3365:
986:
430:
3463:
20:
3352:
1303:
1462:
906:
1563:
684:
578:
970:
1333:
1068:
1621:
In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral. Only nodes
1689:
417:
In 1958, Kaleb
Williams proved that no two hypotenuses will ever coincide, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a
356:
2121:
811:
223:
193:
1825:
605:
499:
459:
407:
383:
247:
45:
1216:
735:
163:
1267:
1240:
1601:
806:
311:
1292:
1130:
1106:
3225:
1729:
1709:
1639:
1016:
775:
755:
706:
625:
519:
139:
1478:
630:
524:
1306:
Philip J. Davis' analytic continuation of the Spiral of
Theodorus, including extension in the opposite direction from the origin (negative nodes numbers).
278:
to
Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories.
911:
3303:
1869:
974:
2606:
2142:
2114:
3150:
2880:
2845:
1021:
3213:
2825:
2616:
267:, which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are
3280:
2107:
3401:
2050:
1833:
1618:
An analytic continuation of Davis' continuous form of the Spiral of
Theodorus extends in the opposite direction from the origin.
3249:
3182:
2815:
2695:
2078:
Heuvers, J.; Moak, D. S.; Boursaw, B (2000), "The functional equation of the square root spiral", in T. M. Rassias (ed.),
3700:
3318:
3075:
2963:
2138:
2958:
1608:
3487:
3027:
2654:
2470:
2445:
1457:{\displaystyle T(x)=\prod _{k=1}^{\infty }{\frac {1+i/{\sqrt {k}}}{1+i/{\sqrt {x+k}}}}\qquad (-1<x<\infty )}
3536:
3189:
3160:
2520:
2375:
3323:
2600:
3057:
1644:
3659:
3285:
3261:
3135:
3070:
3011:
2948:
2938:
2674:
2593:
2455:
2365:
2245:
461:. If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
1875:
3308:
3256:
3155:
2983:
2933:
2918:
2913:
2684:
2485:
2420:
2410:
2360:
117:) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the
3654:
3452:
3394:
3356:
3208:
2993:
2870:
2793:
2741:
2543:
2450:
2300:
1974:
1086:
263:
3364:
3052:
324:
3603:
3333:
3273:
3237:
3080:
2903:
2850:
2820:
2810:
2719:
2582:
2475:
2390:
2345:
2325:
2170:
2155:
1924:
The distribution of natural numbers divisible by 2, 3, 5, 7, 11, 13, and 17 on the square root spiral
198:
3313:
3232:
3220:
3201:
3165:
3003:
2988:
2978:
2928:
2923:
2865:
2734:
2626:
2555:
2490:
2480:
2380:
2350:
2290:
2265:
2190:
2180:
2165:
1898:
Mathematisch-Physikalische
Semesterberichte zur Pflege des Zusammenhangs von Schule und Universität
1473:
168:
106:
79:
1803:
583:
477:
440:
388:
364:
228:
26:
3635:
3583:
3369:
3328:
3268:
3196:
3062:
2855:
2830:
2798:
2395:
2340:
2305:
2200:
2067:
1927:
1779:
1604:
1198:
1082:
711:
144:
3620:
3037:
2636:
1740:
1246:
985:
429:
3598:
3588:
3546:
3442:
3042:
2953:
2803:
2729:
2702:
2465:
2285:
2275:
2210:
2130:
1958:
1829:
1465:
1314:
the discrete points of the spiral of
Theodorus by a smooth curve was proposed and answered by
1222:
268:
1568:
1472:. This function can be characterized axiomatically as the unique function that satisfies the
3695:
3497:
3387:
3244:
3115:
3032:
2788:
2776:
2724:
2460:
2059:
778:
275:
118:
114:
1970:
1909:
1274:
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to
784:
289:
3630:
3482:
3432:
2885:
2875:
2769:
2495:
2036:
1966:
1905:
1745:
1315:
1277:
1115:
1091:
418:
2088:
901:{\displaystyle \varphi \left(k\right)=\sum _{n=1}^{k}\varphi _{n}=2{\sqrt {k}}+c_{2}(k)}
3561:
3447:
3145:
3140:
2968:
2860:
2840:
2668:
2225:
2195:
1714:
1694:
1624:
1612:
1319:
1001:
760:
740:
691:
610:
504:
318:
124:
75:
3689:
3640:
3105:
2973:
2943:
2764:
2572:
2515:
1469:
1323:
1311:
3615:
3610:
3556:
3502:
3047:
2835:
2510:
2270:
2260:
19:
1849:
3705:
3674:
3526:
3477:
2661:
2549:
2255:
2240:
1978:
1078:
314:
3648:
3551:
3531:
3379:
2647:
2566:
2505:
2500:
2440:
2425:
2370:
2355:
2310:
2250:
2235:
2215:
2185:
2150:
421:, they will never pass through any of the other vertices of the total figure.
110:
1085:. Just as the distance between two windings of the Archimedean spiral equals
3669:
3593:
3518:
3492:
3437:
2400:
2385:
2335:
2230:
2220:
2205:
2175:
2099:
1327:
1302:
1135:
The following table shows successive windings of the spiral approaching pi:
91:
409:
hypotenuse belongs to the last triangle that does not overlap the figure.
