2598:
2429:
2389:
3512:
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54:
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42:
2610:
1618:
745:
2409:
753:
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2685:
are made from metal cams whose outer gripping surfaces are shaped as arcs of logarithmic spirals. When the device is inserted into a rock crack, the rotation of these cams expands their combined width to match the width of the crack, while maintaining a constant angle against the surface of the rock
1345:
2520:
The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the Sun (or Moon for nocturnal species) is the only light source and flying that way will result in a practically straight
2121:
1053:
2645:
are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles between the tooth centerline and the radial lines, which gives the meshing transmission more
544:
3706:
1043:
707:
for "miraculous spiral", is another name for the logarithmic spiral. Although this curve had already been named by other mathematicians, the specific name ("miraculous" or "marvelous" spiral) was given to this curve by
2064:
1476:
875:
632:
3041:
C. Q. Yu CQ and M. I. Rosenblatt, "Transgenic corneal neurofluorescence in mice: a new model for in vivo investigation of nerve structure and regeneration," Invest
Ophthalmol Vis Sci. 2007 Apr;48(4):1535-42.
2540:, their pitch angles vary with distance from the galactic center, unlike logarithmic spirals (for which this angle does not vary), and also at variance with the other mathematical spirals used to model them.
218:
2638:) by each machine in the cutting process. To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters.
270:
458:
1883:
1765:
1612:
1246:
2532:
galaxy has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. However, although spiral galaxies have often been modeled as logarithmic spirals,
814:
2499:
2365:
for every 90 degrees of rotation (pitch angle about 17.03239 degrees). It can be approximated by a "Fibonacci spiral", made of a sequence of quarter circles with radii proportional to
712:, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as
686:
1936:
1805:
1236:{\displaystyle L(\varphi _{1},\varphi _{2})={\frac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}={\frac {r(\varphi _{2})-r(\varphi _{1})}{\sin \alpha }}}
1557:
395:
1703:
157:
2321:{\displaystyle z(t)=\underbrace {(kt+b)\;+it} _{\text{line}}\quad \to \quad e^{z(t)}=e^{kt+b}\cdot e^{it}=\underbrace {e^{b}e^{kt}(\cos t+i\sin t)} _{\text{log. spiral}}}
1986:
1669:
463:
2460:
1518:
2346:
941:
895:
348:
1371:
322:
921:
2686:(relative to the center of the spiral, where force is applied). The pitch angle of the spiral is chosen to optimize the friction of the device against the rock.
2116:
961:
291:
2517:, assuming the prey travels in a straight line. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.
2627:
are frequency-independent antennas, that is, antennas whose radiation pattern, impedance and polarization remain largely unmodified over a wide bandwidth.
2098:
exactly maps all lines not parallel with the real or imaginary axis in the complex plane, to all logarithmic spirals in the complex plane with centre at
2603:
A kerf-canceling mechanism leverages the self similarity of the logarithmic spiral to lock in place under rotation, independent of the kerf of the cut.
972:
2436:
shell showing the chambers arranged in an approximately logarithmic spiral. The plotted spiral (dashed blue curve) is based on growth rate parameter
166:
2634:), there can be a loss of precision when the mechanism is fabricated on a different machine due to the difference of material removed (that is, the
3345:
3322:
223:
2547:(this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).
1991:
1389:
832:
3401:
552:
716:. Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as
2505:
In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:
3212:
3112:
3085:
2983:
2956:
2929:
2865:
2825:
2796:
2771:
3366:, an educational website about the science of pattern formation, spirals in nature, and spirals in the mythic imagination.
3236:
Jiang, Jianfeng; Luo, Qingsheng; Wang, Liting; Qiao, Lijun; Li, Minghao (2020). "Review on logarithmic spiral bevel gear".
1340:{\displaystyle \ L(-\infty ,\varphi _{2})={\frac {r(\varphi _{2})}{\sin \alpha }}\quad ({\color {red}{\text{finite !}}})\;}
637:
400:
3374:
1810:
3687:
1708:
1562:
17:
361:
3768:
3288:
2714:, a geometric problem asking for the path followed by mice chasing one another whose solution is a logarithmic spiral
1988:. Hence they are all scaled copies of the red one. But they can also be generated by rotating the red one by angles
3725:
3394:
763:
756:
Animation showing the constant angle between an intersecting circle centred at the origin and a logarithmic spiral.
2465:
93:(1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by
3667:
2682:
2668:
3387:
1908:
31:
1770:
3854:
2580:
2576:
1523:
3903:
3491:
2664:
273:
2569:. In these cases, the reason may be construction from expanding similar shapes, as is the case for
3501:
3332:
3191:
Roumen, Thijs; Apel, Ingo; Shigeyama, Jotaro; Muhammad, Abdullah; Baudisch, Patrick (2020-10-20).
3128:
3058:
2740:
Underweysung der
Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen unnd gantzen corporen
1681:
130:
4026:
2717:
1941:
1624:
825:
46:
3352:
NASA Astronomy
Picture of the Day: Hurricane Isabel vs. the Whirlpool Galaxy (25 September 2003)
3369:
3193:"Kerf-canceling mechanisms: Making laser-cut mechanisms operate across different laser cutters"
2378:
1374:
35:
3073:
2973:
2855:
2439:
1489:
4077:
4072:
4021:
3819:
3761:
3102:
2946:
2919:
2881:
Chin, Gilbert J. (8 December 2000). "Organismal
Biology: Flying Along a Logarithmic Spiral".
