2001:). Next, the radius of this circle is itself subdivided into 12 unit segments (radial units), and a series of concentric circles is constructed, each with radius incremented by one radial unit. Starting with the horizontal diameter and the innermost concentric circle, the point is marked where its radius intersects its circumference; one then moves to the next concentric circle and to the next diameter (moving up to construct a counterclockwise spiral, or down for clockwise) to mark the next point. After all points have been marked, successive points are connected by a line approximating the arithmetic spiral (or by a smooth curve of some sort; see
2032:) to which are attached two slotted arms: one horizontal arm is affixed to (travels up) the screw threads of the vertical shaft at one end, and holds a drawing tool at the other end; another sloped arm is affixed at one end to the top of the screw shaft, and is joined by a pin loosely fitted in its slot to the slot of the horizontal arm. The two arms rotate together and work in consort to produce the arithmetic spiral: as the horizontal arm gradually climbs the screw, that arm’s slotted attachment to the sloped arm gradually shortens the drawing radius. The angle of the sloped arm remains constant throughout (traces a
1065:
2052:
1951:
2608:
1672:
31:
240:
2024:
winds around the fixed pin which does not pivot). Such a method is a simple way to create an arithmetic spiral, arising naturally from use of a string compass with winding pin (not the loose pivot of a common string compass). The string compass drawing tool has various modifications and designs, and this construction method is reminiscent of string-based methods for creating ellipses (with two fixed pins).
541:
1893:
2036:), and setting a different angle varies the pitch of the spiral. This device provides a high degree of precision, depending on the precision with which the device is machined (machining a precise helical screw thread is a related challenge). And of course the use of a screw shaft in this mechanism is reminiscent of
2023:
A mechanical method for constructing the arithmetic spiral uses a modified string compass, where the string wraps and winds (or unwraps/unwinds) about a fixed central pin (that does not pivot), thereby incrementing (or decrementing) the length of the radius (string) as the angle changes (the string
1992:
The
Archimedean Spiral cannot be constructed precisely by traditional compass and straightedge methods, since the arithmetic spiral requires the radius of the curve to be incremented constantly as the angle at the origin is incremented. But an arithmetic spiral can be constructed approximately, to
1732:
a point moves with well-approximated uniform acceleration along the
Archimedean spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity (see contribution from Mikhail
298:
2027:
Yet another mechanical method is a variant of the previous string compass method, providing greater precision and more flexibility. Instead of the central pin and string of the string compass, this device uses a non-rotating shaft (column) with helical threads (screw; see
1947:", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly. Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter.
1184:
1005:
783:
2005:). Depending on the desired degree of precision, this method can be improved by increasing the size of the large outer circle, making more subdivisions of both its circumference and radius, increasing the number of concentric circles (see
1996:
The common traditional construction uses compass and straightedge to approximate the arithmetic spiral. First, a large circle is constructed and its circumference is subdivided by 12 diameters into 12 arcs (of 30 degrees each; see regular
1662:
666:
867:
2400:
Kim, Hyosun; Trejo, Alfonso; Liu, Sheng-Yuan; Sahai, Raghvendra; Taam, Ronald E.; Morris, Mark R.; Hirano, Naomi; Hsieh, I-Ta (March 2017). "The large-scale nebular pattern of a superwind binary in an eccentric orbit".
1983:
shows an approximate
Archimedean spiral in the dust clouds surrounding it, thought to be ejected matter from the star that has been shepherded into a spiral by another companion star as part of a double star system.
1079:
1071:
of the
Archimedean spiral, tangent to the spiral and having the same curvature at the tangent point. The spiral itself is not drawn, but can be seen as the points where the circles are especially close to each
1440:
1338:
536:{\displaystyle {\begin{aligned}|v_{0}|&={\sqrt {v^{2}+\omega ^{2}(vt+c)^{2}}}\\v_{x}&=v\cos \omega t-\omega (vt+c)\sin \omega t\\v_{y}&=v\sin \omega t+\omega (vt+c)\cos \omega t\end{aligned}}}
1055:
908:
684:
587:
303:
1829:
679:
1929:
form
Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record).
1572:
2009:). Approximating the Archimedean Spiral by this method is of course reminiscent of Archimedes’ famous method of approximating π by doubling the sides of successive polygons (see
1579:
582:
125:
790:
1679:
The
Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to
1993:
varying degrees of precision, by various manual drawing methods. One such method uses compass and straightedge; another method uses a modified string compass.
2330:
1343:
1721:. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the
1211:
76:
corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant
1012:
1781:
2162:
2224:
1964:
They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder.
1889:. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs.