3625:
3541:
2537:
2315:
2160:
1558:{\displaystyle f(x+1)=\left(1+{\frac {i}{\sqrt {x+1}}}\right)\cdot f(x),}
1109:
679:{\displaystyle \varphi _{n}=\arctan \left({\frac {1}{\sqrt {n}}}\right).}
95:
51:
3110:
2435:
2430:
2330:
2320:
2295:
2071:
1691:
are numbered in the figure. The dashed circle in the coordinate origin
573:{\displaystyle \tan \left(\varphi _{n}\right)={\frac {1}{\sqrt {n}}}.}
141:
th triangle in the sequence is a right triangle with the side lengths
3410:
2405:
2280:
1896:
Teuffel, Erich (1958), "Eine
Eigenschaft der Quadratwurzelschnecke",
1776:
The ordered distribution of natural numbers on the square root spiral
99:
71:
2063:
1112:, the distance between two consecutive windings quickly approaches
965:{\displaystyle \lim _{k\to \infty }c_{2}(k)=-2.157782996659\ldots }
3418:
3414:
2781:
2759:
2415:
1932:
1784:
1301:
984:
437:
Theodorus stopped his spiral at the triangle with a hypotenuse of
428:
258:
18:
361:
Plato, tutored by
Theodorus, questioned why Theodorus stopped at
23:
The spiral of
Theodorus up to the triangle with a hypotenuse of
3383:
2103:
1108:, as the number of spins of the spiral of Theodorus approaches
998:
The growth of the radius of the spiral at a certain triangle
978:
757:
th triangle. It grows proportionally to the square root of
2048:
Gronau, Detlef (March 2004), "The Spiral of
Theodorus",
1147:
Accuracy of average winding-distance in comparison to π
433:
Colored extended spiral of Theodorus with 110 triangles
1806:
1717:
1697:
1647:
1627:
1571:
1481:
1336:
1280:
1249:
1225:
1201:
1118:
1094:
1024:
1004:
914:
814:
787:
763:
743:
714:
694:
633:
613:
586:
527:
507:
480:
443:
391:
367:
327:
292:
231:
201:
195:. For example, the 16th triangle has sides measuring
171:
147:
127:
29:
3576:
3511:
3470:
3425:
3296:
3174:
3124:
3098:
3020:
3002:
2901:
2894:
2750:
2712:
2529:
2137:
1063:{\displaystyle \Delta r={\sqrt {n+1}}-{\sqrt {n}}.}
2007:
1819:
1723:
1703:
1683:
1633:
1595:
1557:
1456:
1286:
1261:
1234:
1210:
1124:
1100:
1062:
1010:
964:
900:
800:
769:
749:
729:
700:
678:
619:
599:
572:
513:
493:
453:
401:
377:
350:
305:
274:Plato does not attribute the irrationality of the
241:
217:
187:
157:
133:
39:
385:. The reason is commonly believed to be that the
1318:in 2001 by analogy with Euler's formula for the
916:
257:Although all of Theodorus' work has been lost,
3395:
2115:
2090:Analytic Continuation of the Theodorus Spiral
1965:(PhD thesis), Brown University, p. 173,
113:of the prior right triangle (with length the
8:
1641:with the integer value of the polar radius
3508:
3402:
3388:
3380:
3128:
2898:
2122:
2108:
2100:
1828:, Princeton University Press, p. 33,
78:, placed edge-to-edge. It was named after
2019:
1931:
1807:
1805:
1783:
1716:
1696:
1674:
1666:
1664:
1652:
1646:
1626:
1570:
1514:
1480:
1464:which was further studied by his student
1413:
1408:
1390:
1385:
1373:
1367:
1356:
1335:
1279:
1248:
1224:
1200:
1117:
1093:
1050:
1034:
1023:
1003:
935:
919:
913:
883:
869:
857:
847:
836:
813:
792:
786:
762:
742:
713:
693:
657:
638:
632:
612:
591:
585:
555:
542:
526:
506:
485:
479:
444:
442:
392:
390:
368:
366:
341:
332:
326:
297:
291:
232:
230:
208:
200:
172:
170:
148:
146:
126:
30:
28:
16:Polygonal curve made from right triangles
2006:. An alternative derivation is given in
1137:
1769:
1767:
1765:
1763:
1761:
1757:
1684:{\displaystyle r_{n}=\pm {\sqrt {|n|}}}
521:th triangle (or spiral segment), then:
102:. Another right triangle (which is the
3151:Latin translations of the 12th century
2003:
1863:
1861:
2881:Straightedge and compass construction
2080:Functional Equations and Inequalities
1991:
1945:
271:by means of the Spiral of Theodorus.