2738:
2395:
2331:
1898:
926:
880:
327:
113:
1350:
301:
3970:
3165:
2428:
2095:
729:
8:
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3735:
3695:
3479:
3357:
NASA Astronomy
Picture of the Day: Typhoon Rammasun vs. the Pinwheel Galaxy (17 May 2008)
3328:
900:
3169:
3052:
112:
by the fact that the distances between the turnings of a logarithmic spiral increase in
3987:
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3673:
3410:
3307:
3218:
3012:
2900:
2696:
2597:
2533:
2101:
946:
733:
276:
109:
3707:
How Long Is the Coast of
Britain? Statistical Self-Similarity and Fractional Dimension
3197:
Proceedings of the 33rd Annual ACM Symposium on User
Interface Software and Technology
3965:
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3913:
3809:
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3600:
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2514:
295:
125:
94:
90:
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3864:
3754:
3551:
3276:
3271:
Todesco, Gian Marco (2018). "Weird gears". In Emmer, Michele; Abate, Marco (eds.).
3245:
3200:
3173:
3022:
2896:
2892:
2759:
2555:
2419:
2388:
2366:
1483:
539:{\displaystyle x=ae^{k\varphi }\cos \varphi ,\qquad y=ae^{k\varphi }\sin \varphi .}
3997:
3849:
3799:
3474:
3280:
3192:
2763:
2706:
1893:
1038:{\displaystyle \kappa ={\frac {1}{r{\sqrt {1+k^{2}}}}}={\frac {\cos \alpha }{r}}}
713:
709:
98:
78:
3928:
3814:
3662:
3657:
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2624:
2348:
of the logarithmic spiral is the angle between the line and the imaginary axis.
58:
53:
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3000:
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2711:
2630:
When manufacturing mechanisms by subtractive fabrication machines (such as
2362:
41:
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3893:
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can form as the result of wave refraction and diffraction by the coast.
4015:
3918:
3898:
3628:
3623:
3541:
3444:
3238:
Journal of the
Brazilian Society of Mechanical Sciences and Engineering
2975:
The universal book of mathematics: from
Abracadabra to Zeno's paradoxes
2816:
The Golden Ratio: The Story of Phi, The World's Most
Astonishing Number
2059:{\displaystyle -109^{\circ },-173^{\circ },-218^{\circ },-253^{\circ }}
1471:{\displaystyle A={\frac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2}}{4k}}}
870:{\displaystyle \tan \alpha =k\quad ({\color {red}{\text{constant !}}})}
4036:
3960:
3885:
3859:
3804:
3730:
3633:
3379:
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2701:
2529:
725:
721:
627:{\displaystyle (z=x+iy,\,e^{i\varphi }=\cos \varphi +i\sin \varphi )}
160:
1897:: A result of the previous property: A scaled logarithmic spiral is
3992:
3908:
3459:
3424:
3154:"Frequency-independent antennas and broad-band derivatives thereof"
2655:
2566:
2551:
2433:
2077:
1767:, which is the original spiral uniformly scaled (at the origin) by
1617:
1378:
744:
732:" ("Although changed, I shall arise the same."), but, by error, an
717:
3017:
85:
that often appears in nature. The first to describe a logarithmic
3585:
3580:
3568:
3546:
3531:
3439:
3323:
Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves
2570:
2562:
2408:
2399:
2081:
213:{\displaystyle r=ae^{k\varphi },\quad \varphi \in \mathbb {R} ,}
3777:
2544:
2525:
817:
116:, while in an Archimedean spiral these distances are constant.
86:
3375:
YouTube lecture on Zeno's mice problem and logarithmic spirals
2361:
is a logarithmic spiral that grows outward by a factor of the
3785:
3781:
3618:
2066:
resp.. All spirals have no points in common (see property on
724:
heads. Jacob Bernoulli wanted such a spiral engraved on his
704:
82:
752:
3496:
3434:
3429:
3302:
3190:
2635:
2510:
3074:"The form, function, and synthesis of the molluscan shell"
2857:
The history of the calculus and its conceptual development
265:{\displaystyle \varphi ={\frac {1}{k}}\ln {\frac {r}{a}},}
3361:
30:"Spira mirabilis" redirects here. For the orchestra, see
2921:
Discovering Moths: Nighttime Jewels in Your Own Backyard
3275:. Springer International Publishing. pp. 179–193.
2999:
Savchenko, S. S.; Reshetnikov, V. P. (September 2013).
2944:
2758:. Springer International Publishing. pp. 173–175.
2418:
often have the shape of a logarithmic spiral, here the
3076:. In István Hargittai and Clifford A. Pickover (ed.).
1573:
690:
2468:
2442:
2334:
2124:
2104:
1994:
1944:
1911:
1813:
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466:
453:{\displaystyle (x=r\cos \varphi ,\,y=r\sin \varphi )}
403:
364:
330:
304:
279:
226:
169:
133:
108:
The logarithmic spiral can be distinguished from the
2998:
2789:
Divine Proportion: Φ Phi in Art, Nature, and Science
1878:{\displaystyle \;e^{kn2\pi }\;,n=\pm 1,\pm 2,...,\;}
2948:
Spiral structure in galaxies: a density wave theory
2853:
2352:
2813:
2493:
2454:
2340:
2320:
2110:
2058:
1980:
1930:
1877:
1799:
1760:{\displaystyle r=ae^{-k\varphi _{0}}e^{k\varphi }}
1759:
1697:
1663:
1607:{\displaystyle r={\tfrac {1}{a}}e^{-k\varphi }\,.}
1606:
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1339:
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935:
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538:
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389:
342:
316:
285:
264:
212:
151:
3101:Allan Thomas Williams and Anton Micallef (2009).
