22:
718:
The relation between perimeter and diameter for these polygons was proven by
Reinhardt, and rediscovered independently multiple times. The relation between diameter and width was proven by Bezdek and Fodor in 2000; their work also investigates the optimal polygons for this problem when the number of
149:
of the endpoints of these arcs is defined as a
Reinhardt polygon. Necessarily, the vertices of the underlying Reuleaux polygon are also endpoints of arcs and vertices of the Reinhardt polygon, but the Reinhardt polygon may also have additional vertices, interior to the sides of the Reuleaux polygon.
145:. Some Reuleaux polygons have side lengths that are irrational multiples of each other, but if a Reuleaux polygon has sides that can be partitioned into a system of arcs of equal length, then the polygon formed as the
1066:
582:
520:
25:
Four 15-sided
Reinhardt polygons (yellow), inscribed in Reuleaux polygons (curved black outer boundaries). The diameters are shown as blue line segments within each polygon.
796:
446:
1095:
338:
1559:
1167:
1142:
1119:
979:
959:
939:
919:
899:
876:
856:
824:
765:
745:
712:
692:
669:
649:
626:
606:
466:
406:
386:
362:
310:
290:
270:
243:
215:
195:
171:
111:
87:
67:
984:
448:, from which the dimensions of the polygon may be calculated. If the side length of a Reinhardt polygon is 1, then its perimeter is just
468:. The diameter of the polygon (the longest distance between any two of its points) equals the side length of these isosceles triangles,
1651:
137:
is a convex shape with circular-arc sides, each centered on a vertex of the shape and all having the same radius; an example is the
1352:
122:
878:
has two distinct odd prime factors and is not the product of these two factors, sporadic
Reinhardt polygons also exist.
1774:
1754:
2181:
1749:
1706:
1681:
1809:
1734:
523:
142:
118:
1759:
1644:
1318:
798:
to obtain the same polygon. The
Reinhardt polygons that have this sort of rotational symmetry are called
2160:
2100:
1739:
533:
471:
2044:
1814:
1744:
1686:
1101:. However, the number of sporadic Reinhardt polygons is less well-understood, and for most values of
2150:
2125:
2095:
2090:
2049:
1764:
1395:
34:
2155:
1696:
1616:
1475:
1430:
1404:
417:
1144:(counting two polygons as the same when they can be rotated or flipped to form each other) are:
1729:
1637:
138:
770:
121:
for their diameter, and the largest possible width for their perimeter. They are named after
1664:
1600:
1566:
1459:
1414:
1361:
423:
134:
38:
1612:
1578:
1528:
1471:
1426:
1375:
2130:
2110:
2105:
2075:
1794:
1769:
1701:
1608:
1574:
1524:
1467:
1422:
1371:
1071:
527:
222:
42:
315:
1393:
Hare, Kevin G.; Mossinghoff, Michael J. (2019), "Most
Reinhardt polygons are sporadic",
2140:
2120:
2085:
2080:
1711:
1691:
1565:, North-Holland Math. Stud., vol. 87, Amsterdam: North-Holland, pp. 209–214,
1544:
1350:
Mossinghoff, Michael J. (2011), "Enumerating isodiametric and isoperimetric polygons",
1152:
1127:
1104:
1098:
964:
944:
924:
904:
884:
861:
841:
809:
750:
730:
697:
677:
654:
634:
611:
591:
451:
391:
371:
347:
295:
275:
255:
246:
228:
200:
180:
156:
96:
72:
52:
45:, each vertex of a Reinhardt polygon participates in at least one defining pair of the
1570:
2175:
2115:
1966:
1859:
1779:
1721:
1620:
1479:
1434:
341:
340:
smaller arcs. Therefore, the possible numbers of sides of
Reinhardt polygons are the
90:
694:-sided polygons with their perimeter, and the smallest possible perimeter among all
2145:
2015:
1971:
1935:
1925:
1920:
831:
767:-sided regular Reuleaux polygons are symmetric: they can be rotated by an angle of
365:
174:
651:-sided polygons with their diameter, and the smallest possible diameter among all
608:-sided polygons with their diameter, and the smallest possible diameter among all
2054:
1961:
1940:
1930:
835:
146:
21:
1366:
2059:
1915:
1905:
1789:
1463:
1418:
218:
2034:
2024:
2001:
1991:
1981:
1910:
1819:
1784:
827:
114:
1121:
the total number of
Reinhardt polygons is dominated by the sporadic ones.