7:
2846:Incircle and excircles of a triangle
1848:Plato; Dyde, Samuel Walters (1899),
1144:Calculated average winding-distance
708:triangles is called the total angle
109:) is formed, with one leg being the
1963:The generalized Theodorus iteration
1330:function. Davis found the function
688:The sum of the angles of the first
580:Therefore, the growth of the angle
286:Each of the triangles' hypotenuses
2008:Heuvers, Moak & Boursaw (2000)
1448:
1368:
1256:
1205:
1025:
926:
14:
2051:The American Mathematical Monthly
351:{\displaystyle h_{1}={\sqrt {2}}}
3461:
3363:
3350:
1800:An Imaginary Tale: The Story of
261:put Theodorus into his dialogue
121:. The process then repeats; the
2041:Spirals from Theodorus to Chaos
1429:
989:A triangle or section of spiral
3183:A History of Greek Mathematics
2696:The Quadrature of the Parabola
1854:, J. Maclehose, pp. 86–87
1711:is the circle of curvature at
1675:
1667:
1581:
1575:
1549:
1543:
1497:
1485:
1451:
1430:
1346:
1340:
1250:
1226:
1202:
947:
941:
923:
895:
889:
724:
718:
218:{\displaystyle 4={\sqrt {16}}}
90:The spiral is started with an
1:
188:{\displaystyle {\sqrt {n+1}}}
2964:Intersecting secants theorem
1820:{\displaystyle {\sqrt {-1}}}
600:{\displaystyle \varphi _{n}}
494:{\displaystyle \varphi _{n}}
454:{\displaystyle {\sqrt {17}}}
402:{\displaystyle {\sqrt {17}}}
378:{\displaystyle {\sqrt {17}}}
242:{\displaystyle {\sqrt {17}}}
40:{\displaystyle {\sqrt {17}}}
2959:Intersecting chords theorem
2826:Doctrine of proportionality
1871:A Lesson on The Root Spiral
1211:{\displaystyle \to \infty }
1146:
1143:
730:{\displaystyle \varphi (k)}
165:and 1, and with hypotenuse
158:{\displaystyle {\sqrt {n}}}
3722:
2655:On the Sphere and Cylinder
2608:On the Sizes and Distances
94:right triangle, with each
3459:
3357:Ancient Greece portal
3346:
3161:Philosophy of mathematics
3131:
3076:Ptolemy's table of chords
2131:Ancient Greek mathematics
1262:{\displaystyle \to 100\%}
107:automedian right triangle
3028:Aristarchus's inequality
2601:On Conoids and Spheroids
2087:Waldvogel, Jörg (2009),
1235:{\displaystyle \to \pi }
1077:The Spiral of Theodorus
3136:Ancient Greek astronomy
2949:Inscribed angle theorem
2939:Greek geometric algebra
2594:Measurement of a Circle
1922:Hahn, Harry K. (2008),
1851:The Theaetetus of Plato
1798:Nahin, Paul J. (1998),
1774:Hahn, Harry K. (2007),
1596:{\displaystyle f(0)=1,}
1310:The question of how to
3370:Mathematics portal
3156:Non-Euclidean geometry
3111:Mouseion of Alexandria
2984:Tangent-secant theorem
2934:Geometric mean theorem
2919:Exterior angle theorem
2914:Angle bisector theorem
2618:On Sizes and Distances
2043:, A K Peters/CRC Press
1821:
1725:
1705:
1685:
1635:
1597:
1565:the initial condition
1559:
1458:
1372:
1307:
1288:
1263:
1236:
1212:
1126:
1102:
1064:
1012:
990:
966:
902:
852:
802:
771:
751:
731:
702:
680:
621:
601:
574:
515:
495:
455:
434:
403:
379:
352:
307:
243:
225:, 1 and hypotenuse of
219:
189:
159:
135:
47:
41:
3058:Pappus's area theorem
2994:Theorem of the gnomon
2871:Quadratrix of Hippias
2794:Circles of Apollonius
2742:Problem of Apollonius
2720:Constructible numbers
2544:Archimedes Palimpsest
1959:Leader, Jeffery James
1822:
1726:
1706:
1686:
1636:
1598:
1560:
1459:
1352:
1305:
1289:
1264:
1237:
1213:
1127:
1103:
1087:mathematical constant
1065:
1013:
988:
967:
903:
832:
803:
801:{\displaystyle c_{2}}
772:
752:
732:
703:
681:
622:
607:of the next triangle
602:
575:
516:
496:
456:
432:
404:
380:
353:
317:of the corresponding
308:
306:{\displaystyle h_{n}}
244:
220:
190:
160:
136:
42:
22:
3274:prehistoric counting
3071:Ptolemy's inequality
3012:Apollonius's theorem
2851:Method of exhaustion
2821:Diophantine equation
2811:Circumscribed circle
2628:On the Moving Sphere
1804:
1715:
1695:
1645:
1625:
1569:
1479:
1334:
1287:{\displaystyle \pi }
1278:
1247:
1223:
1199:
1125:{\displaystyle \pi }
1116:
1101:{\displaystyle \pi }
1092:
1022:
1002:
912:
812:
785:
761:
741:
712:
692:
631:
611:
584:
525:
505:
501:is the angle of the
478:
441:
389:
365:
325:
290:
229:
199:
169:
145:
125:
27:
3701:Pythagorean theorem
3360: •
3166:Neusis construction
3086:Spiral of Theodorus
2979:Pythagorean theorem
2924:Euclidean algorithm
2866:Lune of Hippocrates
2735:Squaring the circle
2491:Theon of Alexandria
2166:Aristaeus the Elder
1990:In an appendix to (
1474:functional equation
80:Theodorus of Cyrene
56:spiral of Theodorus
3053:Menelaus's theorem
3043:Irrational numbers
2856:Parallel postulate
2831:Euclidean geometry
2799:Apollonian circles
2341:Isidore of Miletus
2082:, pp. 111–117
1817:
1721:
1701:
1681:
1631:
1593:
1555:
1454:
1308:
1284:
1259:
1232:
1208:
1122:
1098:
1083:Archimedean spiral
1073:Archimedean spiral
1060:
1008:
991:
962:
930:
898:
798:
767:
747:
727:
698:
676:
617:
597:
570:
511:
491:
451:
435:
399:
375:
348:
303:
239:
215:
185:
155:
131:
68:Pythagoras's snail
64:Pythagorean spiral
60:square root spiral
48:
37:
3683:
3682:
3572:
3571:
3377:
3376:
3342:
3341:
3094:
3093:
3081:Ptolemy's theorem
2954:Intercept theorem
2804:Apollonian gasket
2730:Doubling the cube
2703:The Sand Reckoner
1948:, pp. 37–38.