1179:{\displaystyle f\colon \theta \mapsto (r\,\cos \theta ,r\,\sin \theta )=(b\,\theta \,\cos \theta ,b\,\theta \,\sin \theta )}
17:
1468:
2137:
141:
From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle
98:
2546:
2378:
2265:
2186:
1954:
1885:, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to
1000:{\displaystyle \tan \left(\left({\sqrt {x^{2}+y^{2}}}-c\right)\cdot {\frac {\omega }{v}}\right)={\frac {y}{x}}}
2017:
2840:
2468:
Jonathan Matt making the
Archimedean spiral interesting - Video : The surprising beauty of Mathematics
2337:
787:
Squaring the two equations and then adding (and some small alterations) results in the
Cartesian equation
68:
is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to
2632:
1976:
1700:
these distances, as well as the distances of the intersection points measured from the origin, form a
2681:
1904:, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals,
778:{\displaystyle {\begin{aligned}x&=(vt+c)\cos \omega t\\y&=(vt+c)\sin \omega t\end{aligned}}}
1940:
2519:
2804:
2510:
Page with Java application to interactively explore the
Archimedean spiral and its related curves
81:
2514:
2509:
2255:
1064:
2855:
2799:
2597:
2539:
1701:
2748:
2420:
2066:
2037:
2029:
1863:
1657:{\displaystyle \kappa ={\frac {\theta ^{2}+2}{b\left(\theta ^{2}+1\right)^{\frac {3}{2}}}}}
671:
159:
2280:
Gilchrist, J. E.; Campbell, J. E.; Donnelly, C. B.; Peeler, J. T.; Delaney, J. M. (1973).
661:{\displaystyle {\begin{aligned}\int v_{x}\,dt&=x\\\int v_{y}\,dt&=y\end{aligned}}}
8:
2850:
2809:
2101:
1882:
73:
2424:
862:{\displaystyle {\sqrt {x^{2}+y^{2}}}={\frac {v}{\omega }}\cdot \arctan {\frac {y}{x}}+c}
2780:
2765:
2436:
2410:
2370:
2132:
2095:
2072:
2057:
1852:
1697:
2485:
2306:
2281:
2228:
2743:
2733:
2691:
2587:
2482:
2440:
2374:
2311:
2261:
2182:
2084:
2051:
1926:
1901:
1841:
1068:
556:
2845:
2642:
2532:
2428:
2301:
2293:
1886:
222:
77:
2297:
2775:
2627:
2577:
2090:
2010:
2006:
1932:
Asking for a patient to draw an
Archimedean spiral is a way of quantifying human
1892:
184:
171:
2203:
1950:
2706:
2592:
2363:
1944:
1922:
1912:
155:
55:
2524:
2467:
2834:
2785:
2107:
2078:
1968:
1908:
of the same size that almost resemble Archimedean spirals, or hybrid curves.
58:
2432:
2760:
2755:
2701:
2647:
2499:
2002:
1671:
2315:
2819:
2671:
2622:
2475:
2152:
128:
2793:
2696:
2676:
2503:
2181:. Princeton, New Jersey: Princeton University Press. pp. 140–142.
1972:
1187:
150:
61:
239:
2814:
2738:
2663:
2637:
2582:
2490:
2471:
1998:
1980:
1958:
30:
2770:
2686:
2415:
1905:
188:
1900:
The Archimedean spiral has a variety of real-world applications.
1737:
1696:), hence the name "arithmetic spiral". In contrast to this, in a
577:
as the velocity components along the x and y axes, respectively.
177:
2020:
is another simple method to approximate the Archimedean Spiral.
1435:{\displaystyle {\frac {b}{2}}\left_{\theta _{1}}^{\theta _{2}}.}
2555:
2480:
2279:
1933:
1693:
1333:{\displaystyle {\frac {b}{2}}\left_{\theta _{1}}^{\theta _{2}}}
180:
is used below to understand the notion of Archimedean spirals.
51:
1936:; this information helps in diagnosing neurological diseases.
2563:
2559:
1919:
2156:
2110: – Various symbols with three-fold rotational symmetry
2033:
165:
1915:, which can be operated over a wide range of frequencies.
72:(the specific arithmetic spiral of Archimedes). It is the
2204:"Fluid compressing device having coaxial spiral members"
2112:
Pages displaying short descriptions of redirect targets
1683:
2151:
1050:{\displaystyle r={\frac {v}{\omega }}\cdot \theta +c.}
232:-axis, then the velocity of the point with respect to
1784:
1582:
1471:
1346:
1214:
1082:
1015:
911:
793:
682:
585:
301:
101:
2360:
2047:
1840:. Other spirals falling into this group include the
34:
Three 360° loops of one arm of an Archimedean spiral
1076:Given the parametrization in cartesian coordinates
2362:
2081: – Self-similar curve related to golden ratio
1824:{\displaystyle r=a+b\cdot \theta ^{\frac {1}{c}}.}
1823:
1656:
1566:
1434:
1332:
1178:
1049:
999:
861:
777:
670:The above equations can be integrated by applying
660:
535:
119:
2282:"Spiral Plate Method for Bacterial Determination"
2104: – Polygonal curve made from right triangles
2075: – Spiral that surrounds equal area per turn
674:, leading to the following parametric equations:
162:states that this spiral was discovered by Conon.
