1322:
1313:
1304:
1295:
1286:
1275:
1266:
1257:
1248:
1239:
29:
435:
Concerning the nonconstructability of the regular icositrigon, A. Baragar (2002) showed it is not possible to construct a regular 23-gon using only a compass and twice-notched straightedge by demonstrating that every point constructible with said method lies in a tower of
1183:
This result establishes, considering prime-power regular polygons less than the 100-gon, that it is impossible to construct the 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, and 89-gons with neusis. But it is not strong enough to decide the cases of the
355:
691:
533:
1412:
859:
249:
1188:, 25-, 31-, 41-, and 61-gons. Elliot Benjamin and Chip Snyder discovered in 2014 that the regular hendecagon (11-gon) is neusis constructible; the remaining cases are still open.
244:
1036:
978:
578:
460:
916:
598:
556:
1174:
1128:
1082:
889:
783:
1148:
1102:
1056:
998:
956:
936:
803:
618:
395:
375:
743:
1413:
https://www.researchgate.net/profile/Pietro_Milici2/publication/257393577_The_Equiangular_Compass/links/5d4c687da6fdcc370a8725e0/The-Equiangular-Compass.pdf
1984:
623:
465:
409:
1363:
Arthur
Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151-164,
1454:
81:
96:
104:
91:
86:
1379:
Benjamin, Elliot; Snyder, C. Mathematical
Proceedings of the Cambridge Philosophical Society 156.3 (May 2014): 409-424.;
1192:
1577:
1557:
811:
73:
1552:
1509:
1484:
1612:
350:{\textstyle A={\frac {23}{4}}a^{2}\cot {\frac {\pi }{23}}=23r^{2}\tan {\frac {\pi }{23}}\simeq 41.8344\,a^{2},}
1537:
1562:
1447:
862:
1963:
1903:
1542:
1204:
535:, being a sequence of nested fields in which the degree of the extension at each step is 2, 3, 5, or 6.
405:
1847:
1617:
1547:
1489:
222:
1953:
1928:
1898:
1893:
1852:
1567:
437:
429:
190:
147:
1003:
961:
561:
443:
1958:
1499:
1208:
1228:
894:
209:
63:
1427:
1938:
1532:
1440:
109:
53:
1396:
583:
541:
1467:
1364:
1321:
1312:
1212:
413:
1303:
1294:
1285:
1274:
1265:
1256:
1247:
1238:
1933:
1913:
1908:
1878:
1597:
1572:
1504:
1153:
1107:
203:
155:
151:
49:
42:
1061:
868:
748:
189:. The icositrigon has the distinction of being the smallest regular polygon that is not
1943:
1923:
1888:
1883:
1514:
1494:
1200:
1133:
1087:
1041:
983:
941:
921:
788:
603:
425:
380:
360:
216:
143:
139:
125:
121:
696:
1978:
1918:
1769:
1662:
1582:
1524:
1224:
1196:
1948:
1818:
1774:
1738:
1728:
1723:
1368:
421:
162:
28:
1857:
1743:
1733:
417:
1380:
1862:
1718:
1708:
1592:
1185:
1837:
1827:
1804:
1794:
1784:
1713:
1622:
1587:
686:{\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K}
528:{\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K}
1842:
1832:
1789:
1748:
1677:
1667:
1657:
1476:
174:
1799:
1779:
1692:
1687:
1682:
1672:
1647:
1602:
1463:
398:
186:
580:
is constructible using a compass and twice-notched straightedge. Then
1607:
1411:
December 1st, 2012, The
Mathematical Intelligencer, Vol. 34, Issue 4
430:
smallest regular polygon that is not constructible even with neusis
1652:
1432:
1428:
Automated
Detection of Interesting Properties in Regular Polygons
1348:
1436:
808:
If we can construct the regular p-gon, then we can construct
1352:
1397:
https://getd.libs.uga.edu/pdfs/lee_hwa-young_201712_ma.pdf
252:
225:
1156:
1136:
1110:
1090:
1064:
1044:
1006:
986:
964:
944:
924:
897:
871:
814:
791:
751:
699:
626:
606:
586:
564:
544:
468:
446:
383:
363:
1871:
1817:
1757:
1701:
1640:
1631:
1523:
1475:
161:
135:
120:
103:
72:
62:
48:
38:
21:
1223:Below is a table of ten regular icositrigrams, or
1195:either, because 23 is not a Pierpont prime, nor a
1168:
1142:
1122:
1096:
1076:
1050:
1030:
992:
972:
950:
930:
910:
883:
853:
797:
777:
737:
685:
612:
592:
572:
550:
527:
454:
389:
369:
349:
238:
745:at each step is 2, 3, 5, or 6. In particular, if
854:{\displaystyle \zeta _{p}=e^{\frac {2\pi i}{p}}}
428:. In addition, the regular icositrigon is the
1448:
1215:; yet this is true for all regular polygons.