2039:
2029:
1986:
1945:
1874:
1864:
1854:
1673:
46:
1996:
1976:
1889:
1884:
1879:
1869:
1844:
1799:
1660:
1604:
250:
1804:
1591:
Bezdek, A.; Fodor, F. (2000), "On convex polygons of maximal width",
858:-sided Reinhardt polygons are periodic. In the remaining cases, when
719:
sides is a power of two (for which
Reinhardt polygons do not exist).
1849:
1629:
1409:
20:
1541:
Larman, D. G.; Tamvakis, N. K. (1984), "The decomposition of the
802:, and Reinhardt polygons without rotational symmetry are called
1633:
1061:{\displaystyle {\frac {p2^{n/p}}{4n}}{\bigl (}1+o(1){\bigr )},}
526:
of the polygon (the shortest distance between any two parallel
1493:
1515:
Vincze, Stephen (1950), "On a geometrical extremum problem",
1450:
Datta, Basudeb (1997), "A discrete isoperimetric problem",
177:, then it is not possible to form a Reinhardt polygon with
368:, or two times a prime number, there is only one shape of
292:
sides may be formed by subdividing each arc of a regular
113:
sides, the
Reinhardt polygons have the largest possible
1561:-sphere and the boundaries of plane convex domains",
1547:
1155:
1130:
1107:
1074:
987:
967:
947:
927:
907:
887:
864:
844:
812:
773:
753:
733:
700:
680:
657:
637:
614:
594:
536:
474:
454:
426:
394:
374:
350:
318:
298:
278:
258:
231:
203:
183:
159:
99:
75:
55:
1498:
Jahresbericht der Deutschen Mathematiker-Vereinigung
416:
The diameter pairs of a Reinhardt polygon form many
2068:
2014:
1954:
1898:
1837:
1828:
1720:
1672:
588:They have the largest possible perimeter among all
1553:
1161:
1136:
1124:The numbers of these polygons for small values of
1113:
1089:
1060:
973:
953:
933:
913:
893:
870:
850:
818:
790:
759:
739:
706:
686:
663:
643:
620:
600:
576:
514:
460:
440:
400:
388:-sided Reinhardt polygon, but all other values of
380:
356:
332:
304:
284:
264:
237:
209:
189:
165:
105:
81:
61:
1321:, the polygons maximizing area for their diameter
69:sides exist, often with multiple forms, whenever
420:with the sides of the triangle, with apex angle
674:They have the largest possible width among all
631:They have the largest possible width among all
408:have Reinhardt polygons with multiple shapes.
1645:
1050:
1025:
8:
1563:Convexity and graph theory (Jerusalem, 1981)
584:. These polygons are optimal in three ways:
1494:"Extremale Polygone gegebenen Durchmessers"
344:, numbers that are not powers of two. When
1834:
1652:
1638:
1630:
1546:
1408:
1365:
1154:
1129:
1106:
1073:
1049:
1048:
1024:
1023:
1002:
998:
988:
986:
966:
946:
926:
906:
886:
863:
843:
811:
780:
772:
752:
732:
699:
679:
656:
636:
613:
593:
560:
540:
535:
498:
478:
473:
453:
430:
425:
393:
373:
349:
322:
317:
297:
277:
257:
230:
202:
182:
158:
117:for their diameter, the largest possible
98:
74:
54:
1146:
901:, there are only finitely many distinct
245:sides is a Reinhardt polygon. Any other
49:of the polygon. Reinhardt polygons with
1331:
981:-sided periodic Reinhardt polygons is
747:-sided Reinhardt polygons formed from
530:) equals the height of this triangle,
628:-sided polygons with their perimeter.