2832:
2399:
2554:
2369:. Mathematical Association of America. p.
1740:asymptotically approaches a circle with radius
1675:Archimedean spiral represented on a polar graph
148:Archimedes described such a spiral in his book
2515:Online exploration using JSXGraph (JavaScript)
1778:is used for the more general group of spirals
2540:
2361:Walser, H.; Hilton, P.; Pedersen, J. (2000).
2016:Compass and straightedge construction of the
1769:
1707:The Archimedean spiral has two arms, one for
2656:
1059:
2547:
2533:
1979:) are Archimedean. For instance, the star
1943:(DLP) projection systems to minimize the "
1833:The normal Archimedean spiral occurs when
545:As shown in the figure alongside, we have
2414:
2328:
2305:
2163:On-Line Encyclopedia of Integer Sequences
2131:
1490:
1366:
1234:
1163:
1159:
1143:
1139:
1117:
1101:
637:
603:
251:(anticlockwise) about the origin in time
27:Spiral with constant distance from itself
1949:
1891:
1670:
1063:
238:
166:Derivation of general equation of spiral
29:
2201:
1987:
209:, the object was at an arbitrary point
14:
2833:
2253:
274:is the position of the object at time
2528:
2481:
2354:
2176:
1939:Archimedean spirals are also used in
1736:As the Archimedean spiral grows, its
138:controls the distance between loops.
2222:
2202:Sakata, Hirotsugu; Okuda, Masayuki.
2170:
1911:Archimedean spirals can be found in
1567:{\displaystyle {\frac {b}{2}}\left.}
183:Suppose a point object moves in the
95:it can be described by the equation
2329:Peressini, Tony (3 February 2009).
2322:
2087: – Spiral asymptotic to a line
24:
2138:Dictionary of Scientific Biography
1967:Many dynamic spirals (such as the
1666:
25:
2867:
2520:Archimedean spiral at "mathcurve"
2461:
2098: – Self-similar growth curve
263:is the position of the object at
2606:
2050:
1896:Mechanism of a scroll compressor
1725:-axis will yield the other arm.
555:representing the modulus of the
120:{\displaystyle r=b\cdot \theta }
2447:
2393:
2069: – Water pumping mechanism
1876:
54:named after the 3rd-century BC
2273:
2247:
2216:
2195:
2145:
2125:
1955:Atacama Large Millimeter Array
1925:and the grooves of very early
1173:
1133:
1127:
1095:
1092:
756:
741:
712:
697:
514:
499:
442:
427:
374:
358:
322:
307:
221:plane rotates with a constant
13:
1:
2298:10.1128/AEM.25.2.244-252.1973
2225:"Early Development of the LP"
2118:
2257:Handbook for Sound Engineers
559:of the particle at any time
7:
2331:"Joan's Paper Roll Problem"
2260:, CRC Press, p. 1586,
2141:. Vol. 3. p. 391.
2043:
1975:, or the pattern made by a
1576:The curvature is given by
198:-axis, with respect to the
10:
2872:
2153:Sloane, N. J. A.
2011:Polygon approximation of π
1770:General Archimedean spiral
247:plane rotates to an angle
169:
2721:
2615:
2604:
2570:
1060:Arc length and curvature
236:-axis may be written as:
194:directed parallel to the
134:. Changing the parameter
2179:A History of Mathematics
1941:digital light processing
158:was a friend of his and
2433:10.1038/s41550-017-0060
2177:Boyer, Carl B. (1968).
2157:"Sequence A091154"
1961:
1897:
1825:
1676:
1658:
1568:
1442:The total length from
1436:
1334:
1180:
1073:
1051:
1001:
863:
779:
662:
537:
293:
121:
35:
2254:Ballou, Glen (2008),
2240:. See the passage on
1953:
1906:involutes of a circle
1895:
1826:
1702:geometric progression
1674:
1659:
1569:
1437:
1335:
1181:
1067:
1052:
1002:
869:(using the fact that
864:
780:
663:
538:
242:
122:
33:
2486:"Archimedes' Spiral"
2286:Applied Microbiology
2135:. "Conon of Samos".