3005:Monthly Notices of the Royal Astronomical Society
2402:shows an approximately logarithmic spiral pattern
4054:
3235:
2971:
3776:
3273:Imagine Math 6: Between Culture and Mathematics
2754:Hammer, Øyvind (2016). "Dürer's dirty secret".
2076:Logarithmic spirals are congruent to their own
358:The logarithmic spiral with the polar equation
3370:Online exploration using JSXGraph (JavaScript)
3071:
2917:
2786:
2736:
2587:
943:would be 0 and the curve a circle with radius
97:(1638), and later extensively investigated by
3762:
3395:
1167:
1119:
809:{\displaystyle r=ae^{k\varphi }\;,\;k\neq 0,}
3199:. Virtual Event USA: ACM. pp. 293–303.
2494:{\displaystyle \arctan b\approx 10^{\circ }}
397:can be represented in Cartesian coordinates
3878:
3050:
3001:"Pitch angle variations in spiral galaxies"
2860:. Courier Dover Publications. p. 133.
2844:, J. W. Edwards (1952), "Evolutes". p. 206.
1905:The diagram shows spirals with slope angle
353:
3769:
3755:
3402:
3388:
2162:
1874:
1834:
1814:
1336:
793:
789:
3104:Beach management: principles and practice
3026:
3016:
2842:A Handbook on Curves and Their Properties
1600:
580:
428:
203:
2924:. Down East Enterprise Inc. p. 63.
2427:
1616:
897:(see diagram and animation).(In case of
751:
743:
52:
40:
3270:
2671:, with logarithmic spiral cam surfaces
1373:. This property was first realized by
14:
4055:
3409:
2753:
2583:is an example of such a type of beach.
3750:
3383:
3303:
3151:
2811:
1901:(by rotation) to the original curve.
681:{\displaystyle z=ae^{(k+i)\varphi }.}
2978:. John Wiley and Sons. p. 188.
2880:
2805:
748:Definition of slope angle and sector
3688:The Chemical Basis of Morphogenesis
2565:structures including the shells of
1931:{\displaystyle \alpha =20^{\circ }}
24:
3333:Spira Mirabilis - Wonderful Spiral
3035:
1800:{\displaystyle e^{-k\varphi _{0}}}
1262:
816:has the following properties (see
25:
4089:
3339:
3080:. World Scientific. p. 370.
2756:The Perfect Shape: Spiral Stories
2379:Patterns in nature § Spirals
1325:
856:
3828:
3510:
2945:G. Bertin and C. C. Lin (1996).
2654:
2608:
2596:
2407:
2387:
2353:Special cases and approximations
1552:{\displaystyle r=ae^{k\varphi }}
390:{\displaystyle r=ae^{k\varphi }}
3264:
3229:
3184:
3145:
3121:
3094:
3065:
3044:
2992:
2965:
2188:
2184:
1678:: Rotating the spiral by angle
1320:
851:
501:
195:
3325:, University of Georgia (1999)
2938:
2911:
2897:10.1126/science.290.5498.1857c
2874:
2847:
2834:
2780:
2747:
2730:
2303:
2276:
2203:
2197:
2185:
2159:
2144:
2134:
2128:
1520:) maps the logarithmic spiral
1496:
1448:
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1405:
1333:
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1256:
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1203:
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864:
852:
667:
655:
621:
556:
447:
404:
146:
134:
61:following a logarithmic spiral
13:
1:
3054:Treatise on physics, Volume 1
2723:
2683:spring-loaded camming devices
1559:onto the logarithmic spiral
739:
119:
3281:10.1007/978-3-319-93949-0_16
2854:Carl Benjamin Boyer (1949).
2820:. New York: Broadway Books.
2764:10.1007/978-3-319-47373-4_41
2669:spring-loaded camming device
2615:A logarithmic spiral antenna
2372:
2092:Complex exponential function
2068:complex exponential function
1698:{\displaystyle \varphi _{0}}
152:{\displaystyle (r,\varphi )}
34:. For the Italian film, see
7:
3129:"kerf-canceling mechanisms"
2690:
2588:In engineering applications
1981:{\displaystyle a=1,2,3,4,5}
1664:{\displaystyle a=1,2,3,4,5}
105:, "the marvelous spiral".
32:Spira Mirabilis (orchestra)
10:
4094:
3250:10.1007/s40430-020-02488-y
2791:. Sterling Publishing Co.
2581:Half Moon Bay (California)
2577:Logarithmic spiral beaches
2462:, resulting in a pitch of
2376:
736:was placed there instead.
29:
3943:
3837:
3826:
3792:
3718:
3668:D'Arcy Wentworth Thompson
3611:
3519:
3508:
3417:
3107:. Earthscan. p. 14.
2972:David J. Darling (2004).
2951:. MIT Press. p. 78.
2074:Relation to other curves:
163:spiral can be written as
27:Self-similar growth curve
2455:{\displaystyle b=0.1759}
1513:{\displaystyle r\to 1/r}
354:In Cartesian coordinates
3205:10.1145/3379337.3415895
3158:Proceedings of the IEEE
3072:Michael Cortie (1992).
2918:John Himmelman (2002).
2787:Priya Hemenway (2005).
2737:Albrecht Dürer (1525).
2341:{\displaystyle \alpha }
2088:based on their centers.
936:{\displaystyle \alpha }
890:{\displaystyle \alpha }
760:The logarithmic spiral
728:along with the phrase "
343:{\displaystyle k\neq 0}
3057:. Churchill. pp.