1815:
1724:{\displaystyle O}
1704:{\displaystyle O}
1679:
1634:{\displaystyle n}
1530:
1529:
1427:
1424:
1395:
1272:
1271:
1055:
1045:
1011:{\displaystyle n}
915:
874:
770:{\displaystyle k}
750:{\displaystyle k}
701:{\displaystyle k}
667:
666:
620:{\displaystyle n}
565:
564:
514:{\displaystyle n}
449:
397:
373:
346:
237:
213:
183:
153:
134:{\displaystyle n}
58:(also called the
35:
3713:
3509:
3488:Boerdijk–Coxeter
3465:
3464:
3404:
3397:
3390:
3381:
3368:
3367:
3355:
3354:
3353:
3129:
3116:Platonic Academy
3063:Problem II.8 of
3033:Crossbar theorem
2989:Thales's theorem
2929:Euclid's theorem
2899:
2816:Commensurability
2777:Axiomatic system
2725:Angle trisection
2690:
2680:
2642:
2632:
2622:
2612:
2588:
2578:
2561:
2124:
2117:
2110:
2101:
2096:
2095:
2083:
2074:
2044:
2023:
2020:Waldvogel (2009)
2017:
2011:
2001:
1995:
1988:
1982:
1981:
1955:
1949:
1943:
1937:
1936:
1935:
1919:
1913:
1912:
1893:
1887:
1886:
1885:
1883:
1878:on 11 April 2013
1874:, archived from
1865:
1856:
1855:
1845:
1839:
1838:
1826:
1824:
1823:
1818:
1816:
1808:
1795:
1789:
1788:
1787:
1771:
1730:
1728:
1727:
1722:
1710:
1708:
1707:
1702:
1690:
1688:
1687:
1682:
1680:
1678:
1670:
1665:
1657:
1656:
1640:
1638:
1637:
1632:
1602:
1600:
1599:
1594:
1564:
1562:
1561:
1556:
1536:
1532:
1531:
1519:
1515:
1463:
1461:
1460:
1455:
1428:
1426:
1425:
1414:
1412:
1397:
1396:
1391:
1389:
1374:
1371:
1366:
1298:Continuous curve
1293:
1291:
1290:
1285:
1268:
1266:
1265:
1260:
1241:
1239:
1238:
1233:
1217:
1215:
1214:
1209:
1138:
1131:
1129:
1128:
1123:
1107:
1105:
1104:
1099:
1069:
1067:
1066:
1061:
1056:
1051:
1046:
1035:
1017:
1015:
1014:
1009:
981:
971:
969:
968:
963:
940:
939:
929:
907:
905:
904:
899:
888:
887:
875:
870:
862:
861:
851:
846:
828:
807:
805:
804:
799:
797:
796:
781:correction term
776:
774:
773:
768:
756:
754:
753:
748:
736:
734:
733:
728:
707:
705:
704:
699:
685:
683:
682:
677:
672:
668:
662:
658:
643:
642:
626:
624:
623:
618:
606:
604:
603:
598:
596:
595:
579:
577:
576:
571:
566:
560:
556:
551:
547:
546:
520:
518:
517:
512:
500:
498:
497:
492:
490:
489:
460:
458:
457:
452:
450:
445:
408:
406:
405:
400:
398:
393:
384:
382:
381:
376:
374:
369:
357:
355:
354:
349:
347:
342:
337:
336:
312:
310:
309:
304:
302:
301:
276:square root of 2
253:History and uses
248:
246:
245:
240:
238:
233:
224:
222:
221:
216:
214:
209:
194:
192:
191:
186:
184:
173:
164:
162:
161:
156:
154:
149:
140:
138:
137:
132:
119:square root of 3
115:square root of 2
46:
44:
43:
38:
36:
31:
3721:
3720:
3716:
3715:
3714:
3712:
3711:
3710:
3686:
3685:
3684:
3679:
3568:
3522:
3507:
3466:
3462:
3457:
3421:
3408:
3378:
3373:
3362:
3351:
3349:
3338:
3304:Arabian/Islamic
3292:
3281:numeral systems
3170:
3120:
3090:
3038:Heron's formula
3016:
2998:
2890:
2886:Triangle center
2876:Regular polygon
2753:and definitions
2752:
2746:
2708:
2688:
2678:
2640:
2630:
2620:
2610:
2586:
2576:
2559:
2525:
2496:Theon of Smyrna
2141:
2133:
2128:
2093:
2086:
2077:
2064:10.2307/4145130
2047:
2035:
2032:
2030:Further reading
2027:
2026:
2018:
2014:
2002:
1998:
1989:
1985:
1957:
1956:
1952:
1944:
1940:
1921:
1920:
1916:
1895:
1894:
1890:
1881:
1879:
1867:
1866:
1859:
1847:
1846:
1842:
1836:
1802:
1801:
1797:
1796:
1792:
1773:
1772:
1759:
1754:
1746:List of spirals
1741:Fermat's spiral
1737:
1713:
1712:
1693:
1692:
1648:
1643:
1642:
1623:
1622:
1567:
1566:
1507:
1503:
1477:
1476:
1398:
1375:
1332:
1331:
1316:Philip J. Davis
1300:
1276:
1275:
1245:
1244:
1221:
1220:
1197:
1196:
1114:
1113:
1090:
1089:
1075:
1020:
1019:
1000:
999:
996:
973:
931:
910:
909:
879:
853:
818:
810:
809:
788:
783:
782:
759:
758:
739:
738:
710:
709:
690:
689:
653:
634:
629:
628:
609:
608:
587:
582:
581:
538:
534:
523:
522:
503:
502:
481:
476:
475:
472:
467:
439:
438:
427:
415:
387:
386:
363:
362:
328:
323:
322:
293:
288:
287:
284:
255:
227:
226:
197:
196:
167:
166:
143:
142:
123:
122:
88:
76:right triangles
25:
24:
17:
12:
11:
5:
3719:
3717:
3709:
3708:
3703:
3698:
3688:
3687:
3681:
3680:
3678:
3677:
3672:
3667:
3662:
3657:
3652:
3645:
3644:
3643:
3633:
3628:
3623:
3618:
3613:
3608:
3607:
3606:
3601:
3596:
3586:
3580:
3578:
3574:
3573:
3570:
3569:
3567:
3566:
3565:
3564:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3520:
3515:
3513:
3506:
3505:
3500:
3495:
3490:
3485:
3480:
3474:
3472:
3468:
3467:
3460:
3458:
3456:
3455:
3450:
3445:
3440:
3435:
3429:
3427:
3423:
3422:
3409:
3407:
3406:
3399:
3392:
3384:
3375:
3374:
3347:
3344:
3343:
3340:
3339:
3337:
3336:
3331:
3326:
3321:
3316:
3311:
3306:
3300:
3298:
3297:Other cultures
3294:
3293:
3291:
3290:
3289:
3288:
3278:
3277:
3276:
3266:
3265:
3264:
3254:
3253:
3252:
3242:
3241:
3240:
3230:
3229:
3228:
3218:
3217:
3216:
3206:
3205:
3204:
3194:
3193:
3192:
3178:
3176:
3172:
3171:
3169:
3168:
3163:
3158:
3153:
3148:
3146:Greek numerals
3143:
3141:Attic numerals
3138:
3132:
3126:
3122:
3121:
3119:
3118:
3113:
3108:
3102:
3100:
3096:
3095:
3092:
3091:
3089:
3088:
3083:
3078:
3073:
3068:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3024:
3022:
3018:
3017:
3015:
3014:
3008:
3006:
3000:
2999:
2997:
2996:
2991:
2986:
2981:
2976:
2971:
2969:Law of cosines
2966:
2961:
2956:
2951:
2946:
2941:
2936:
2931:
2926:
2921:
2916:
2910:
2908:
2896:
2892:
2891:
2889:
2888:
2883:
2878:
2873:
2868:
2863:
2861:Platonic solid
2858:
2853:
2848:
2843:
2841:Greek numerals
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2807:
2806:
2801:
2791:
2786:
2785:
2784:
2774:
2773:
2772:
2767:
2756:
2754:
2748:
2747:
2745:
2744:
2739:
2738:
2737:
2732:
2727:
2716:
2714:
2710:
2709:
2707:
2706:
2699:
2692:
2682:
2672:
2669:Planisphaerium
2665:
2658:
2651:
2644:
2634:
2624:
2614:
2604:
2597:
2590:
2580:
2570:
2563:
2553:
2546:
2541:
2533:
2531:
2527:
2526:
2524:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2398:
2393:
2388:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2323:
2318:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2258:
2253:
2248:
2243:
2238:
2233:
2228:
2223:
2218:
2213:
2208:
2203:
2198:
2193:
2188:
2183:
2178:
2173:
2168:
2163:
2158:
2153:
2147:
2145:
2139:Mathematicians
2135:
2134:
2129:
2127:
2126:
2119:
2112:
2104:
2098:
2097:
2084:
2075:
2058:(3): 230–237,
2045:
2031:
2028:
2025:
2024:
2012:
1996:
1983:
1950:
1938:
1914:
1888:
1857:
1840:
1834:
1814:
1811:
1790:
1756:
1755:
1753:
1750:
1749:
1748:
1743:
1736:
1733:
1720:
1700:
1677:
1673:
1669:
1663:
1660:
1655:
1651:
1630:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1554:
1551:
1548:
1545:
1542:
1539:
1535:
1528:
1525:
1522:
1518:
1513:
1510:
1506:
1502:
1499:
1496:
1493:
1490:
1487:
1484:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1423:
1420:
1417:
1411:
1407:
1404:
1401:
1394:
1388:
1384:
1381:
1378:
1370:
1365:
1362:
1359:
1355:
1351:
1348:
1345:
1342:
1339:
1320:gamma function
1299:
1296:
1283:
1270:
1269:
1258:
1255:
1252:
1242:
1231:
1228:
1218:
1207:
1204:
1193:
1192:
1189:
1186:
1182:
1181:
1178:
1175:
1171:
1170:
1167:
1164:
1160:
1159:
1156:
1153:
1149:
1148:
1145:
1142:
1121:
1097:
1074:
1071:
1059:
1054:
1049:
1044:
1041:
1038:
1033:
1030:
1027:
1007:
995:
992:
961:
958:
957:2.157782996659
955:
952:
949:
946:
943:
938:
934:
928:
925:
922:
918:
897:
894:
891:
886:
882:
878:
873:
868:
865:
860:
856:
850:
845:
842:
839:
835:
831:
827:
824:
821:
817:
795:
791:
766:
746:
726:
723:
720:
717:
697:
675:
671:
665:
661:
656:
652:
649:
646:
641:
637:
616:
594:
590:
569:
563:
559:
554:
550:
545:
541:
537:
533:
530:
510:
488:
484:
471:
468:
466:
463:
448:
426:
423:
414:
411:
396:
372:
345:
340:
335:
331:
319:natural number
300:
296:
283:
280:
254:
251:
236:
212:
207:
204:
182:
179:
176:
152:
130:
87:
84:
34:
15:
13:
10:
9:
6:
4:
3:
2:
3718:
3707:
3704:
3702:
3699:
3697:
3694:
3693:
3691:
3676:
3673:
3671:
3668:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3650:
3646:
3642:
3639:
3638:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3605:
3602:
3600:
3597:
3595:
3592:
3591:
3590:
3587:
3585:
3582:
3581:
3579:
3575:
3563:
3560:
3559:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3517:
3516:
3514:
3510:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3475:
3473:
3469:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3430:
3428:
3424:
3420:
3416:
3412:
3405:
3400:
3398:
3393:
3391:
3386:
3385:
3382:
3372:
3371:
3366:
3359:
3358:
3345:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3301:
3299:
3295:
3287:
3284:
3283:
3282:
3279:
3275:
3272:
3271:
3270:
3267:
3263:
3260:
3259:
3258:
3255:
3251:
3248:
3247:
3246:
3243:
3239:
3236:
3235:
3234:
3231:
3227:
3224:
3223:
3222:
3219:
3215:
3212:
3211:
3210:
3207:
3203:
3200:
3199:
3198:
3195:
3191:
3187:
3186:
3185:
3184:
3180:
3179:
3177:
3173:
3167:
3164:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3133:
3130:
3127:
3123:
3117:
3114:
3112:
3109:
3107:
3104:
3103:
3101:
3097:
3087:
3084:
3082:
3079:
3077:
3074:
3072:
3069:
3067:
3066:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3025:
3023:
3019:
3013:
3010:
3009:
3007:
3005:
3001:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2975:
2974:Pons asinorum
2972:
2970:
2967:
2965:
2962:
2960:
2957:
2955:
2952:
2950:
2947:
2945:
2944:Hinge theorem
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2909:
2907:
2906:
2900:
2897:
2893:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2805:
2802:
2800:
2797:
2796:
2795:
2792:
2790:
2787:
2783:
2780:
2779:
2778:
2775:
2771:
2768:
2766:
2763:
2762:
2761:
2758:
2757:
2755:
2749:
2743:
2740:
2736:
2733:
2731:
2728:
2726:
2723:
2722:
2721:
2718:
2717:
2715:
2711:
2705:
2704:
2700:
2698:
2697:
2693:
2691:
2687:
2683:
2681:
2677:
2673:
2671:
2670:
2666:
2664:
2663:
2659:
2657:
2656:
2652:
2650:
2649:
2645:
2643:
2639:
2635:
2633:
2629:
2625:
2623:
2619:
2615:
2613:
2611:(Aristarchus)
2609:
2605:
2603:
2602:
2598:
2596:
2595:
2591:
2589:
2585:
2581:
2579:
2575:
2571:
2569:
2568:
2564:
2562:
2558:
2554:
2552:
2551:
2547:
2545:
2542:
2540:
2539:
2535:
2534:
2532:
2528:
2522:
2519:
2517:
2516:Zeno of Sidon
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2339:
2337:
2334:
2332:
2329:
2327:
2324:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2148:
2146:
2144:
2140:
2136:
2132:
2125:
2120:
2118:
2113:
2111:
2106:
2105:
2102:
2092:
2091:
2085:
2081:
2076:
2073:
2069:
2065:
2061:
2057:
2053:
2052:
2046:
2042:
2038:
2034:
2033:
2029:
2021:
2016:
2013:
2009:
2005:
2004:Gronau (2004)
2000:
1997:
1993:
1987:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1954:
1951:
1947:
1942:
1939:
1934:
1929:
1925:
1918:
1915:
1911:
1907:
1903:
1899:
1892:
1889:
1877:
1873:
1872:
1864:
1862:
1858:
1853:
1852:
1844:
1841:
1837:
1835:0-691-02795-1
1831:
1827:
1812:
1809:
1794:
1791:
1786:
1781:
1777:
1770:
1768:
1766:
1764:
1762:
1758:
1751:
1747:
1744:
1742:
1739:
1738:
1734:
1732:
1718:
1698:
1671:
1661:
1658:
1653:
1649:
1628:
1619:
1616:
1614:
1610:
1606:
1590:
1587:
1584:
1578:
1572:
1552:
1546:
1540:
1537:
1533:
1526:
1523:
1520:
1516:
1511:
1508:
1504:
1500:
1494:
1491:
1488:
1482:
1475:
1471:
1467:
1445:
1442:
1439:
1436:
1433:
1421:
1418:
1415:
1409:
1405:
1402:
1399:
1392:
1386:
1382:
1379:
1376:
1363:
1360:
1357:
1353:
1349:
1343:
1337:
1329:
1325:
1321:
1317:
1313:
1304:
1297:
1295:
1281:
1253:
1243:
1229:
1219:
1195:
1194:
1190:
1187:
1184:
1183:
1179:
1176:
1173:
1172:
1168:
1165:
1162:
1161:
1157:
1154:
1151:
1150:
1141:Winding No.:
1140:
1139:
1136:
1133:
1119:
1111:
1095:
1088:
1084:
1080:
1072:
1070:
1057:
1052:
1047:
1042:
1039:
1036:
1031:
1028:
1005:
993:
987:
983:
980:
976:
959:
956:
953:
950:
944:
936:
932:
920:
892:
884:
880:
876:
871:
866:
863:
858:
854:
848:
843:
840:
837:
833:
829:
825:
822:
819:
815:
793:
789:
780:
764:
744:
721:
715:
695:
686:
673:
669:
663:
659:
654:
650:
647:
644:
639:
635:
614:
592:
588:
567:
561:
557:
552:
548:
543:
539:
535:
531:
528:
508:
486:
482:
469:
464:
462:
446:
431:
424:
422:
420:
412:
410:
394:
370:
359:
343:
338:
333:
329:
320:
316:
298:
294:
281:
279:
277:
272:
270:
266:
265:
260:
252:
250:
234:
210:
205:
202:
180:
177:
174:
150:
128:
120:
116:
112:
108:
105:
101:
97:
93:
85:
83:
81:
77:
73:
69:
65:
61:
57:
53:
32:
21:
3664:
3647:
3512:Biochemistry
3361:
3348:
3190:Thomas Heath
3181:
3085:
3064:
3048:Law of sines
2904:
2836:Golden ratio
2701:
2694:
2685:
2679:(Theodosius)
2675:
2667:
2660:
2653:
2646:
2637:
2627:
2621:(Hipparchus)
2617:
2607:
2599:
2592:
2583:
2573:
2565:
2560:(Apollonius)
2556:
2548:
2536:
2511:Zeno of Elea
2271:Eratosthenes
2261:Dionysodorus
2089:
2079:
2055:
2049:
2040:
2037:Davis, P. J.
2015:
1999:
1986:
1962:
1953:
1946:Davis (2001)
1941:
1923:
1917:
1901:
1897:
1891:
1880:, retrieved
1876:the original
1870:
1868:Long, Kate,
1850:
1843:
1799:
1793:
1775:
1620:
1617:
1605:monotonicity
1309:
1273:
1134:
1079:approximates
1076:
997:
687:
473:
436:
416:
360:
285:
273:
262:
256:
103:
98:having unit
89:
86:Construction
74:composed of
67:
63:
59:
55:
49:
3660:Pitch angle
3636:Logarithmic
3584:Archimedean
3547:Polyproline
3257:mathematics
3065:Arithmetica
2662:Ostomachion
2631:(Autolycus)
2550:Arithmetica
2326:Hippocrates
2256:Dinostratus
2241:Dicaearchus
2171:Aristarchus
1904:: 148–152,
1324:interpolant
1312:interpolate
465:Growth rate
413:Overlapping
315:square root
3690:Categories
3649:On Spirals
3599:Hyperbolic
3309:Babylonian
3209:arithmetic
3175:History of
3004:Apollonius
2689:(Menelaus)
2648:On Spirals
2567:Catoptrics
2506:Xenocrates
2501:Thymaridas
2486:Theodosius
2471:Theaetetus
2451:Simplicius
2441:Pythagoras
2426:Posidonius
2411:Philonides
2371:Nicomachus
2366:Metrodorus
2356:Menaechmus
2311:Hipparchus
2301:Heliodorus
2251:Diophantus
2236:Democritus
2216:Chrysippus
2186:Archimedes
2181:Apollonius
2151:Anaxagoras
2143:(timeline)
1992:Davis 2001
1752:References
1191:99.97447%
1180:99.91453%
1169:99.91245%
1166:3.1443455
1158:99.44255%
1155:3.1592037
313:gives the
282:Hypotenuse
269:irrational
264:Theaetetus
111:hypotenuse
3670:Spirangle
3665:Theodorus
3604:Poinsot's
3594:Epispiral
3438:Curvature
3433:Algebraic
2770:Inscribed
2530:Treatises
2521:Zenodorus
2481:Theodorus
2456:Sosigenes
2401:Philolaus
2386:Oenopides
2381:Nicoteles
2376:Nicomedes
2336:Hypsicles
2231:Ctesibius
2221:Cleomedes
2206:Callippus
2191:Autolycus
2176:Aristotle
