575:
874:
856:
737:
847:
838:
773:
755:
746:
983:
976:
969:
962:
955:
948:
939:
566:
1038:
1031:
1024:
1017:
1010:
1003:
994:
865:
654:
647:
829:
764:
640:
424:
661:
1072:
728:
29:
1186:
1175:
1164:
1079:
1086:
1241:
1227:
1213:
676:
631:
374:
427:
Symmetries of a regular icositetragon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the
241:
1122:
with equal edge lengths. In 3-dimensions it will be a zig-zag skew icositetragon and can be seen in the vertices and side edges of a dodecagonal antiprism with the same D
267:
1093:
A regular skew icositetragon is seen as zig-zagging edges of a dodecagonal antiprism, a dodecagrammic antiprism, and a dodecagrammic crossed-antiprism.
604:-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the
1926:
1126:, symmetry, order 48. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossed-antiprism, s{2,24/7} also have regular skew dodecagons.
1296:
1320:
The
Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994),
1391:
1107:
with 24 vertices and edges but not existing on the same plane. The interior of such an icositetragon is not generally defined. A
114:
104:
81:
1921:
96:
122:
109:
86:
91:
1514:
1494:
73:
1489:
1446:
1421:
1364:
535:
when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as
1549:
574:
915:{12} and dodecagram {12/5}. These also generate two quasitruncations: t{12/11}={24/11}, and t{12/7}={24/7}.
1474:
1499:
1384:
1900:
1840:
1479:
1288:
1139:
612:=12, and it can be divided into 66: 6 squares and 5 sets of 12 rhombs. This decomposition is based on a
400:
242:
Trigonometric constants expressed in real radicals § 7.5°: regular icositetragon (24-sided polygon)
227:
1784:
1554:
1484:
1426:
1890:
1865:
1835:
1830:
1789:
1504:
1284:
1190:
1179:
982:
975:
968:
961:
954:
947:
938:
565:
165:
1037:
1030:
1023:
1016:
1009:
1002:
993:
1895:
1436:
1168:
692:
512:
1325:
688:
369:{\displaystyle A=6t^{2}\cot {\frac {\pi }{24}}={6}t^{2}(2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}}).}
223:
63:
1875:
1469:
1377:
1337:
1292:
1119:
873:
855:
736:
437:
423:
127:
53:
1404:
1201:
1152:
864:
846:
837:
772:
754:
745:
653:
646:
828:
763:
639:
542:
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the
1870:
1850:
1845:
1815:
1534:
1509:
1441:
1354:
1071:
908:
660:
404:
247:
217:
173:
169:
49:
42:
1269:
1880:
1860:
1825:
1820:
1451:
1431:
1280:
1135:
782:
613:
523:. The dihedral symmetries are divided depending on whether they pass through vertices (
161:
157:
143:
139:
1299:(Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
1915:
1855:
1599:
1519:
1461:
1245:
1231:
1217:
1143:
547:
691:: {24/5}, {24/7}, and {24/11}. There are also 7 regular star figures using the same
1885:
1755:
1711:
1675:
1665:
1660:
1340:
1104:
684:
680:
A regular triangle, octagon, and icositetragon can completely fill a plane vertex.
515:
labels these by a letter and group order. The full symmetry of the regular form is
473:
180:
727:
596:-gon whose opposite sides are parallel and of equal length) can be dissected into
28:
1794:
1701:
1680:
1670:
511:
These 16 symmetries can be seen in 22 distinct symmetries on the icositetragon.
1185:
1799:
1655:
1645:
1529:
1774:
1764:
1741:
1731:
1721:
1650:
1559:
1524:
1345:
1174:
1163:
1085:
1078:
912:
412:
408:
388:
230:
675:
1359:
1779:
1769:
1726:
1685:
1614:
1604:
1594:
1413:
192:
1240:
1226:
1212:
1736:
1716:
1629:
1624:
1619:
1609:
1584:
1539:
1400:
1308:
617:
589:
585:
384:
234:
204:
391:(12-gon), tetracontaoctagon (48-gon), and enneacontahexagon (96-gon).
1544:
250:
icositetragon is 165°, meaning that one exterior angle would be 15°.
630:
379:
The icositetragon appeared in
Archimedes' polygon approximation of
1589:
1369:
1052:
207:. The sum of any icositetragon's interior angles is 3960 degrees.