2502:
2495:
2456:
2342:
2322:
2112:
2060:
1982:
1932:
1879:
1801:
1761:
1699:
1671:
1665:
1608:
1553:
1514:
1472:
1375:Evangelista Torricelli
1367:
1366:{\displaystyle k>0}
1341:
1237:
1039:
957:
937:
917:
891:
871:
810:
757:
749:
682:
628:
540:
454:
391:
350:being real constants.
344:
318:
317:{\displaystyle a>0}
287:
266:
214:
153:
62:
50:
36:Spira Mirabilis (film)
3502:Widmanstätten pattern
3028:10.1093/mnras/stt1627
2812:Livio, Mario (2002).
2558:, such as hurricanes.
2496:
2457:
2431:
2396:extratropical cyclone
2377:Further information:
2343:
2323:
2113:
2061:
1983:
1933:
1880:
1802:
1762:
1700:
1666:
1620:
1609:
1554:
1515:
1473:
1368:
1342:
1238:
1040:
958:
938:
918:
892:
872:
811:
755:
747:
683:
629:
541:
455:
392:
345:
319:
288:
267:
215:
154:
114:geometric progression
56:
44:
3308:"Logarithmic Spiral"
3152:Mayes, P.E. (1992).
3051:Andrew Gray (1901).
2466:
2440:
2332:
2122:
2102:
2096:exponential function
1992:
1942:
1909:
1811:
1771:
1709:
1682:
1625:
1563:
1524:
1490:
1390:
1351:
1247:
1054:
973:
947:
927:
901:
881:
833:
764:
730:Eadem mutata resurgo
638:
553:
464:
401:
362:
328:
302:
277:
224:
167:
131:
45:Logarithmic spiral (
3736:Mathematics and art
3726:Pattern recognition
3696:Aristid Lindenmayer
3329:Alexander Bogomolny
3170:1992IEEEP..80..103M
2718:Tait–Kneser theorem
2534:Archimedean spirals
2524:The arms of spiral
916:{\displaystyle k=0}
694:and Jacob Bernoulli
3674:On Growth and Form
3574:Logarithmic spiral
3411:Patterns in nature
3305:Weisstein, Eric W.
2697:Archimedean spiral
2643:spiral bevel gears
2543:The nerves of the
2538:hyperbolic spirals
2509:The approach of a
2503:
2491:
2452:
2338:
2318:
2317:
2310:
2183:
2176:
2108:
2056:
1978:
1928:
1875:
1797:
1757:
1705:yields the spiral
1695:
1672:
1661:
1604:
1582:
1549:
1510:
1468:
1381:had been invented.
1363:
1337:
1331:
1233:
1035:
953:
933:
913:
887:
867:
862:
806:
758:
750:
734:Archimedean spiral
678:
624:
536:
450:
387:
340:
314:
296:natural logarithms
294:being the base of
283:
262:
210:
149:
110:Archimedean spiral
71:equiangular spiral
67:logarithmic spiral
63:
51:
18:Logarithmic Spiral
4050:
4049:
3939:
3938:
3744:
3743:
3701:Benoît Mandelbrot
3601:Self-organization
3537:Natural selection
3527:Pattern formation
3335:, at cut-the-knot
3214:978-1-4503-7514-6
3114:978-1-84407-435-8
3087:978-981-02-0615-4
2985:978-0-471-27047-8
2958:978-0-262-02396-2
2931:978-0-89272-528-1
2867:978-0-486-60509-8
2827:978-0-7679-0815-3
2798:978-1-4027-3522-6
2773:978-3-319-47372-7
2556:tropical cyclones
2515:classical pursuit
2367:Fibonacci numbers
2315:
2251:
2249:
2181:
2142:
2140:
2111:{\displaystyle 0}
1676:Rotating, scaling
1581:
1466:
1329:
1318:
1252:
1231:
1115:
1111:
1033:
1012:
1009:
956:{\displaystyle a}
877:with pitch angle
860:
286:{\displaystyle e}
257:
241:
126:polar coordinates
57:A section of the
16:(Redirected from
4085:
3876:
3875:
3855:Boerdijk–Coxeter
3832:
3831:
3771:
3764:
3757:
3748:
3747:
3552:Sexual selection
3514:
3404:
3397:
3390:
3381:
3380:
3348:history and math
3318:
3317:
3295:
3294:
3268:
3262:
3261:
3233:
3227:
3226:
3188:
3182:
3181:
3178:10.1109/5.119570
3149:
3143:
3142:
3140:
3139:
3125:
3119:
3118:
3098:
3092:
3091:
3069:
3063:
3062:
3048:
3042:
3039:
3033:
3032:
3030:
3020:
3011:(2): 1074–1083.
