:, a
175:Self
65:and
52:Type
2749:doi
2584:doi
2580:500
2535:doi
2473:in
2409:20p
2405:50p
1010:sin
908:BOC
894:'s
786:cos
744:cos
639:of
633:− 1
629:− 2
574:cos
461:sin
429:is
384:of
325:cot
208:of
189:or
181:In
81:{7}
3365::
2757:MR
2755:.
2747:.
2735:19
2733:.
2719:.
2673:^
2578:.
2574:.
2541:.
2531:95
2529:.
2525:.
2481:.
2477:,
2466:.
2431:.
2227:7.
2152:A
2137:5.
1811:.
1791:0.
1723:,
1095:7h
1042:.
877:).
841:1.
669:A
651:),
644:2π
625:+
240:A
162:,
158:,
154:,
150:,
120:(D
2851:)
2847:(
2837:e
2830:t
2823:v
2763:.
2751::
2741::
2667:.
2620:.
2590:.
2586::
2537::
2223:/
2216:4
2196:,
2193:7
2189:/
2182:2
2179:,
2176:7
2172:/
2134:=
2127:2
2123:c
2117:2
2113:a
2107:+
2100:2
2096:b
2090:2
2086:c
2080:+
2073:2
2069:a
2063:2
2059:b
2030:,
2027:c
2024:b
2018:=
2013:2
2009:c
2000:2
1996:a
1972:,
1969:b
1966:a
1963:=
1958:2
1954:b
1945:2
1941:c
1917:,
1914:c
1911:a
1908:=
1903:2
1899:a
1890:2
1886:b
1859:.
1856:a
1844:c
1840:,
1837:a
1825:b
1788:=
1785:1
1782:+
1779:t
1771:2
1767:t
1763:2
1755:3
1751:t
1737:b
1735:/
1733:a
1729:a
1727:/
1725:c
1721:c
1719:/
1717:b
1700:,
1697:0
1694:=
1689:3
1685:b
1681:+
1676:2
1672:b
1668:a
1662:b
1657:2
1653:a
1649:2
1641:3
1637:a
1615:,
1612:0
1609:=
1604:3
1600:a
1596:+
1591:2
1587:a
1583:c
1577:a
1572:2
1568:c
1564:2
1556:3
1552:c
1530:,
1527:0
1524:=
1519:3
1515:c
1506:2
1502:c
1498:b
1492:c
1487:2
1483:b
1479:2
1476:+
1471:3
1467:b
1440:,
1437:c
1434:b
1431:=
1428:c
1425:a
1422:+
1419:b
1416:a
1401:)
1383:c
1380:1
1375:+
1370:b
1367:1
1362:=
1357:a
1354:1
1331:,
1328:)
1325:b
1322:+
1319:a
1316:(
1313:b
1310:=
1305:2
1301:c
1279:,
1276:)
1273:a
1270:+
1267:c
1264:(
1261:a
1258:=
1253:2
1249:b
1227:,
1224:)
1221:b
1215:c
1212:(
1209:c
1206:=
1201:2
1197:a
1183:c
1179:b
1175:a
1171:c
1167:b
1160:a
1152:c
1148:b
1144:a
1123:h
1119:2
1115:v
1111:7
1107:7
1101:(
1093:D
1021:7
1007:2
977:2
972:3
942:C
939:B
934:2
931:1
926:=
923:D
920:B
904:A
867::
834:)
829:)
823:3
818:3
814:(
802:3
799:1
793:(
781:7
776:2
773:=
769:)
764:7
757:2
751:(
741:6
708:6
705:=
696:A
693:O
648:7
631:x
627:x
623:x
591:7
584:2
571:2
517:;
512:2
508:R
484:,
478:7
471:2
455:2
449:2
445:R
441:7
427:R
403:,
400:7
396:/
357:.
352:2
348:a
336:7
320:2
316:a
310:4
307:7
302:=
299:A
286:a
282:A
262:7
258:4
136:)
132:(
122:7
71:7
20:)
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