8:
1381:https://dx.doi.org/10.1017/S0305004113000753
1637:
1455:
1441:
1433:
1155:
1135:
1109:
1089:
1063:
1043:
1019:
1008:
1007:
1005:
985:
966:
965:
963:
943:
923:
902:
896:
870:
832:
819:
813:
790:
768:
767:
750:
720:
707:
698:
671:
652:
639:
628:
627:
625:
605:
585:
566:
565:
563:
543:
513:
494:
481:
470:
469:
467:
448:
447:
445:
382:
362:
338:
333:
317:
305:
285:
273:
259:
251:
226:
224:
1233:
1338:
1227:23-gons, labeled with their respective
18:
7:
980:, where the only primes that divide
1203:. It can be constructed using the
14:
1150:must be divisible by 11, and for
1346:Tomahawk-nonconstructible n-gons
1320:
1311:
1302:
1293:
1284:
1273:
1264:
1255:
1246:
1237:
785:, then the only primes dividing
94:
89:
84:
79:
27:
1985:Polygons by the number of sides
805:are 2, 3, and 5. (Theorem 5.1)
620:that lies in a tower of fields
404:The regular icositrigon is not
239:{\textstyle {\frac {3780}{23}}}
1369:10.1080/00029890.2002.11919848
1025:
1012:
772:
758:
732:
700:
1:
1393:Origami-Constructible Numbers
1282:
1235:
1031:{\displaystyle \mathbb {Q} }
973:{\displaystyle \mathbb {Q} }
573:{\displaystyle \mathbb {C} }
455:{\displaystyle \mathbb {Q} }
2001:
911:{\displaystyle \zeta _{p}}
861:, which is the root of an
215:A regular icositrigon has
185:) or 23-gon is a 23-sided
246:degrees, with an area of
26:
1353:https://oeis.org/A048136
1180:must be divisible by 7.
410:compass and straightedge
1409:The equiangular compass
593:{\displaystyle \alpha }
551:{\displaystyle \alpha }
74:CoxeterâDynkin diagrams
1395:University of Georgia
1191:An icositrigon is not
1170:
1144:
1124:
1098:
1078:
1052:
1032:
994:
974:
952:
932:
912:
885:
863:irreducible polynomial
855:
799:
779:
739:
687:
614:
594:
574:
552:
529:
456:
391:
371:
351:
240:
1407:P. Milici, R. Dawson
1391:Young Lee, H. (2017)
1205:quadratrix of Hippias
1193:origami constructible
1171:
1145:
1125:
1104:. In particular, for
1099:
1079:
1053:
1033:
1000:are 2, 3, and 5. But
995:
975:
953:
933:
913:
886:
856:
800:
780:
740:
688:
615:
595:
575:
553:
530:
457:
392:
372:
352:
241:
33:A regular icositrigon
16:Polygon with 23 sides
1688:Nonagon/Enneagon (9)
1618:Tangential trapezoid
1231:{23/q}, 2 ⤠q ⤠11.