7:
1388:
1386:
1384:
1445:
1443:
1345:
1343:
1341:
1339:
1337:
1335:
16:Polygon with many longest diagonals
14:
714:-sided polygons with their width.
671:-sided polygons with their width.
941:is the smallest prime factor of
577:{\displaystyle 1/2\tan(\pi /2n)}
515:{\displaystyle 1/2\sin(\pi /2n)}
1517:Acta Universitatis Szegediensis
1353:Journal of Combinatorial Theory
272:, and a Reinhardt polygon with
1084:
1078:
1045:
1039:
961:, then the number of distinct
921:-sided Reinhardt polygons. If
571:
554:
509:
492:
1:
1571:10.1016/S0304-0208(08)72828-7
312:-sided Reuleaux polygon into
125:, who studied them in 1922.
129:Definition and construction
2198:
1367:10.1016/j.jcta.2011.03.004
93:. Among all polygons with
1419:10.1007/s10711-018-0326-5
412:Dimensions and optimality
1492:Reinhardt, Karl (1922),
723:Symmetry and enumeration
524:curves of constant width
143:curves of constant width
1464:10.1023/A:1004997002327
791:{\displaystyle 2\pi /d}
1555:
1319:Biggest little polygon
1163:
1138:
1115:
1091:
1062:
975:
955:
935:
915:
895:
872:
852:
830:, or the product of a
820:
792:
761:
741:
708:
688:
665:
645:
622:
602:
578:
516:
462:
442:
441:{\displaystyle \pi /n}
402:
382:
358:
334:
306:
286:
266:
239:
211:
191:
167:
107:
83:
63:
26:
1593:Archiv der Mathematik
1556:
1164:
1139:
1116:
1092:
1063:
976:
956:
936:
916:
896:
873:
853:
821:
793:
762:
742:
709:
689:
666:
646:
623:
603:
579:
517:
463:
443:
403:
383:
359:
335:
307:
287:
267:
240:
212:
192:
168:
108:
84:
64:
24:
1885:Nonagon/Enneagon (9)
1815:Tangential trapezoid
1545:
1153:
1128:
1105:
1090:{\displaystyle o(1)}
1072:
985:
965:
945:
925:
905:
885:
862:
842:
810:
771:
751:
731:
698:
678:
655:
635:
612:
592:
534:
472:
452:
424:
392:
372:
348:
316:
296:
276:
256:
229:
201:
181:
157:
97:
73:
53:
1997:Megagon (1,000,000)
1765:Isosceles trapezoid
1452:Geometriae Dedicata
1396:Geometriae Dedicata
418:isosceles triangles
333:{\displaystyle n/d}
141:. These shapes are
35:equilateral polygon
1967:Icositetragon (24)
1605:10.1007/PL00000413
1551:
1159:
1134:
1111:
1087:
1058:
971:
951:
931:
911:
891:
868:
848:
816:
788:
757:
737:
704:
684:
661:
641:
618:
598:
574:
512:
458:
438:
398:
378:
354:
330:
302:
282:
262:
235:
207:
187:
163:
103:
91:not a power of two
79:
59:
27:
2182:Types of polygons
2169:
2168:
2010:
2009:
1987:Myriagon (10,000)
1972:Triacontagon (30)
1936:Heptadecagon (17)
1926:Pentadecagon (15)
1921:Tetradecagon (14)
1860:Quadrilateral (4)
1730:Antiparallelogram
1554:{\displaystyle n}
1310:
1309:
1162:{\displaystyle n}
1137:{\displaystyle n}
1114:{\displaystyle n}
1099:little O notation
1021:
974:{\displaystyle n}
954:{\displaystyle n}
934:{\displaystyle p}
914:{\displaystyle n}
894:{\displaystyle n}
871:{\displaystyle n}
851:{\displaystyle n}
819:{\displaystyle n}
760:{\displaystyle d}
740:{\displaystyle n}
707:{\displaystyle n}
687:{\displaystyle n}
664:{\displaystyle n}
644:{\displaystyle n}
621:{\displaystyle n}
601:{\displaystyle n}
461:{\displaystyle n}
401:{\displaystyle n}
381:{\displaystyle n}
357:{\displaystyle n}