2156:Anthemius
1979:303808219
1933:0801.4422
1810:−
1785:0712.2184
1662:±
1538:⋅
1449:∞
1434:−
1369:∞
1354:∏
1328:factorial
1282:π
1257:%
1251:→
1230:π
1227:→
1206:∞
1203:→
1188:3.142395
1120:π
1096:π
1048:−
1026:Δ
960:…
954:−
927:∞
924:→
855:φ
834:∑
816:φ
777:, with a
716:φ
651:
636:φ
589:φ
540:φ
532:
483:φ
425:Extension
92:isosceles
3626:Involute
3621:Fermat's
3562:Collagen
3498:Symmetry
3334:Japanese
3319:Egyptian
3262:timeline
3250:timeline
3238:timeline
3233:geometry
3226:timeline
3221:calculus
3214:timeline
3202:timeline
2905:Elements
2751:Concepts
2713:Problems
2686:Spherics
2676:Spherics
2641:(Euclid)
2587:(Euclid)
2584:Elements
2577:(Euclid)
2538:Almagest
2446:Serenus
2421:Porphyry
2361:Menelaus
2316:Hippasus
2291:Eutocius
2266:Domninus
2161:Archytas
2039:(2001),
1975:ProQuest
1961:(1990),
1882:30 April
1735:See also
1609:argument
1607:in both
1326:for the
1177:3.14428
1110:infinity
737:for the
52:geometry
3696:Spirals
3655:Padovan
3589:Cotes's
3577:Spirals
3483:Antenna
3471:Helices
3443:Gallery
3419:helices
3411:Spirals
3314:Chinese
3269:numbers
3197:algebra
3125:Related
3099:Centers
2895:Results
2765:Central
2436:Ptolemy
2431:Proclus
2396:Perseus
2351:Marinus
2331:Hypatia
2321:Hippias
2296:Geminus
2286:Eudoxus
2276:Eudemus
2246:Diocles
2072:4145130
1971:2685516
1910:0096160
1613:modulus
1470:Iserles
1468:and by
979:A105459
977::
779:bounded
321:, with
70:) is a
3641:Golden
3557:Triple
3537:Double
3503:Triple
3453:Topics
3426:Curves
3415:curves
3329:Indian
3106:Cyrene
2638:Optics
2557:Conics
2476:Theano
2466:Thales
2461:Sporus
2406:Philon
2391:Pappus
2281:Euclid
2211:Carpus
2201:Bryson
2070:
1977:
1969:
1908:
1832:
1466:Leader
1322:as an
994:Radius
908:where
648:arctan
100:length
72:spiral
54:, the
3616:Euler
3611:Doyle
3552:Super
3527:Alpha
3478:Angle
3324:Incan
3245:logic
3021:Other
2789:Chord
2782:Axiom
2760:Angle
2416:Plato
2306:Heron
2226:Conon
2094:(PDF)
2068:JSTOR
1928:arXiv
1780:arXiv
470:Angle
259:Plato
66:, or
3675:Ulam
3631:List
3532:Beta
3493:Hemi
3448:List
3417:and
3286:list
2574:Data
2346:Leon
2196:Bion
1884:2008
1830:ISBN
1611:and
1603:and
1446:<
1440:<
1081:the
975:OEIS
627:is:
419:line
104:only
3188:by
2902:In
2060:doi
2056:111
1254:100
1018:is
982:).
917:lim
529:tan
474:If
96:leg
50:In
3706:Pi
3692::
3542:Pi
3521:10
3413:,
2066:,
2054:,
1973:,
1967:MR
1926:,
1906:MR
1900:,
1860:^
1778:,
1760:^
1731:.
1615:.
1294:.
1185:5
1174:4
1163:3
1152:2
1132:.
808::
447:17
395:17
371:17
358:.
249:.
235:17
211:16
82:.
62:,
33:17
3519:3
3403:e
3396:t
3389:v
2123:e
2116:t
2109:v
2062::
2022:.
2010:.
1994:)
1930::
1902:6
1813:1
1782::
1719:O
1699:O
1676:|
1672:n
1668:|
1659:=
1654:n
1650:r
1629:n
1591:,
1588:1
1585:=
1582:)
1579:0
1576:(
1573:f
1553:,
1550:)
1547:x
1544:(
1541:f
1534:)
1527:1
1524:+
1521:x
1517:i
1512:+
1509:1
1505:(
1501:=
1498:)
1495:1
1492:+
1489:x
1486:(
1483:f
1452:)
1443:x
1437:1
1431:(
1422:k
1419:+
1416:x
1410:/
1406:i
1403:+
1400:1
1393:k
1387:/
1383:i
1380:+
1377:1
1364:1
1361:=
1358:k
1350:=
1347:)
1344:x
1341:(
1338:T
1058:.
1053:n
1043:1
1040:+
1037:n
1032:=
1029:r
1006:n
972:(
951:=
948:)
945:k
942:(
937:2
933:c
921:k
896:)
893:k
890:(
885:2
881:c
877:+
872:k
867:2
864:=
859:n
849:k
844:1
841:=
838:n
830:=
826:)
823:k
820:(
794:2
790:c
765:k
745:k
725:)
722:k
719:(
696:k
674:.
670:)
664:n
660:1
655:(
645:=
640:n
615:n
593:n
568:.
562:n
558:1
553:=
549:)
544:n
536:(
509:n
487:n
344:2
339:=
334:1
330:h
299:n
295:h
206:=
203:4
181:1
178:+
175:n
151:n
129:n
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