1169:{\displaystyle p=29}
1154:
1134:
1123:{\displaystyle p=23}
1108:
1088:
1062:
1042:
1004:
984:
962:
942:
922:
895:
869:
812:
789:
749:
697:
693:for which the index
624:
604:
584:
562:
542:
466:
444:
416:, on account of the
397:is the inradius, or
381:
361:
250:
223:
191:neusis constructible
1800:Megagon (1,000,000)
1568:Isosceles trapezoid
1077:{\displaystyle p-1}
884:{\displaystyle p-1}
600:belongs to a field
377:is side length and
197:Regular icositrigon
22:Regular icositrigon
1770:Icositetragon (24)
1209:Archimedean spiral
1166:
1140:
1120:
1094:
1074:
1048:
1028:
990:
970:
948:
928:
908:
891:. By Theorem 5.1,
881:
851:
795:
778:{\displaystyle N=}
775:
735:
683:
610:
590:
570:
548:
525:
452:
387:
367:
347:
236:
208:is represented by
1972:
1971:
1813:
1812:
1790:Myriagon (10,000)
1775:Triacontagon (30)
1739:Heptadecagon (17)
1729:Pentadecagon (15)
1724:Tetradecagon (14)
1663:Quadrilateral (4)
1533:Antiparallelogram
1330:
1329:
1143:{\displaystyle N}
1097:{\displaystyle N}
1051:{\displaystyle K}
1038:is a subfield of
993:{\displaystyle N}
951:{\displaystyle N}
931:{\displaystyle K}
848:
798:{\displaystyle N}
613:{\displaystyle K}
390:{\displaystyle r}
370:{\displaystyle a}
325:
293:
267:
234:
171:
170:
1992:
1785:Chiliagon (1000)
1765:Icositrigon (23)
1744:Octadecagon (18)
1734:Hexadecagon (16)
1638:
1457:
1450:
1443:
1434:
1415:
1405:
1399:
1389:
1383:
1377:
1371:
1361:
1355:
1343:
1324:
1315:
1306:
1297:
1288:
1277:
1268:
1259:
1250:
1241:
1234:
1213:auxiliary curves
1175:
1173:
1172:
1167:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1103:
1101:
1100:
1095:
1083:
1081:
1080:
1075:
1057:
1055:
1054:
1049:
1037:
1035:
1034:
1029:
1024:
1023:
1011:
999:
997:
996:
991:
979:
977:
976:
971:
969:
957:
955:
954:
949:
937:
935:
934:
929:
918:lies in a field
917:
915:
914:
909:
907:
906:
890:
888:
887:
882:
860:
858:
857:
852:
850:
849:
844:
833:
824:
823:
804:
802:
801:
796:
784:
782:
781:
776:
771:
744:
742:
741:
738:{\displaystyle }
736:
731:
730:
712:
711:
692:
690:
689:
684:
676:
675:
657:
656:
644:
643:
631:
619:
617:
616:
611:
599:
597:
596:
591:
579:
577:
576:
571:
569:
557:
555:
554:
549:
534:
532:
531:
526:
518:
517:
499:
498:
486:
485:
473:
461:
459:
458:
453:
451:
420:being neither a
414:angle trisection
396:
394:
393:
388:
376:
374:
373:
368:
356:
354:
353:
348:
343:
342:
326:
318:
310:
309:
294:
286:
278:
277:
268:
260:
245:
243:
242:
237:
235:
227:
99:
98:
97:
93:
92:
88:
87:
83:
82:
31:
19:
2000:
1999:
1995:
1994:
1993:
1991:
1990:
1989:
1975:
1974:
1973:
1968:
1867:
1821:
1809:
1753:
1719:Tridecagon (13)
1709:Hendecagon (11)
1697:
1633:
1627:
1598:Right trapezoid
1519:
1471:
1461:
1424:
1419:
1418:
1406:
1402:
1390:
1386:
1378:
1374:
1362:
1358:
1344:
1340:
1335:
1325:
1316:
1307:
1298:
1289:
1278:
1269:
1260:
1251:
1242:
1229:Schläfli symbol
1221:
1219:Related figures
1152:
1151:
1132:
1131:
1106:
1105:
1086:
1085:
1060:
1059:
1040:
1039:
1015:
1002:
1001:
982:
981:
960:
959:
940:
939:
920:
919:
898:
893:
892:
867:
866:
834:
828:
815:
810:
809:
787:
786:
747:
746:
716:
703:
695:
694:
667:
648:
635:
622:
621:
602:
601:
582:
581:
560:
559:
540:
539:
509:
490:
477:
464:
463:
442:
441:
379:
378:
359:
358:
334:
301:
269:
248:
247:
221:
220:
217:internal angles
210:Schläfli symbol
199:
115:
95:
90:
85:
80:
78:
64:Schläfli symbol
43:Regular polygon
34:
17:
12:
11:
5:
1998:
1996:
1988:
1987:
1977:
1976:
1970:
1969:
1967:
1966:
1961:
1956:
1951:
1946:
1941:
1936:
1931:
1926:
1924:Pseudotriangle
1921:
1916:
1911:
1906:
1901:
1896:
1891:
1886:
1881:
1875:
1873:
1869:
1868:
1866:
1865:
1860:
1855:
1850:
1845:
1840:
1835:
1830:
1824:
1822:
1815:
1814:
1811:
1810:
1808:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1761:
1759:
1755:
1754:
1752:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1714:Dodecagon (12)
1711:
1705:
1703:
1699:
1698:
1696:
1695:
1690:
1685:
1680:
1675:
1670:
1665:
1660:
1655:
1650:
1644:
1642:
1635:
1629:
1628:
1626:
1625:
1620:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1535:
1529:
1527:
1525:Quadrilaterals
1521:
1520:
1518:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1481:
1479:
1473:
1472:
1462:
1460:
1459:
1452:
1445:
1437:
1431:
1430:
1423:
1422:External links
1420:
1417:
1416:
1400:
1384:
1372:
1356:
1337:
1336:
1334:
1331:
1328:
1327:
1318:
1309:
1300:
1291:
1281:
1280:
1271:
1262:
1253:
1244:
1220:
1217:
1165:
1162:
1159:
1139:
1119:
1116:
1113:
1093:
1073:
1070:
1067:
1047:
1027:
1022:
1018:
1014:
1010:
989:
968:
947:
927:
905:
901:
880:
877:
874:
847:
843:
840:
837:
831:
827:
822:
818:
794:
774:
770:
766:
763:
760:
757:
754:
734:
729:
726:
723:
719:
715:
710:
706:
702:
682:
679:
674:
670:
666:
663:
660:
655:
651:
647:
642:
638:
634:
630:
609:
589:
568:
547:
524:
521:
516:
512:
508:
505:
502:
497:
493:
489:
484:
480:
476:
472:
450:
426:Pierpont prime
386:
366:
346:
341:
337:
332:
329:
324:
321:
316:
313:
308:
304:
300:
297:
292:
289:
284:
281:
276:
272:
266:
263:
258:
255:
233:
230:
198:
195:
183:icosikaitrigon
169:
168:
165:
159:
158:
137:
133:
132:
129:
122:Internal angle
118:
117:
113:
107:
105:Symmetry group
101:
100:
76:
70:
69:
66:
60:
59:
56:
46:
45:
40:
36:
35:
32:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
1997:
1986:
1983:
1982:
1980:
1965:
1964:Weakly simple
1962:
1960:
1957:
1955:
1952:
1950:
1947:
1945:
1942:
1940:
1937:
1935:
1932:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1904:Infinite skew
1902:
1900:
1897:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1876:
1874:
1870:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1825:
1823:
1820:
1819:Star polygons
1816:
1806:
1805:Apeirogon (â)
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1762:
1760:
1756:
1750:
1749:Icosagon (20)
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1706:
1704:
1700:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1643:
1639:
1636:
1630:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1583:Parallelogram
1581:
1579:
1578:Orthodiagonal
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1558:Ex-tangential
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1530:
1528:
1526:
1522:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1482:
1480:
1478:
1474:
1469:
1465:
1458:
1453:
1451:
1446:
1444:
1439:
1438:
1435:
1429:
1426:
1425:
1421:
1414:
1410:
1404:
1401:
1398:
1394:
1388:
1385:
1382:
1376:
1373:
1370:
1366:
1360:
1357:
1354:
1350:
1347:
1342:
1339:
1332:
1323:
1319:
1314:
1310:
1305:
1301:
1296:
1292:
1287:
1283:
1276:
1272:
1267:
1263:
1258:
1254:
1249:
1245:
1240:
1236:
1232:
1230:
1226:
1218:
1216:
1214:
1210:
1206:
1202:
1198:
1194:
1189:
1187:
1181:
1179:
1163:
1160:
1157:
1137:
1117:
1114:
1111:
1091:
1071:
1068:
1065:
1045:
1020:
1016:
987:
945:
925:
903:
899:
878:
875:
872:
864:
845:
841:
838:
835:
829:
825:
820:
816:
806:
792:
764:
761:
755:
752:
727:
724:
721:
717:
713:
708:
704:
680:
677:
672:
668:
664:
661:
658:
653:
649:
645:
640:
636:
632:
607:
587:
545:
536:
522:
519:
514:
510:
506:
503:
500:
495:
491:
487:
482:
478:
474:
439:
433:
431:
427:
423:
419:
415:
411:
407:
406:constructible
402:
400:
384:
364:
344:
339:
335:
330:
327:
322:
319:
314:
311:
306:
302:
298:
295:
290:
287:
282:
279:
274:
270:
264:
261:
256:
253:
231:
228:
218:
213:
211:
207:
205:
196:
194:
192:
188:
184:
180:
176:
166:
164:
160:
157:
153:
149:
145:
141:
138:
134:
130:
127:
123:
119:
116:), order 2Ă23
111:
108:
106:
102:
77:
75:
71:
67:
65:
61:
57:
55:
51:
47:
44:
41:
37:
30:
25:
20:
1764:
1758:>20 sides
1693:Decagon (10)
1678:Heptagon (7)
1668:Pentagon (5)
1658:Triangle (3)
1553:Equidiagonal
1408:
1403:
1392:
1387:
1375:
1359:
1345:
1341:
1222:
1211:, and other
1197:power of two
1190:
1182:
1177:
807:
537:
434:
403:
214:
202:
200:
182:
178:
172:
163:Dual polygon
1954:Star-shaped
1929:Rectilinear
1899:Equilateral
1894:Equiangular
1858:Hendecagram
1702:11â20 sides
1683:Octagon (8)
1673:Hexagon (6)
1648:Monogon (1)
1490:Equilateral
206:icositrigon
179:icositrigon
148:equilateral
1959:Tangential
1863:Dodecagram
1641:1â10 sides
1632:By number
1613:Tangential
1593:Right kite
1333:References
938:of degree
865:of degree
462:such that
136:Properties
1939:Reinhardt
1848:Enneagram
1838:Heptagram
1828:Pentagram
1795:65537-gon
1653:Digon (2)
1623:Trapezoid
1588:Rectangle
1538:Bicentric
1500:Isosceles
1477:Triangles
1069:−
1017:ζ
900:ζ
876:−
839:π
817:ζ
725:−
665:⊂
662:⋯
659:⊂
646:⊂
588:α
546:α
507:⊂
504:⋯
501:⊂
488:⊂
418:number 23
328:≃
320:π
315:
288:π
283:
131:â164.348°
1979:Category
1914:Isotoxal
1909:Isogonal
1853:Decagram
1843:Octagram
1833:Hexagram
1634:of sides
1563:Harmonic
1464:Polygons
1326:{23/11}
1317:{23/10}
1084:divides
538:Suppose
175:geometry
156:isotoxal
152:isogonal
110:Dihedral
54:vertices
1934:Regular
1879:Concave
1872:Classes
1780:257-gon
1603:Rhombus
1543:Crossed
1308:{23/9}
1299:{23/8}
1290:{23/7}
1279:{23/6}
1270:{23/5}
1261:{23/4}
1252:{23/3}
1243:{23/2}
408:with a
399:apothem
331:41.8344
204:regular
187:polygon
126:degrees
1944:Simple
1889:Cyclic
1884:Convex
1608:Square
1548:Cyclic
1510:Obtuse
1505:Kepler
438:fields
422:Fermat
357:where
212:{23}.
144:cyclic
140:Convex
1919:Magic
1515:Right
1495:Ideal
1485:Acute
1201:three
1058:, so
958:over
440:over
177:, an
50:Edges
1949:Skew
1573:Kite
1468:List
1349:OEIS
1225:star
424:nor
229:3780
181:(or
167:Self
68:{23}
52:and
39:Type
1365:doi
1199:or
1186:11-
558:in
412:or
312:tan
280:cot
219:of
173:In
1981::
1351:;
1207:,
1176:,
1164:29
1130:,
1118:23
432:.
401:.
323:23
299:23
291:23
262:23
232:23
201:A
193:.
154:,
150:,
146:,
142:,
114:23
112:(D
58:23
1470:)
1466:(
1456:e
1449:t
1442:v
1367::
1178:N
1161:=
1158:p
1138:N
1115:=
1112:p
1092:N
1072:1
1066:p
1046:K
1026:]
1021:p
1013:[
1009:Q
988:N
967:Q
946:N
926:K
904:p
879:1
873:p
846:p
842:i
836:2
830:e
826:=
821:p
793:N
773:]
769:Q
765::
762:K
759:[
756:=
753:N
733:]
728:1
722:j
718:K
714::
709:j
705:K
701:[
681:K
678:=
673:n
669:K
654:1
650:K
641:0
637:K
633:=
629:Q
608:K
567:C
523:K
520:=
515:n
511:K
496:1
492:K
483:0
479:K
475:=
471:Q
449:Q
385:r
365:a
345:,
340:2
336:a
307:2
303:r
296:=
275:2
271:a
265:4
257:=
254:A
128:)
124:(
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