305:{\displaystyle d}
285:{\displaystyle n}
265:{\displaystyle d}
249:must have an odd
238:{\displaystyle n}
210:{\displaystyle n}
190:{\displaystyle n}
166:{\displaystyle n}
139:Reuleaux triangle
106:{\displaystyle n}
82:{\displaystyle n}
62:{\displaystyle n}
31:Reinhardt polygon
2189:
1982:Chiliagon (1000)
1962:Icositrigon (23)
1941:Octadecagon (18)
1931:Hexadecagon (16)
1835:
1654:
1647:
1640:
1631:
1624:
1623:
1588:
1582:
1581:
1560:
1558:
1557:
1552:
1538:
1532:
1531:
1512:
1506:
1505:
1489:
1483:
1482:
1447:
1438:
1437:
1412:
1390:
1379:
1378:
1369:
1360:(6): 1801–1815,
1347:
1168:
1166:
1165:
1160:
1147:
1143:
1141:
1140:
1135:
1120:
1118:
1117:
1112:
1096:
1094:
1093:
1088:
1067:
1065:
1064:
1059:
1054:
1053:
1029:
1028:
1022:
1020:
1012:
1011:
1010:
1006:
989:
980:
978:
977:
972:
960:
958:
957:
952:
940:
938:
937:
932:
920:
918:
917:
912:
900:
898:
897:
892:
877:
875:
874:
869:
857:
855:
854:
849:
825:
823:
822:
817:
797:
795:
794:
789:
784:
766:
764:
763:
758:
746:
744:
743:
738:
713:
711:
710:
705:
693:
691:
690:
685:
670:
668:
667:
662:
650:
648:
647:
642:
627:
625:
624:
619:
607:
605:
604:
599:
583:
581:
580:
575:
564:
544:
528:supporting lines
521:
519:
518:
513:
502:
482:
467:
465:
464:
459:
447:
445:
444:
439:
434:
407:
405:
404:
399:
387:
385:
384:
379:
363:
361:
360:
355:
339:
337:
336:
331:
326:
311:
309:
308:
303:
291:
289:
288:
283:
271:
269:
268:
263:
244:
242:
241:
236:
216:
214:
213:
208:
196:
194:
193:
188:
172:
170:
169:
164:
135:Reuleaux polygon
112:
110:
109:
104:
88:
86:
85:
80:
68:
66:
65:
60:
43:regular polygons
39:Reuleaux polygon
2197:
2196:
2192:
2191:
2190:
2188:
2187:
2186:
2172:
2171:
2170:
2165:
2064:
2018:
2006:
1950:
1916:Tridecagon (13)
1906:Hendecagon (11)
1894:
1830:
1824:
1795:Right trapezoid
1716:
1668:
1658:
1628:
1627:
1590:
1589:
1585:
1543:
1542:
1540:
1539:
1535:
1514:
1513:
1509:
1491:
1490:
1486:
1449:
1448:
1441:
1392:
1391:
1382:
1349:
1348:
1333:
1328:
1315:
1151:
1150:
1126:
1125:
1103:
1102:
1070:
1069:
1013:
994:
990:
983:
982:
963:
962:
943:
942:
923:
922:
903:
902:
883:
882:
860:
859:
840:
839:
808:
807:
769:
768:
749:
748:
729:
728:
725:
696:
695:
676:
675:
653:
652:
633:
632:
610:
609:
590:
589:
532:
531:
470:
469:
450:
449:
422:
421:
414:
390:
389:
370:
369:
346:
345:
314:
313:
294:
293:
274:
273:
254:
253:
227:
226:
223:regular polygon
199:
198:
179:
178:
155:
154:
131:
95:
94:
71:
70:
51:
50:
37:inscribed in a
29:In geometry, a
17:
12:
11:
5:
2195:
2193:
2185:
2184:
2174:
2173:
2167:
2166:
2164:
2163:
2158:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2121:Pseudotriangle
2118:
2113:
2108:
2103:
2098:
2093:
2088:
2083:
2078:
2072:
2070:
2066:
2065:
2063:
2062:
2057:
2052:
2047:
2042:
2037:
2032:
2027:
2021:
2019:
2012:
2011:
2008:
2007:
2005:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1964:
1958:
1956:
1952:
1951:
1949:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1913:
1911:Dodecagon (12)
1908:
1902:
1900:
1896:
1895:
1893:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1841:
1839:
1832:
1826:
1825:
1823:
1822:
1817:
1812:
1807:
1802:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1726:
1724:
1722:Quadrilaterals
1718:
1717:
1715:
1714:
1709:
1704:
1699:
1694:
1689:
1684:
1678:
1676:
1670:
1669:
1659:
1657:
1656:
1649:
1642:
1634:
1626:
1625:
1583:
1550:
1533:
1507:
1484:
1439:
1380:
1330:
1329:
1327:
1324:
1323:
1322:
1314:
1311:
1308:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1237:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1158:
1133:
1110:
1086:
1083:
1080:
1077:
1057:
1052:
1047:
1044:
1041:
1038:
1035:
1032:
1027:
1019:
1016:
1009:
1005:
1001:
997:
993:
970:
950:
930:
910:
890:
867:
847:
815:
787:
783:
779:
776:
756:
736:
724:
721:
716:
715:
703:
683:
672:
660:
640:
629:
617:
597:
573:
570:
567:
563:
559:
556:
553:
550:
547:
543:
539:
511:
508:
505:
501:
497:
494:
491:
488:
485:
481:
477:
457:
437:
433:
429:
413:
410:
397:
377:
353:
342:polite numbers
329:
325:
321:
301:
281:
261:
247:natural number
234:
206:
186:
162:
130:
127:
123:Karl Reinhardt
102:
78:
58:
15:
13:
10:
9:
6:
4:
3:
2:
2194:
2183:
2180:
2179:
2177:
2162:
2161:Weakly simple
2159:
2157:
2154:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2101:Infinite skew
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2073:
2071:
2067:
2061:
2058:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2022:
2020:
2017:
2016:Star polygons
2013:
2003:
2002:Apeirogon (∞)
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1959:
1957:
1953:
1947:
1946:Icosagon (20)
1944:
1942:
1939:
1937:
1934:
1932:
1929:
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1924:
1922:
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1914:
1912:
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1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1780:Parallelogram
1778:
1776:
1775:Orthodiagonal
1773:
1771:
1768:
1766:
1763:
1761:
1758:
1756:
1755:Ex-tangential
1753:
1751:
1748:
1746:
1743:
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1736:
1733:
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1728:
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1234:
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1198:
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1189:
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1177:
1174:
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1081:
1075:
1055:
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1017:
1014:
1007:
1003:
999:
995:
991:
968:
948:
928:
908:
888:
879:
865:
845:
837:
833:
829:
813:
805:
801:
785:
781:
777:
774:
754:
734:
722:
720:
701:
681:
673:
658:
638:
630:
615:
595:
587:
586:
585:
568:
565:
561:
557:
551:
548:
545:
541:
537:
529:
525:
506:
503:
499:
495:
489:
486:
483:
479:
475:
455:
435:
431:
427:
419:
411:
409:
395:
375:
367:
351:
343:
327:
323:
319:
299:
279:
259:
252:
248:
232:
224:
220:
204:
184:
176:
160:
151:
148:
144:
140:
136:
128:
126:
124:
120:
116:
100:
92:
76:
56:
48:
44:
40:
36:
32:
23:
19:
2135:
1955:>20 sides
1890:Decagon (10)
1875:Heptagon (7)
1865:Pentagon (5)
1855:Triangle (3)