2108:Triple spiral symbol
1988:Construction methods
1782:
1580:
1469:
1344:
1212:
1080:
1013:
909:
791:
680:
672:integration by parts
583:
299:
202:-plane. Let at time
99:
18:Spiral of Archimedes
2841:Squaring the circle
2425:2017NatAs...1E..60K
2133:Bulmer-Thomas, Ivor
2102:Spiral of Theodorus
2018:Spiral of Theodorus
1883:squaring the circle
1774:Sometimes the term
1428:
1329:
278:, at a distance of
80:. Equivalently, in
2500:archimedean spiral
2483:Weisstein, Eric W.
2343:on 3 November 2013
2231:on 5 November 2005
2166:. OEIS Foundation.
2096:Logarithmic spiral
2058:Mathematics portal
1962:
1927:gramophone records
1902:Scroll compressors
1898:
1821:
1776:Archimedean spiral
1698:logarithmic spiral
1677:
1654:
1564:
1432:
1357:
1340:or, equivalently:
1330:
1225:
1176:
1074:
1069:Osculating circles
1047:
1009:Its polar form is
997:
859:
775:
773:
658:
656:
533:
531:
294:
117:
70:Archimedes' spiral
66:Archimedean spiral
44:Archimedes' spiral
40:Archimedean spiral
36:
2828:
2827:
2717:
2716:
2085:Hyperbolic spiral
2067:Archimedes' screw
2038:Archimedes’ screw
2030:Archimedes’ screw
1977:Catherine's wheel
1842:hyperbolic spiral
1815:
1652:
1648:
1549:
1509:
1480:
1385:
1355:
1293:
1253:
1223:
1030:
995:
977:
953:
851:
832:
819:
383:
178:physical approach
145:as time elapses.
82:polar coordinates
48:arithmetic spiral
16:(Redirected from
2863:
2654:
2653:
2633:Boerdijk–Coxeter
2610:
2609:
2549:
2542:
2535:
2526:
2525:
2496:
2495:
2455:
2451:
2445:
2444:
2418:
2403:Nature Astronomy
2397:
2391:
2390:
2388:
2387:
2368:
2358:
2352:
2351:
2349:
2348:
2342:
2336:. Archived from
2335:
2326:
2320:
2319:
2309:
2277:
2271:
2270:
2251:
2245:
2239:
2237:
2236:
2227:. Archived from
2220:
2214:
2213:
2211:
2210:
2199:
2193:
2192:
2174:
2168:
2167:
2149:
2143:
2142:
2129:
2113:
2060:
2055:
2054:
2007:Polygonal Spiral
1887:trisect an angle
1872:
1861:
1850:
1839:
1830:
1828:
1827:
1822:
1817:
1816:
1808:
1765:
1764:
1762:
1761:
1756:
1753:
1751:
1731:
1724:
1720:
1713:
1691:
1687:
1663:
1661:
1660:
1655:
1653:
1651:
1650:
1649:
1641:
1639:
1635:
1628:
1627:
1608:
1601:
1600:
1590:
1573:
1571:
1570:
1565:
1560:
1556:
1555:
1551:
1550:
1548:
1547:
1532:
1510:
1508:
1507:
1492:
1481:
1473:
1464:
1451:
1441:
1439:
1438:
1433:
1427:
1426:
1425:
1415:
1414:
1413:
1403:
1399:
1386:
1384:
1383:
1368:
1356:
1348:
1339:
1337:
1336:
1331:
1328:
1327:
1326:
1316:
1315:
1314:
1304:
1300:
1299:
1295:
1294:
1292:
1291:
1276:
1254:
1252:
1251:
1236:
1224:
1216:
1207:
1198:
1185:
1183:
1182:
1177:
1056:
1054:
1053:
1048:
1031:
1023:
1006:
1004:
1003:
998:
996:
988:
983:
979:
978:
970:
965:
961:
954:
952:
951:
939:
938:
929:
904:
903:
901:
900:
895:
892:
878:
868:
866:
865:
860:
852:
844:
833:
825:
820:
818:
817:
805:
804:
795:
784:
782:
781:
776:
774:
667:
665:
664:
659:
657:
636:
635:
602:
601:
576:
569:
562:
554:
542:
540:
539:
534:
532:
470:
469:
398:
397:
384:
382:
381:
357:
356:
344:
343:
334:
325:
320:
319:
310:
291:
277:
273:
269:
262:
254:
250:
246:
235:
231:
227:
223:angular velocity
220:
216:
208:
201:
197:
193:
187:with a constant
185:Cartesian system
144:
137:
133:
126:
124:
123:
118:
94:
78:angular velocity
21:
2871:
2870:
2866:
2865:
2864:
2862:
2861:
2860:
2831:
2830:
2829:
2824:
2713:
2667:
2652:
2611:
2607:
2602:
2566:
2553:
2464:
2459:
2458:
2452:
2448:
2398:
2394:
2385:
2383:
2381:
2359:
2355:
2346:
2344:
2340:
2333:
2327:
2323:
2278:
2274:
2268:
2252:
2248:
2242:Variable Groove
2234:
2232:
2223:Penndorf, Ron.