911:
icositetragrams constructed as deeper truncations of the regular
1311:, Mathematical recreations and Essays, Thirteenth edition, p.141
254:
1373:
380:
444:, order 48. There are 7 subgroup dihedral symmetries: (Dih
921:
Isogonal truncations of regular dodecagon and dodecagram
1111:
has vertices alternating between two parallel planes.
695:: 2{12}, 3{8}, 4{6}, 6{4}, 8{3}, 3{8/3}, and 2{12/5}.
557:
546:
subgroup has no degrees of freedom but can be seen as
270:
1197:
1148:
1808:
1754:
1694:
1638:
1577:
1568:
1460:
1412:
179:
153:
138:
121:
72:
62:
48:
38:
21:
701:Icositetragrams as star polygons and star figures
368:
1138:for many higher-dimensional polytopes, seen as
237:, tt{6}, or thrice-truncated triangle, ttt{3}.
1385:
8:
1574:
1392:
1378:
1370:
917:
697:
399:As 24 = 2 Ă 3, a regular icositetragon is
1209:
1160:
990:
627:
353:
343:
333:
318:
309:
296:
284:
269:
622:
422:
1262:
1054:3 regular skew zig-zag icositetragons
226:{24} and can also be constructed as a
18:
687:. There are 3 regular forms given by
257:of a regular icositetragon is: (with
7:
1291:, (2008) The Symmetries of Things,
539:for their central gyration orders.
411:, it can be constructed by an edge-
203:) or 24-gon is a twenty-four-sided
14:
1134:The regular icositetragon is the
1239:
1225:
1211:
1184:
1173:
1162:
1084:
1077:
1070:
1036:
1029:
1022:
1015:
1008:
1001:
992:
981:
974:
967:
960:
953:
946:
937:
872:
863:
854:
845:
836:
827:
771:
762:
753:
744:
735:
726:
683:An icositetragram is a 24-sided
674:
659:
652:
645:
638:
629:
573:
564:
112:
107:
102:
94:
89:
84:
79:
27:
1927:Polygons by the number of sides
1035:
991:
980:
936:
862:
826:
761:
725:
360:
324:
233:, t{12}, or a twice-truncated
1:
1355:Naming Polygons and Polyhedra
935:
881:
822:
780:
721:
562:
519:and no symmetry is labeled
1943:
1116:regular skew icositetragon
1109:skew zig-zag icositetragon
920:
700:
239:
68:{24}, t{12}, tt{6}, ttt{3}
1322:Metamorphoses of polygons
1200:
1151:
1092:
928:
812:
711:
531:for perpendiculars), and
26:
527:for diagonal) or edges (
415:of a regular dodecagon.
246:One interior angle in a
559:24-gon with 264 rhombs
74:CoxeterâDynkin diagrams
33:A regular icositetragon
1922:Constructible polygons
1140:orthogonal projections
429:
370:
1289:Chaim Goodman-Strauss
1270:Constructible Polygon
1191:Omnitruncated 24-cell
606:regular icositetragon
434:regular icositetragon
426:
371:
240:Further information:
211:Regular icositetragon
22:Regular icositetragon
16:Polygon with 24 edges
1625:Nonagon/Enneagon (9)
1555:Tangential trapezoid
268:
1737:Megagon (1,000,000)
1505:Isosceles trapezoid
1169:Bitruncated 24-cell
1055:
625:
560:
1707:Icositetragon (24)
1338:Weisstein, Eric W.