1750:Equidiagonal
1599:(1): 75–80,
1596:
1592:
1586:
1562:
1536:
1520:
1516:
1510:
1501:
1497:
1487:
1458:(1): 55–68,
1455:
1451:
1400:
1394:
1357:
1356:, Series A,
1351:
1123:
880:
834:with an odd
832:power of two
803:
799:
726:
717:
415:
366:prime number
175:power of two
152:
132:
41:. As in the
30:
28:
18:
2151:Star-shaped
2126:Rectilinear
2096:Equilateral
2091:Equiangular
2055:Hendecagram
1899:11–20 sides
1880:Octagon (8)
1870:Hexagon (6)
1845:Monogon (1)
1687:Equilateral
1523:: 136–142,
838:, then all
836:prime power
221:, then the
147:convex hull
2156:Tangential
2060:Dodecagram
1838:1–10 sides
1829:By number
1810:Tangential
1790:Right kite
1326:References
1097:term uses
1068:where the
364:is an odd
219:odd number
197:sides. If
2136:Reinhardt
2045:Enneagram
2035:Heptagram
2025:Pentagram
1992:65537-gon
1850:Digon (2)
1820:Trapezoid
1785:Rectangle
1735:Bicentric
1697:Isosceles
1674:Triangles
1621:123299791
1504:: 251–270
1480:118797507
1435:119629098
1410:1405.5233
881:For each
828:semiprime
778:π
558:π
552:
496:π
490:
428:π
115:perimeter
2176:Category
2111:Isotoxal
2106:Isogonal
2050:Decagram
2040:Octagram
2030:Hexagram
1831:of sides
1760:Harmonic
1661:Polygons
1403:: 1–18,
1313:See also
804:sporadic
800:periodic
47:diameter
2131:Regular
2076:Concave
2069:Classes
1977:257-gon
1800:Rhombus
1740:Crossed
1613:1728365
1579:0791034
1529:0038087
1472:1432534
1427:3933447
1376:2793611
251:divisor
2141:Simple
2086:Cyclic
2081:Convex
1805:Square
1745:Cyclic
1707:Obtuse
1702:Kepler
1619:
1611:
1577:
1527:
1478:
1470:
1433:
1425:
1374:
522:. The
217:is an
33:is an
2116:Magic
1712:Right
1692:Ideal
1682:Acute
1617:S2CID
1476:S2CID
1431:S2CID
1405:arXiv
826:is a
806:. If
225:with
173:is a
119:width
2146:Skew
1770:Kite
1665:List
727:The
1601:doi
1567:doi
1460:doi
1415:doi
1401:198
1362:doi
1358:118
1306:12
1235:24
549:tan
487:sin
153:If
89:is
2178::
1615:,
1609:MR
1607:,
1597:74
1595:,
1575:MR
1573:,
1525:MR
1521:12
1519:,
1502:31
1500:,
1496:,
1474:,
1468:MR
1466:,
1456:64
1454:,
1442:^
1429:,
1423:MR
1421:,
1413:,
1399:,
1383:^
1372:MR
1370:,
1334:^
1297:10
1240:#:
1232:23
1229:22
1226:21
1223:20
1220:19
1217:18
1214:17
1211:16
1208:15
1205:14
1202:13
1199:12
1196:11
1193:10
133:A
1667:)
1663:(
1653:e
1646:t
1639:v
1603::
1569::
1549:n
1462::
1417::
1407::
1364::
1303:1
1300:1
1294:2
1291:1
1288:5
1285:1
1282:0
1279:5
1276:1
1273:1
1270:2
1267:1
1264:1
1261:2
1258:0
1255:1
1252:1
1249:1
1246:0
1243:1
1190:9
1187:8
1184:7
1181:6
1178:5
1175:4
1172:3
1169::
1157:n
1132:n
1109:n
1085:)
1082:1
1079:(
1076:o
1056:,
1051:)
1046:)
1043:1
1040:(
1037:o
1034:+
1031:1
1026:(
1018:n
1015:4
1008:p
1004:/
1000:n
996:2
992:p
969:n
949:n
929:p
909:n
889:n
866:n
846:n
814:n
786:d
782:/
775:2
755:d
735:n
702:n
682:n
659:n
639:n
616:n
596:n
572:)
569:n
566:2
562:/
555:(
546:2
542:/
538:1
510:)
507:n
504:2
500:/
493:(
484:2
480:/
476:1
456:n
436:n
432:/
396:n
376:n
352:n
328:d
324:/
320:n
300:d
280:n
260:d
233:n
205:n
185:n
161:n
101:n
77:n
57:n
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