2221:
2217:
2208:
2206:
2200:
2196:
2189:
2175:
2171:
2150:
2146:
2130:
2126:
2121:
2116:
2111:
2091:List of spirals
2073:Fermat's spiral
2056:
2049:
2046:
1990:
1923:balance springs
1879:
1867:
1856:
1853:Fermat's spiral
1845:
1834:
1807:
1803:
1783:
1780:
1779:
1772:
1757:
1754:
1747:
1745:
1744:
1742:
1741:
1729:
1722:
1715:
1708:
1692:is measured in
1689:
1680:
1669:
1667:Characteristics
1640:
1623:
1619:
1618:
1614:
1613:
1609:
1596:
1592:
1591:
1589:
1581:
1578:
1577:
1543:
1539:
1531:
1524:
1520:
1503:
1499:
1491:
1486:
1482:
1472:
1470:
1467:
1466:
1459:
1453:
1449:
1443:
1421:
1417:
1416:
1409:
1405:
1404:
1379:
1375:
1367:
1362:
1358:
1347:
1345:
1342:
1341:
1322:
1318:
1317:
1310:
1306:
1305:
1287:
1283:
1275:
1268:
1264:
1247:
1243:
1235:
1230:
1226:
1215:
1213:
1210:
1209:
1206:
1200:
1197:
1191:
1081:
1078:
1077:
1062:
1022:
1014:
1011:
1010:
987:
969:
947:
943:
934:
930:
928:
927:
923:
922:
918:
910:
907:
906:
896:
893:
888:
887:
885:
880:
870:
843:
824:
813:
809:
800:
796:
794:
792:
789:
788:
772:
771:
734:
728:
727:
690:
683:
681:
678:
677:
655:
654:
644:
631:
627:
621:
620:
610:
597:
593:
586:
584:
581:
580:
575:
571:
568:
564:
560:
557:position vector
546:
530:
529:
471:
465:
461:
458:
457:
399:
393:
389:
386:
385:
377:
373:
352:
348:
339:
335:
333:
326:
321:
315:
311:
306:
302:
300:
297:
296:
279:
275:
271:
264:
256:
252:
248:
244:
233:
229:
225:
218:
210:
203:
199:
195:
191:
174:
172:Circular motion
168:
142:
135:
131:
100:
97:
96:
84:
42:(also known as
28:
23:
22:
15:
12:
11:
5:
2869:
2859:
2858:
2853:
2848:
2843:
2826:
2825:
2823:
2822:
2817:
2812:
2807:
2802:
2797:
2790:
2789:
2788:
2778:
2773:
2768:
2763:
2758:
2753:
2752:
2751:
2746:
2741:
2731:
2725:
2723:
2719:
2718:
2715:
2714:
2712:
2711:
2710:
2709:
2699:
2694:
2689:
2684:
2679:
2674:
2669:
2665:
2660:
2658:
2651:
2650:
2645:
2640:
2635:
2630:
2625:
2619:
2617:
2613:
2612:
2605:
2603:
2601:
2600:
2595:
2590:
2585:
2580:
2574:
2572:
2568:
2567:
2552:
2551:
2544:
2537:
2529:
2523:
2522:
2517:
2512:
2507:
2497:
2478:
2463:
2462:External links
2460:
2457:
2456:
2446:
2392:
2379:
2353:
2321:
2272:
2266:
2246:
2215:
2194:
2187:
2169:
2144:
2123:
2122:
2120:
2117:
2115:
2114:
2105:
2099:
2093:
2088:
2082:
2076:
2070:
2063:
2062:
2061:
2045:
2042:
1989:
1986:
1945:rainbow effect
1913:spiral antenna
1881:One method of
1878:
1875:
1820:
1814:
1811:
1806:
1802:
1799:
1796:
1793:
1790:
1787:
1771:
1768:
1668:
1665:
1647:
1644:
1638:
1634:
1631:
1626:
1622:
1617:
1612:
1607:
1604:
1599:
1595:
1588:
1585:
1563:
1559:
1554:
1546:
1542:
1538:
1535:
1530:
1527:
1523:
1519:
1516:
1513:
1506:
1502:
1498:
1495:
1489:
1485:
1479:
1476:
1457:
1447:
1431:
1424:
1420:
1412:
1408:
1402:
1398:
1395:
1392:
1389:
1382:
1378:
1374:
1371:
1365:
1361:
1354:
1351:
1325:
1321:
1313:
1309:
1303:
1298:
1290:
1286:
1282:
1279:
1274:
1271:
1267:
1263:
1260:
1257:
1250:
1246:
1242:
1239:
1233:
1229:
1222:
1219:
1204:
1195:
1175:
1172:
1169:
1166:
1162:
1158:
1155:
1152:
1149:
1146:
1142:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1116:
1113:
1110:
1107:
1104:
1100:
1097:
1094:
1091:
1088:
1085:
1061:
1058:
1046:
1043:
1040:
1037:
1034:
1029:
1026:
1021:
1018:
994:
991:
986:
982:
976:
973:
968:
964:
960:
957:
950:
946:
942:
937:
933:
926:
921:
917:
914:
858:
855:
850:
847:
842:
839:
836:
831:
828:
823:
816:
812:
808:
803:
799:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
735:
733:
730:
729:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
691:
689:
686:
685:
653:
650:
647:
645:
643:
640:
634:
630:
626:
623:
622:
619:
616:
613:
611:
609:
606:
600:
596:
592:
589:
588:
573:
566:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
472:
468:
464:
460:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
400:
396:
392:
388:
387:
380:
376:
372:
369:
366:
363:
360:
355:
351:
347:
342:
338:
332:
329:
327:
324:
318:
314:
309:
305:
304:
167:
164:
156:Conon of Samos
116:
113:
110:
107:
104:
26:
9:
6:
4:
3:
2:
2868:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2838:
2836:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2795:
2791:
2787:
2784:
2783:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2750:
2747:
2745:
2742:
2740:
2737:
2736:
2735:
2732:
2730:
2727:
2726:
2724:
2720:
2708:
2705:
2704:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2662:
2661:
2659:
2655:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2620:
2618:
2614:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2581:
2579:
2576:
2575:
2573:
2569:
2565:
2561:
2557:
2550:
2545:
2543:
2538:
2536:
2531:
2530:
2527:
2521:
2518:
2516:
2513:
2511:
2508:
2505:
2501:
2498:
2493:
2492:
2487:
2484:
2479:
2477:
2473:
2469:
2466:
2465:
2454:
2450:
2442:
2438:
2434:
2430:
2426:
2422:
2417:
2412:
2408:
2404:
2396:
2382:
2380:9780883855324
2376:
2372:
2367:
2366:
2357:
2339:
2332:
2325:
2317:
2313:
2308:
2303:
2299:
2295:
2292:(2): 244–52.
2291:
2287:
2283:
2276:
2269:
2267:9780240809694
2263:
2259:
2258:
2250:
2243:
2230:
2226:
2219:
2205:
2198:
2190:
2188:0-691-02391-3
2184:
2180:
2173:
2165:
2164:
2158:
2154:
2148:
2140:
2139:
2134:
2128:
2124:
2109:
2106:
2103:
2100:
2097:
2094:
2092:
2089:
2086:
2083:
2080:
2079:Golden spiral
2077:
2074:
2071:
2068:
2065:
2064:
2059:
2053:
2048:
2041:
2039:
2035:
2031:
2025:
2021:
2019:
2014:
2012:
2008:
2004:
2000:
1994:
1985:
1982:
1978:
1974:
1970:
1969:Parker spiral
1965:
1960:
1956:
1952:
1948:
1946:
1942:
1937:
1935:
1930:
1928:
1924:
1921:
1918:The coils of
1916:
1914:
1909:
1907:
1903:
1894:
1890:
1888:
1884:
1874:
1870:
1865:
1859:
1854:
1848:
1843:
1837:
1831:
1818:
1812:
1809:
1804:
1800:
1797:
1794:
1791:
1788:
1785:
1777:
1767:
1760:
1750:
1739:
1734:
1733:Gaichenkov).