1180:Runcinated 24-cell
1101:skew icositetragon
1053:
1049:Skew icositetragon
693:vertex arrangement
623:
588:states that every
558:
430:
366:
222:is represented by
1909:
1908:
1750:
1749:
1727:Myriagon (10,000)
1712:Triacontagon (30)
1676:Heptadecagon (17)
1666:Pentadecagon (15)
1661:Tetradecagon (14)
1600:Quadrilateral (4)
1470:Antiparallelogram
1297:978-1-56881-220-5
1254:
1253:
1196:
1195:
1120:vertex-transitive
1097:
1096:
1046:
1045:
987:t{12/11}={24/11}
905:
904:
667:
666:
583:
582:
407:. As a truncated
383:, along with the
358:
348:
338:
304:
189:
188:
1934:
1722:Chiliagon (1000)
1702:Icositrigon (23)
1681:Octadecagon (18)
1671:Hexadecagon (16)
1575:
1394:
1387:
1380:
1371:
1360:(simple) polygon
1351:
1350:
1328:
1318:
1312:
1306:
1300:
1278:
1272:
1267:
1243:
1229:
1215:
1198:
1188:
1177:
1166:
1149:
1088:
1081:
1074:
1056:
1040:
1033:
1026:
1019:
1012:
1005:
996:
985:
978:
971:
964:
957:
950:
941:
918:
876:
867:
860:{24/10}=2{12/5}
858:
849:
840:
831:
775:
766:
757:
748:
739:
730:
698:
689:Schläfli symbols
678:
670:Related polygons
663:
656:
649:
642:
633:
626:
616:projection of a
577:
568:
561:
375:
373:
372:
367:
359:
354:
349:
344:
339:
334:
323:
322:
313:
305:
297:
289:
288:
201:icosikaitetragon
117:
116:
115:
111:
110:
106:
105:
99:
98:
97:
93:
92:
88:
87:
83:
82:
31:
19:
1942:
1941:
1937:
1936:
1935:
1933:
1932:
1931:
1912:
1911:
1910:
1905:
1804:
1758:
1746:
1690:
1656:Tridecagon (13)
1646:Hendecagon (11)
1634:
1570:
1564:
1535:Right trapezoid
1456:
1408:
1398:
1341:"Icositetragon"
1336:
1335:
1332:
1331:
1326:Branko GrĂźnbaum
1319:
1315:
1307:
1303:
1279:
1275:
1268:
1264:
1259:
1249:
1244:
1235:
1230:
1221:
1216:
1205:
1189:
1178:
1167:
1156:
1132:
1130:Petrie polygons
1125:
1051:
1042:t{12/7}={24/7}
1041:
998:t{12/5}={24/5}
997:
986:
942:
907:There are also
883:Interior angle
877:
868:
859:
850:
841:
832:
776:
767:
758:
749:
740:
731:
709:Convex polygon
679:
672:
634:
578:
569:
556:
507:
503:
499:
495:
491:
487:
483:
479:
471:
467:
463:
459:
455:
451:
447:
441:
421:
405:angle trisector
397:
314:
280:
266:
265:
261:= edge length)
244:
224:Schläfli symbol
213:
133:
113:
108:
103:
101:
100:
95:
90:
85:
80:
78:
64:Schläfli symbol
43:Regular polygon
34:
17:
12:
11:
5:
1940:
1938:
1930:
1929:
1924:
1914:
1913:
1907:
1906:
1904:
1903:
1898:
1893:
1888:
1883:
1878:
1873:
1868:
1863:
1861:Pseudotriangle
1858:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1812:
1810:
1806:
1805:
1803:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1761:
1759:
1752:
1751:
1748:
1747:
1745:
1744:
1739:
1734:
1729:
1724:
1719:
1714:
1709:
1704:
1698:
1696:
1692:
1691:
1689:
1688:
1683:
1678:
1673:
1668:
1663:
1658:
1653:
1651:Dodecagon (12)
1648:
1642:
1640:
1636:
1635:
1633:
1632:
1627:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1587:
1581:
1579:
1572:
1566:
1565:
1563:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1466:
1464:
1462:Quadrilaterals
1458:
1457:
1455:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1418:
1416:
1410:
1409:
1399:
1397:
1396:
1389:
1382:
1374:
1368:
1367:
1362:
1357:
1352:
1330:
1329:
1313:
1301:
1281:John H. Conway
1273:
1261:
1260:
1258:
1255:
1252:
1251:
1247:
1237:
1233:
1223:
1219:
1208:
1207:
1203:
1194:
1193:
1182:
1171:
1159:
1158:
1154:
1144:Coxeter planes
1136:Petrie polygon
1131:
1128:
1123:
1095:
1094:
1090:
1089:
1082:
1075:
1067:
1066:
1063:
1060:
1050:
1047:
1044:
1043:
1034:
1027:
1020:
1013:
1006:
999:
989:
988:
979:
972:
965:
958:
951:
944:
934:
933:
930:
927:
923:
922:
903:
902:
899:
896:
893:
890:
887:
884:
880:
879:
878:{24/12}=12{2}
870:
861:
852:
851:{24/9}=3{8/3}
843:
834:
825:
821:
820:
817:
814:
811:
808:
804:
803:
800:
797:
794:
791:
788:
785:
783:Interior angle
779:
778:
769:
760:
751:
742:
733:
724:
720:
719:
716:
713:
710:
707:
703:
702:
671:
668:
665:
664:
657:
650:
643:
636:
614:Petrie polygon
581:
580:
571:
555:
552:
548:directed edges
505:
501:
497:
493:
489:
485:
481:
477:
476:symmetries: (Z
469:
465:
461:
457:
453:
449:
445:
439:
420:
417:
396:
393:
377:
376:
365:
362:
357:
352:
347:
342:
337:
332:
329:
326:
321:
317:
312:
308:
303:
300:
295:
292:
287:
283:
279:
276:
273:
212:
209:
187:
186:
183:
177:
176:
155:
151:
150:
147:
140:Internal angle
136:
135:
131:
125:
123:Symmetry group
119:
118:
76:
70:
69:
66:
60:
59:
56:
46:
45:
40:
36:
35:
32:
24:
23:
15:
13:
10:
9:
6:
4:
3:
2:
1939:
1928:
1925:
1923:
1920:
1919:
1917:
1902:
1901:Weakly simple
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1841:Infinite skew
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1813:
1811:
1807:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1762:
1760:
1757:
1756:Star polygons
1753:
1743:
1742:Apeirogon (â)
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1699:
1697:
1693:
1687:
1686:Icosagon (20)
1684:
1682:
1679:
1677:
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1643:
1641:
1637:
1631:
1628:
1626:
1623:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1582:
1580:
1576:
1573:
1567:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1520:Parallelogram
1518:
1516:
1515:Orthodiagonal
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1495:Ex-tangential
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1467:
1465:
1463:
1459:
1453:
1450:
1448:
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1419:
1417:
1415:
1411:
1406:
1402:
1395:
1390:
1388:
1383:
1381:
1376:
1375:
1372:
1366:
1365:icosatetragon
1363:
1361:
1358:
1356:
1353:
1348:
1347:
1342:
1339:
1334:
1333:
1327:
1323:
1317:
1314:
1310:
1305:
1302:
1298:
1294:
1290:
1286:
1285:Heidi Burgiel
1282:
1277:
1274:
1271:
1266:
1263:
1256:
1250:
1242:
1238:
1236:
1228:
1224:
1222:
1214:
1210:
1206:
1199:
1192:
1187:
1183:
1181:
1176:
1172:
1170:
1165:
1161:
1157:
1150:
1147:
1146:, including:
1145:
1141:
1137:
1129:
1127:
1121:
1117:
1112:
1110:
1106:
1102:
1091:
1087:
1083:
1080:
1076:
1073:
1069:
1068:
1064:
1061:
1058:
1057:
1048:
1039:
1032:
1028:
1025:
1021:
1018:
1014:
1011:
1007:
1004:
1000:
995:
984:
977:
973:
970:
966:
963:
959:
956:
952:
949:
945:
940:
932:Quasiregular
931:
926:Quasiregular
925:
924:
919:
916:
914:
910:
900:
897:
894:
891:
888:
885:
882:
875:
871:
866:
857:
853:
848:
844:
839:
835:
830:
823:
818:
816:Star polygon
815:
810:Star polygon
809:
806:
805:
801:
798:
795:
792:
789:
786:
784:
781:
774:
770:
765:
756:
752:
747:
743:
741:{24/2}=2{12}
738:
734:
729:
722:
717:
715:Star polygon
714:
708:
705:
704:
699:
696:
694:
690:
686:
681:
677:
669:
662:
658:
655:
651:
648:
644:
641:
637:
632:
628:
621:
619:
615:
611:
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603:
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595:
591:
587:
576:
572:
567:
563:
553:
551:
549:
545:
540:
538:
534:
530:
526:
522:
518:
514:
509:
475:
443:
435:
425:
418:
416:
414:
410:
406:
402:
401:constructible
394:
392:
390:
386:
382:
363:
355:
350:
345:
340:
335:
330:
327:
319:
315:
310:
306:
301:
298:
293:
290:
285:
281:
277:
274:
271:
264:
263:
262:
260:
256:
251:
249:
243:
238:
236:
232:
229:
225:
221:
220:icositetragon
219:
210:
208:
206:
202:
198:
197:icositetragon
194:
184:
182:
178:
175:
171:
167:
163:
159:
156:
152:
148:
145:
141:
137:
134:), order 2Ă24
129:
126:
124:
120:
77:
75:
71:
67:
65:
61:
57:
55:
51:
47:
44:
41:
37:
30:
25:
20:
1706:
1695:>20 sides
1630:Decagon (10)
1615:Heptagon (7)
1605:Pentagon (5)
1595:Triangle (3)
1490:Equidiagonal
1344:
1321:
1316:
1304:
1276:
1265:
1133:
1115:
1113:
1108:
1105:skew polygon
1100:
1098:
906:
842:{24/8}=8{3}
777:{24/6}=6{4}
759:{24/4}=4{6}
750:{24/3}=3{8}
732:{24/1}={24}
685:star polygon
682:
673:
609:
605:
601:
597:
593:
584:
543:
541:
536:
532:
528:
524:
520:
516:
510:
474:cyclic group
433:
431:
398:
395:Construction
378:
258:
252:
245:
216:
214:
200:
196:
190:
181:Dual polygon
1891:Star-shaped
1866:Rectilinear
1836:Equilateral
1831:Equiangular
1795:Hendecagram
1639:11â20 sides
1620:Octagon (8)
1610:Hexagon (6)
1585:Monogon (1)
1427:Equilateral
1065:{12/7}#{ }
1062:{12/5}#{ }
943:t{12}={24}
513:John Conway
456:), and (Dih
166:equilateral
1916:Categories
1896:Tangential
1800:Dodecagram
1578:1â10 sides
1569:By number
1550:Tangential
1530:Right kite
1257:References
813:Compounds
712:Compounds
554:Dissection
154:Properties
1876:Reinhardt
1785:Enneagram
1775:Heptagram
1765:Pentagram
1732:65537-gon
1590:Digon (2)
1560:Trapezoid
1525:Rectangle
1475:Bicentric
1437:Isosceles
1414:Triangles
1346:MathWorld
1059:{12}#{ }
929:Isogonal
913:dodecagon
819:Compound
718:Compound
624:Examples
579:Isotoxal
492:), and (Z
472:), and 8
413:bisection
409:dodecagon
403:using an
389:dodecagon
387:(6-gon),
299:π
294:
231:dodecagon
228:truncated
1851:Isotoxal
1846:Isogonal
1790:Decagram
1780:Octagram
1770:Hexagram
1571:of sides
1500:Harmonic
1401:Polygons
909:isogonal
869:{24/11}
635:12-cube
570:regular
442:symmetry
419:Symmetry
193:geometry
174:isotoxal
170:isogonal
128:Dihedral
54:vertices
1871:Regular
1816:Concave
1809:Classes
1717:257-gon
1540:Rhombus
1480:Crossed
1309:Coxeter
833:{24/7}
768:{24/5}
618:12-cube
590:zonogon
586:Coxeter
428:center.
385:hexagon
248:regular
235:hexagon
218:regular
205:polygon
144:degrees
1881:Simple
1826:Cyclic
1821:Convex
1545:Square
1485:Cyclic
1447:Obtuse
1442:Kepler
1295:
824:Image
723:Image
162:cyclic
158:Convex
1856:Magic
1452:Right
1432:Ideal
1422:Acute
1103:is a
807:Form
799:105°
796:120°
793:135°
790:150°
787:165°
706:Form
464:, Dih
460:, Dih
452:, Dih
448:, Dih
195:, an
50:Edges
1886:Skew
1510:Kite
1405:List
1293:ISBN
898:15°
895:30°
892:45°
889:60°
886:75°
802:90°
592:(a 2
436:has
432:The
255:area
253:The
215:The
199:(or
185:Self
149:165°
52:and
39:Type
1142:in
1124:12d
1118:is
901:0°
544:g24
517:r48
508:).
504:, Z
500:, Z
496:, Z
488:, Z
484:, Z
480:, Z
468:Dih
438:Dih
291:cot
191:In
1918::
1343:.
1324:,
1287:,
1283:,
1248:42
1234:41
1220:21
1153:2F
1114:A
1099:A
620:.
608:,
550:.
521:a1
482:12
478:24
446:12
440:24
381:pi
302:24
172:,
168:,
164:,
160:,
132:24
130:(D
58:24
1407:)
1403:(
1393:e
1386:t
1379:v
1349:.
1246:1
1232:2
1218:4
1204:8
1202:E
1155:4
610:m
602:m
600:(
598:m
594:m
537:g
533:i
529:p
525:d
506:1
502:2
498:4
494:8
490:3
486:6
470:1
466:2
462:4
458:8
454:3
450:6
364:.
361:)
356:6
351:+
346:3
341:+
336:2
331:+
328:2
325:(
320:2
316:t
311:6
307:=
286:2
282:t
278:6
275:=
272:A
259:t
146:)
142:(
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