1726:
1718:
1711:
1705:
1703:
1699:
1695:
1686:
1685:
1673:
1664:
1645:
1642:
1636:
1632:
1629:
1624:
1620:
1615:
1610:
1605:
1602:
1597:
1593:
1586:
1583:
1574:
1561:
1557:
1552:
1544:
1540:
1536:
1533:
1528:
1525:
1521:
1517:
1514:
1511:
1504:
1500:
1496:
1493:
1487:
1483:
1477:
1474:
1465:is therefore
1463:
1456:
1446:
1429:
1422:
1418:
1410:
1406:
1400:
1396:
1393:
1390:
1387:
1380:
1376:
1372:
1369:
1363:
1359:
1352:
1349:
1323:
1319:
1311:
1307:
1301:
1296:
1288:
1284:
1280:
1277:
1272:
1269:
1265:
1261:
1258:
1255:
1248:
1244:
1240:
1237:
1231:
1227:
1220:
1217:
1203:
1194:
1189:
1170:
1167:
1164:
1160:
1156:
1153:
1150:
1147:
1144:
1140:
1136:
1130:
1124:
1121:
1118:
1114:
1111:
1108:
1105:
1102:
1098:
1089:
1086:
1083:
1070:
1066:
1057:
1044:
1041:
1038:
1035:
1032:
1027:
1024:
1019:
1016:
1007:
992:
989:
984:
980:
974:
971:
966:
962:
958:
955:
948:
944:
940:
935:
931:
924:
919:
915:
912:
899:
891:
883:
877:
873:
856:
853:
848:
845:
840:
837:
834:
829:
826:
821:
814:
810:
806:
801:
797:
785:
768:
765:
762:
759:
753:
750:
747:
744:
738:
736:
731:
724:
721:
718:
715:
709:
706:
703:
700:
694:
692:
687:
675:
673:
668:
651:
648:
646:
641:
638:
632:
628:
624:
617:
614:
612:
607:
604:
598:
594:
590:
578:
558:
553:
549:
543:
526:
523:
520:
517:
511:
508:
505:
502:
496:
493:
490:
487:
484:
481:
478:
475:
473:
466:
462:
454:
451:
448:
445:
439:
436:
433:
430:
424:
421:
418:
415:
412:
409:
406:
403:
401:
394:
390:
378:
370:
367:
364:
361:
353:
349:
345:
340:
336:
330:
328:
316:
312:
290:
286:
282:
267:
260:
241:
237:
224:
214:
206:
190:
186:
181:
179:
173:
163:
161:
157:
153:
152:
146:
139:
130:
114:
111:
108:
105:
102:
92:
88:
83:
79:
75:
71:
67:
63:
60:
59:mathematician
57:
53:
49:
45:
41:
32:
19:
2856:Plane curves
2792:
2728:
2657:Biochemistry
2489:
2453:
2449:
2406:
2402:
2395:
2384:. Retrieved
2364:
2356:
2345:. Retrieved
2338:the original
2324:
2289:
2285:
2275:
2256:
2249:
2241:
2233:. Retrieved
2229:the original
2218:
2207:. Retrieved
2197:
2178:
2172:
2160:
2147:
2136:
2127:
2026:
2022:
2015:
2003:French Curve
1995:
1991:
1966:
1963:
1938:
1931:
1917:
1910:
1899:
1880:
1877:Applications
1868:
1857:
1846:
1835:
1832:
1775:
1773:
1758:
1748:
1735:
1727:
1716:
1714:and one for
1709:
1706:
1682:
1678:
1575:
1461:
1454:
1444:
1201:
1192:
1075:
1008:
897:
889:
881:
875:
871:
786:
676:
669:
579:
551:
547:
544:
295:
288:
284:
280:
265:
258:
212:
204:
182:
175:
149:
147:
140:
90:
86:
69:
65:
47:
43:
39:
37:
2805:Pitch angle
2781:Logarithmic
2729:Archimedean
2692:Polyproline
2476:Green Farms
2409:(3): 0060.
1862:), and the
129:real number
64:. The term
2851:Archimedes
2835:Categories
2794:On Spirals
2744:Hyperbolic
2504:PlanetMath
2472:TedX Talks
2416:1704.00449
2386:2014-10-06
2347:2014-10-06
2235:2005-11-25
2209:2006-11-25
2119:References
1973:solar wind
1728:For large
1188:arc length
228:about the
170:See also:
151:On Spirals
62:Archimedes
2815:Spirangle
2810:Theodorus
2749:Poinsot's
2739:Epispiral
2583:Curvature
2578:Algebraic
2491:MathWorld
2441:119433782
1999:dodecagon
1981:LL Pegasi
1959:LL Pegasi
1957:image of
1805:θ
1801:⋅
1621:θ
1594:θ
1584:κ
1541:θ
1526:θ
1518:
1501:θ
1488:θ
1419:θ
1407:θ
1397:θ
1394:
1377:θ
1364:θ
1320:θ
1308:θ
1285:θ
1270:θ
1262:
1245:θ
1232:θ
1171:θ
1168:
1161:θ
1151:θ
1148:
1141:θ
1125:θ
1122:
1109:θ
1106:
1093:↦
1090:θ
1087::
1036:θ
1033:⋅
1028:ω
972:ω
967:⋅
956:−
916:
884:= arctan
841:
835:⋅
830:ω
766:ω
763:
722:ω
719:
625:∫
591:∫
524:ω
521:
497:ω
488:ω
485:
452:ω
449:
425:ω
422:−
416:ω
413:
350:ω
217:. If the
115:θ
112:⋅
2771:Involute
2766:Fermat's
2707:Collagen
2643:Symmetry
2365:Symmetry
2044:See also
189:velocity
2846:Spirals
2800:Padovan
2734:Cotes's
2722:Spirals
2628:Antenna
2616:Helices
2588:Gallery
2564:helices
2556:Spirals
2421:Bibcode
2316:4632851
2155:(ed.).
1971:of the
1763:
1743:
1738:evolute
1694:radians
902:
886:
563:, with
215:, 0, 0)
50:) is a
2786:Golden
2702:Triple
2682:Double
2648:Triple
2598:Topics
2571:Curves
2560:curves
2439:
2377:
2314:
2307:380780
2304:
2264:
2185:
1934:tremor
1864:lituus
1752:|
1746:|
1719:< 0
1712:> 0
1391:arsinh
1072:other.
838:arctan
160:Pappus
52:spiral
46:, the
2761:Euler
2756:Doyle
2697:Super
2672:Alpha
2623:Angle
2437:S2CID
2411:arXiv
2341:(PDF)
2334:(PDF)
1920:watch
1190:from
905:) or
127:with
74:locus
56:Greek
2820:Ulam
2776:List
2677:Beta
2638:Hemi
2593:List
2562:and
2375:ISBN
2312:PMID
2262:ISBN
2183:ISBN
2161:The
2034:cone
1871:= −2
1849:= −1
1186:the
879:and
570:and
261:, 0)
243:The
38:The
2502:at
2429:doi
2302:PMC
2294:doi
2013:).
1873:).
1860:= 2
1851:),
1838:= 1
1688:if
1452:to
1450:= 0
1208:is
1199:to
1165:sin
1145:cos
1119:sin
1103:cos
913:tan
760:sin
716:cos
518:cos
482:sin
446:sin
410:cos
268:= 0
207:= 0
154:.
2837::
2687:Pi
2666:10
2558:,
2488:.
2474:,
2470:-
2435:.
2427:.
2419:.
2405:.
2373:.
2371:27
2310:.
2300:.
2290:25
2288:.
2284:.
2159:.
2040:.
1766:.
1704:.
1684:πb
1515:ln
1460:=
1259:ln
874:=
872:ωt
550:+
548:vt
287:+
285:vt
283:=
270:.
255:.
249:ωt
245:xy
219:xy
200:xy
176:A
89:,
2664:3
2548:e
2541:t
2534:v
2506:.
2494:.
2443:.
2431::
2423::
2413::
2407:1
2389:.
2350:.
2318:.
2296::
2244:.
2238:.
2212:.
2191:.
1869:c
1866:(
1858:c
1855:(
1847:c
1844:(
1836:c
1819:.
1813:c
1810:1
1798:b
1795:+
1792:a
1789:=
1786:r
1759:ω
1755:/
1749:v
1730:θ
1723:y
1717:θ
1710:θ
1690:θ
1681:2
1646:2
1643:3
1637:)
1633:1
1630:+
1625:2
1616:(
1611:b
1606:2
1603:+
1598:2
1587:=
1562:.
1558:]
1553:)
1545:2
1537:+
1534:1
1529:+
1522:(
1512:+
1505:2
1497:+
1494:1
1484:[
1478:2
1475:b
1462:θ
1458:2
1455:θ
1448:1
1445:θ
1430:.
1423:2
1411:1
1401:]
1388:+
1381:2
1373:+
1370:1
1360:[
1353:2
1350:b
1324:2
1312:1
1302:]
1297:)
1289:2
1281:+
1278:1
1273:+
1266:(
1256:+
1249:2
1241:+
1238:1
1228:[
1221:2
1218:b
1205:2
1202:θ
1196:1
1193:θ
1174:)
1157:b
1154:,
1137:b
1134:(
1131:=
1128:)
1115:r
1112:,
1099:r
1096:(
1084:f
1045:.
1042:c
1039:+
1025:v
1020:=
1017:r
993:x
990:y
985:=
981:)
975:v
963:)
959:c
949:2
945:y
941:+
936:2
932:x
925:(
920:(
898:x
894:/
890:y
882:θ
876:θ
857:c
854:+
849:x
846:y
827:v
822:=
815:2
811:y
807:+
802:2
798:x
769:t
757:)
754:c
751:+
748:t
745:v
742:(
739:=
732:y
725:t
713:)
710:c
707:+
704:t
701:v
698:(
695:=
688:x
652:y
649:=
642:t
639:d
633:y
629:v
618:x
615:=
608:t
605:d
599:x
595:v
574:y
572:v
567:x
565:v
561:t
552:c
527:t
515:)
512:c
509:+
506:t
503:v
500:(
494:+
491:t
479:v
476:=
467:y
463:v
455:t
443:)
440:c
437:+
434:t
431:v
428:(
419:t
407:v
404:=
395:x
391:v
379:2
375:)
371:c
368:+
365:t
362:v
359:(
354:2
346:+
341:2
337:v
331:=
323:|
317:0
313:v
308:|
292:.
289:c
281:R
276:t
272:P
266:t
259:c
257:(
253:t
234:z
230:z
226:ω
213:c
211:(
205:t
196:x
192:v
143:θ
136:b
132:b
109:b
106:=
103:r
93:)
91:θ
87:r
85:(
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