2996:
2990:
2989:
2969:
2963:
2962:
2942:
2936:
2935:
2915:
2909:
2908:
2878:
2872:
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2838:
2832:
2831:
2819:
2809:
2803:
2802:
2784:
2778:
2777:
2751:
2745:
2744:
2734:
2658:
2612:
2600:
2500:
2498:
2497:
2492:
2490:
2489:
2461:
2459:
2458:
2453:
2420:Whirlpool Galaxy
2411:
2391:
2347:
2345:
2344:
2339:
2328:The pitch angle
2327:
2325:
2324:
2319:
2316:
2313:
2311:
2306:
2275:
2274:
2262:
2261:
2245:
2244:
2229:
2228:
2207:
2206:
2182:
2179:
2177:
2172:
2117:
2115:
2114:
2109:
2065:
2063:
2062:
2057:
2055:
2054:
2039:
2038:
2023:
2022:
2007:
2006:
1987:
1985:
1984:
1979:
1937:
1935:
1934:
1929:
1927:
1926:
1884:
1882:
1881:
1876:
1833:
1832:
1806:
1804:
1803:
1798:
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1766:
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1704:
1702:
1701:
1696:
1694:
1693:
1670:
1668:
1667:
1662:
1613:
1611:
1610:
1605:
1599:
1598:
1583:
1574:
1558:
1556:
1555:
1550:
1548:
1547:
1519:
1517:
1516:
1511:
1506:
1484:Circle inversion
1477:
1475:
1474:
1469:
1467:
1465:
1457:
1456:
1455:
1446:
1445:
1427:
1426:
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1400:
1372:
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1327:
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1306:
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1301:
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1277:
1276:
1250:
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1234:
1232:
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1219:
1215:
1214:
1193:
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1176:
1171:
1170:
1161:
1160:
1139:
1138:
1123:
1122:
1116:
1104:
1103:
1094:
1093:
1085:
1084:
1072:
1071:
1044:
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1041:
1036:
1034:
1029:
1018:
1013:
1011:
1010:
1008:
1007:
992:
983:
962:
960:
959:
954:
942:
940:
939:
934:
922:
920:
919:
914:
896:
894:
893:
888:
876:
874:
873:
868:
863:
861:
858:
815:
813:
812:
807:
788:
787:
687:
685:
684:
679:
674:
673:
633:
631:
630:
625:
593:
592:
545:
543:
542:
537:
523:
522:
488:
487:
459:
457:
456:
451:
396:
394:
393:
388:
386:
385:
349:
347:
346:
341:
323:
321:
320:
315:
292:
290:
289:
284:
271:
269:
268:
263:
258:
250:
242:
234:
219:
217:
216:
211:
206:
191:
190:
158:
156:
155:
150:
101:, who called it
21:
4093:
4092:
4088:
4087:
4086:
4084:
4083:
4082:
4053:
4052:
4051:
4046:
3935:
3889:
3874:
3833:
3829:
3824:
3788:
3775:
3745:
3740:
3714:
3607:
3515:
3506:
3413:
3408:
3346:Spira mirabilis
3342:
3299:
3298:
3291:
3269:
3265:
3234:
3230:
3215:
3189:
3185:
3150:
3146:
3137:
3135:
3127:
3126:
3122:
3115:
3099:
3095:
3088:
3078:Spiral symmetry
3070:
3066:
3049:
3045:
3040:
3036:
2997:
2993:
2986:
2970:
2966:
2959:
2943:
2939:
2932:
2916:
2912:
2879:
2875:
2868:
2852:
2848:
2839:
2835:
2828:
2810:
2806:
2799:
2785:
2781:
2774:
2752:
2748:
2735:
2731:
2726:
2707:List of spirals
2693:
2674:
2673:
2672:
2666:
2661:
2660:
2659:
2625:spiral antennas
2620:
2619:
2618:
2617:
2616:
2613:
2605:
2604:
2601:
2590:
2513:to its prey in
2485:
2481:
2467:
2464:
2463:
2441:
2438:
2437:
2426:
2425:
2424:
2423:
2422:
2416:spiral galaxies
2412:
2404:
2403:
2392:
2381:
2375:
2355:
2333:
2330:
2329:
2312:
2267:
2263:
2257:
2253:
2252:
2250:
2237:
2233:
2215:
2211:
2193:
2189:
2178:
2143:
2141:
2123:
2120:
2119:
2103:
2100:
2099:
2050:
2046:
2034:
2030:
2018:
2014:
2002:
1998:
1993:
1990:
1989:
1943:
1940:
1939:
1922:
1918:
1910:
1907:
1906:
1894:Self-similarity
1819:
1815:
1812:
1809:
1808:
1789:
1785:
1778:
1774:
1772:
1769:
1768:
1748:
1744:
1736:
1732:
1725:
1721:
1710:
1707:
1706:
1689:
1685:
1683:
1680:
1679:
1626:
1623:
1622:
1588:
1584:
1572:
1564:
1561:
1560:
1540:
1536:
1525:
1522:
1521:
1502:
1491:
1488:
1487:
1458:
1451:
1447:
1441:
1437:
1422:
1418:
1412:
1408:
1401:
1399:
1391:
1388:
1387:
1352:
1349:
1348:
1326:
1324:
1307:
1297:
1293:
1286:
1284:
1272:
1268:
1248:
1245:
1244:
1220:
1210:
1206:
1188:
1184:
1177:
1175:
1166:
1165:
1156:
1152:
1134:
1130:
1118:
1117:
1099:
1095:
1092:
1080:
1076:
1067:
1063:
1055:
1052:
1051:
1019:
1017:
1003:
999:
991:
987:
982:
974:
971:
970:
948:
945:
944:
928:
925:
924:
902:
899:
898:
882:
879:
878:
857:
855:
834:
831:
830:
780:
776:
765:
762:
761:
742:
714:self-similarity
710:Jacob Bernoulli
700:Spira mirabilis
696:
692:Spira mirabilis
654:
650:
639:
636:
635:
585:
581:
554:
551:
550:
515:
511:
480:
476:
465:
462:
461:
402:
399:
398:
378:
374:
363:
360:
359:
356:
329:
326:
325:
303:
300:
299:
278:
275:
274:
249:
233:
225:
222:
221:
202:
183:
179:
168:
165:
164:
132:
129:
128:
122:
103:Spira mirabilis
99:Jacob Bernoulli
39:
28:
23:
22:
15:
12:
11:
5:
4091:
4081:
4080:
4075:
4070:
4065:
4048:
4047:
4045:
4044:
4039:
4034:
4029:
4024:
4019:
4012:
4011:
4010:
4000:
3995:
3990:
3985:
3980:
3975:
3974:
3973:
3968:
3963:
3953:
3947:
3945:
3941:
3940:
3937:
3936:
3934:
3933:
3932:
3931:
3921:
3916:
3911:
3906:
3901:
3896:
3891:
3887:
3882:
3880:
3873:
3872:
3867:
3862:
3857:
3852:
3847:
3841:
3839:
3835:
3834:
3827:
3825:
3823:
3822:
3817:
3812:
3807:
3802:
3796:
3794:
3790:
3789:
3774:
3773:
3766:
3759:
3751:
3742:
3741:
3739:
3738:
3733:
3728:
3722:
3720:
3716:
3715:
3713:
3712:
3711:
3710:
3698:
3693:
3692:
3691:
3679:
3678:
3677:
3665:
3663:Wilson Bentley
3660:
3658:Joseph Plateau
3655:
3650:
3645:
3644:
3643:
3631:
3626:
3621:
3615:
3613:
3609:
3608:
3606:
3605:
3604:
3603:
3598:
3596:Plateau's laws
3593:
3591:Fluid dynamics
3588:
3578:
3577:
3576:
3571:
3566:
3556:
3555:
3554:
3549:
3544:
3539:
3529:
3523:
3521:
3517:
3516:
3509:
3507:
3505:
3504:
3499:
3494:
3489:
3484:
3483:
3482:
3477:
3472:
3467:
3457:
3452:
3447:
3442:
3437:
3432:
3427:
3421:
3419:
3415:
3414:
3407:
3406:
3399:
3392:
3384:
3378:
3377:
3372:
3367:
3363:SpiralZoom.com
3359:
3354:
3349:
3341:
3340:External links
3338:
3337:
3336:
3326:
3319:
3297:
3296:
3289:
3263:
3228:
3213:
3183:
3164:(1): 103–112.
3144:
3120:
3113:
3093:
3086:
3064:
3043:
3034:
2991:
2984:
2964:
2957:
2937:
2930:
2910:
2891:(5498): 1857.
2873:
2866:
2846:
2840:Yates, R. C.:
2833:
2826:
2804:
2797:
2779:
2772:
2746:
2728:
2727:
2725:
2722:
2721:
2720:
2715:
2709:
2704:
2699:
2692:
2689:
2688:
2687:
2663:
2662:
2653:
2652:
2651:
2650:
2649:
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2647:
2639:
2628:
2614:
2607:
2606:
2602:
2595:
2594:
2593:
2592:
2591:
2589:
2586:
2585:
2584:
2574:
2559:
2548:
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2302:
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2270:
2266:
2260:
2256:
2248:
2243:
2240:
2236:
2232:
2227:
2224:
2221:
2218:
2214:
2210:
2205:
2202:
2199:
2196:
2192:
2187:
2175:
2171:
2168:
2165:
2161:
2158:
2155:
2152:
2149:
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2139:
2136:
2133:
2130:
2127:
2107:
2089:
2071:
2053:
2049:
2045:
2042:
2037:
2033:
2029:
2026:
2021:
2017:
2013:
2010:
2005:
2001:
1997:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1925:
1921:
1917:
1914:
1890:
1873:
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1855:
1852:
1849:
1846:
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1818:
1792:
1788:
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1335:
1323:
1316:
1313:
1310:
1305:
1300:
1296:
1292:
1289:
1283:
1280:
1275:
1271:
1267:
1264:
1261:
1258:
1255:
1229:
1226:
1223:
1218:
1213:
1209:
1205:
1202:
1199:
1196:
1191:
1187:
1183:
1180:
1174:
1169:
1164:
1159:
1155:
1151:
1148:
1145:
1142:
1137:
1133:
1129:
1126:
1121:
1114:
1110:
1107:
1102:
1098:
1091:
1088:
1083:
1079:
1075:
1070:
1066:
1062:
1059:
1045:
1032:
1028:
1025:
1022:
1016:
1006:
1002:
998:
995:
990:
986:
981:
978:
964:
952:
932:
912:
909:
906:
886:
866:
854:
850:
847:
844:
841:
838:
805:
802:
799:
796:
792:
786:
783:
779:
775:
772:
769:
741:
738:
695:
689:
677:
672:
669:
666:
663:
660:
657:
653:
649:
646:
643:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
591:
588:
584:
579:
576:
573:
570:
567:
564:
561:
558:
535:
532:
529:
526:
521:
518:
514:
510:
507:
504:
500:
497:
494:
491:
486:
483:
479:
475:
472:
469:
449:
446:
443:
440:
437:
434:
431:
427:
424:
421:
418:
415:
412:
409:
406:
384:
381:
377:
373:
370:
367:
355:
352:
339:
336:
333:
313:
310:
307:
282:
261:
256:
253:
248:
245:
240:
237:
232:
229:
209:
205:
201:
198:
194:
189:
186:
182:
178:
175:
172:
148:
145:
142:
139:
136:
121:
118:
91:Albrecht Dürer
59:Mandelbrot set
26:
9:
6:
4:
3:
2:
4090:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4060:
4058:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4017:
4013:
4009:
4006:
4005:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3972:
3969:
3967:
3964:
3962:
3959:
3958:
3957:
3954:
3952:
3949:
3948:
3946:
3942:
3930:
3927:
3926:
3925:
3922:
3920:
3917:
3915:
3912:
3910:
3907:
3905:
3902:
3900:
3897:
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3877:
3871:
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3866:
3863:
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3846:
3843:
3842:
3840:
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3808:
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3801:
3798:
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3709:
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3685:
3684:
3683:
3680:
3676:
3675:
3671:
3670:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3653:Ernst Haeckel
3651:
3649:
3648:Adolf Zeising
3646:
3642:
3641:
3637:
3636:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3614:
3610:
3602:
3599:
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3594:
3592:
3589:
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3584:
3583:
3582:
3579:
3575:
3572:
3570:
3567:
3565:
3562:
3561:
3560:
3557:
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3548:
3545:
3543:
3540:
3538:
3535:
3534:
3533:
3530:
3528:
3525:
3524:
3522:
3518:
3513:
3503:
3500:
3498:
3495:
3493:
3492:Vortex street
3490:
3488:
3485:
3481:
3478:
3476:
3473:
3471:
3470:Quasicrystals
3468:
3466:
3463:
3462:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3422:
3420:
3416:
3412:
3405:
3400:
3398:
3393:
3391:
3386:
3385:
3382:
3376:
3373:
3371:
3368:
3365:
3364:
3360:
3358:
3355:
3353:
3350:
3347:
3344:
3343:
3334:
3330:
3327:
3324:
3320:
3315:
3314:
3309:
3306:
3301:
3300:
3292:
3290:9783319939490
3286:
3282:
3278:
3274:
3267:
3259:
3255:
3251:
3247:
3243:
3239:
3232:
3224:
3220:
3216:
3210:
3206:
3202:
3198:
3194:
3187:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3148:
3134:
3130:
3124:
3116:
3110:
3106:
3105:
3097:
3089:
3083:
3079:
3075:
3068:
3060:
3056:
3055:
3047:
3038:
3029:
3024:
3019:
3014:
3010:
3006:
3002:
2995:
2987:
2981:
2977:
2976:
2968:
2960:
2954:
2950:
2949:
2941:
2933:
2927:
2923:
2922:
2914:
2906:
2902:
2898:
2894:
2890:
2886:
2885:
2877:
2869:
2863:
2859:
2858:
2850:
2843:
2837:
2829:
2823:
2818:
2817:
2808:
2800:
2794:
2790:
2783:
2775:
2769:
2765:
2761:
2757:
2750:
2742:
2741:
2733:
2729:
2719:
2716:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2694:
2684:
2680:
2679:rock climbing
2676:
2675:
2670:
2665:
2657:
2644:
2640:
2637:
2633:
2632:laser cutters
2629:
2626:
2622:
2621:
2611:
2599:
2582:
2578:
2575:
2572:
2568:
2564:
2560:
2557:
2553:
2549:
2546:
2542:
2539:
2535:
2531:
2527:
2523:
2519:
2516:
2512:
2508:
2507:
2506:
2486:
2482:
2478:
2475:
2472:
2469:
2449:
2446:
2443:
2435:
2432:Cutaway of a
2430:
2421:
2417:
2410:
2401:
2397:
2390:
2380:
2370:
2368:
2364:
2360:
2359:golden spiral
2335:
2307:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2271:
2268:
2264:
2258:
2254:
2246:
2241:
2238:
2234:
2230:
2225:
2222:
2219:
2216:
2212:
2208:
2200:
2194:
2190:
2173:
2169:
2166:
2163:
2156:
2153:
2150:
2147:
2137:
2131:
2125:
2105:
2097:
2093:
2090:
2087:
2083:
2079:
2075:
2072:
2069:
2051:
2047:
2043:
2040:
2035:
2031:
2027:
2024:
2019:
2015:
2011:
2008:
2003:
1999:
1995:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1923:
1919:
1915:
1912:
1904:
1900:
1896:
1895:
1891:
1888:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1829:
1826:
1823:
1820:
1816:
1807:. Scaling by
1790:
1786:
1782:
1779:
1775:
1752:
1749:
1745:
1737:
1733:
1729:
1726:
1722:
1718:
1715:
1712:
1690:
1686:
1677:
1674:
1673:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1621:Examples for
1619:
1601:
1595:
1592:
1589:
1585:
1578:
1575:
1569:
1566:
1544:
1541:
1537:
1533:
1530:
1527:
1507:
1503:
1499:
1493:
1485:
1482:
1479:
1462:
1459:
1452:
1442:
1438:
1431:
1428:
1423:
1413:
1409:
1402:
1396:
1393:
1386:
1383:
1380:
1376:
1360:
1357:
1354:
1314:
1311:
1308:
1298:
1294:
1287:
1281:
1273:
1269:
1265:
1259:
1253:
1227:
1224:
1221:
1211:
1207:
1200:
1197:
1189:
1185:
1178:
1172:
1157:
1153:
1146:
1143:
1135:
1131:
1124:
1112:
1108:
1105:
1100:
1096:
1089:
1081:
1077:
1073:
1068:
1064:
1057:
1049:
1046:
1030:
1026:
1023:
1020:
1014:
1004:
1000:
996:
993:
988:
984:
979:
976:
968:
965:
950:
930:
910:
907:
904:
884:
848:
845:
842:
839:
836:
828:
827:
823:
822:
821:
819:
803:
800:
797:
794:
790:
784:
781:
777:
773:
770:
767:
754:
746:
737:
735:
731:
727:
723:
719:
715:
711:
706:
702:
701:
693:
688:
675:
670:
664:
661:
658:
651:
647:
644:
641:
618:
615:
612:
609:
606:
603:
600:
597:
594:
589:
586:
582:
577:
574:
571:
568:
565:
562:
559:
549:
548:complex plane
533:
530:
527:
524:
519:
516:
512:
508:
505:
502:
498:
495:
492:
489:
484:
481:
477:
473:
470:
467:
444:
441:
438:
435:
432:
429:
425:
422:
419:
416:
413:
410:
407:
382:
379:
375:
371:
368:
365:
351:
337:
334:
331:
311:
308:
305:
297:
293:
280:
259:
254:
251:
246:
243:
238:
235:
230:
227:
207:
199:
196:
192:
187:
184:
180:
176:
173:
170:
162:
143:
140:
137:
127:
117:
115:
111:
106:
104:
100:
96:
92:
88:
84:
80:
76:
75:growth spiral
72:
68:
60:
55:
48:
43:
37:
33:
19:
4078:Plane curves
4073:Exponentials
4014:
4002:
3879:Biochemistry
3705:
3686:
3672:
3638:
3573:
3564:Chaos theory
3487:Tessellation
3362:
3321:Jim Wilson,
3311:
3272:
3266:
3241:
3237:
3231:
3196:
3186:
3161:
3157:
3147:
3136:. Retrieved
3132:
3123:
3103:
3096:
3077:
3067:
3053:
3046:
3037:
3008:
3004:
2994:
2974:
2967:
2947:
2940:
2920:
2913:
2888:
2882:
2876:
2856:
2849:
2841:
2836:
2815:
2807:
2788:
2782:
2755:
2749:
2739:
2732:
2712:Mice problem
2641:Logarithmic
2623:Logarithmic
2504:
2414:The arms of
2363:golden ratio
2356:
2091:
2086:pedal curves
2073:
2067:
1902:
1892:
1886:
1675:
1480:
1385:Sector area:
1384:
1377:even before
1243:Especially:
1047:
966:
824:
759:
699:
698:
697:
691:
357:
123:
107:
102:
79:self-similar
74:
70:
66:
64:
4027:Pitch angle
4003:Logarithmic
3951:Archimedean
3914:Polyproline
3682:Alan Turing
3640:Liber Abaci
3559:Mathematics
3465:in crystals
3455:Soap bubble
3450:Phyllotaxis
2314:log. spiral
826:Pitch angle
720:shells and
161:logarithmic
4068:Logarithms
4057:Categories
4016:On Spirals
3966:Hyperbolic
3629:Empedocles
3624:Pythagoras
3542:Camouflage
3480:in biology
3475:in flowers
3445:Parastichy
3244:(8): 400.
3138:2020-12-26
2724:References
2646:stability.
2563:biological
2084:, and the
1885:gives the
1481:Inversion:
1048:Arc length
859:constant !
740:Properties
120:Definition
4037:Spirangle
4032:Theodorus
3971:Poinsot's
3961:Epispiral
3805:Curvature
3800:Algebraic
3731:Emergence
3634:Fibonacci
3313:MathWorld
3258:1678-5878
3223:222805227
3018:1309.4308
2905:180484583
2702:Epispiral
2571:polygonal
2530:Milky Way
2487:∘
2479:≈
2473:
2373:In nature
2336:α
2308:⏟
2298:
2283:
2231:⋅
2186:→
2174:⏟
2078:involutes
2052:∘
2044:−
2036:∘
2028:−
2020:∘
2012:−
2004:∘
1996:−
1924:∘
1913:α
1899:congruent
1854:±
1845:±
1830:π
1787:φ
1780:−
1753:φ
1734:φ
1727:−
1687:φ
1596:φ
1590:−
1545:φ
1497:→
1439:φ
1429:−
1410:φ
1315:α
1312:
1295:φ
1270:φ
1263:∞
1260:−
1228:α
1225:
1208:φ
1198:−
1186:φ
1154:φ
1144:−
1132:φ
1078:φ
1065:φ
1027:α
1024:
977:κ
967:Curvature
931:α
885:α
843:α
840:
798:≠
785:φ
726:headstone
722:sunflower
671:φ
619:φ
616:
604:φ
601:
590:φ
531:φ
528:
520:φ
496:φ
493:
485:φ
445:φ
442:
423:φ
420:
383:φ
335:≠
247:
228:φ
200:∈
197:φ
188:φ
144:φ
95:Descartes
3993:Involute
3988:Fermat's
3929:Collagen
3865:Symmetry
3460:Symmetry
3418:Patterns
2691:See also
2573:figures.
2567:mollusks
2526:galaxies
2434:nautilus
2082:evolutes
1903:Example:
1379:calculus
1328:finite !
718:nautilus
4063:Spirals
4022:Padovan
3956:Cotes's
3944:Spirals
3850:Antenna
3838:Helices
3810:Gallery
3786:helices
3778:Spirals
3719:Related
3586:Crystal
3581:Physics
3569:Fractal
3547:Mimicry
3532:Biology
3440:Meander
3166:Bibcode
2884:Science
2400:Iceland
546:In the
81:spiral
4008:Golden
3924:Triple
3904:Double
3870:Triple
3820:Topics
3793:Curves
3782:curves
3612:People
3520:Causes
3287:
3256:
3221:
3211:
3133:hpi.de
3111:
3084:
2982:
2955:
2928:
2903:
2864:
2824:
2795:
2770:
2545:cornea
2528:. The
2470:arctan
2450:0.1759
2094:: The
1889:curve.
1347:, if
1251:
923:angle
818:Spiral
298:, and
87:spiral
3983:Euler
3978:Doyle
3919:Super
3894:Alpha
3845:Angle
3619:Plato
3425:Crack
3219:S2CID
3061:–357.
3013:arXiv
2901:S2CID
2561:Many
2552:bands
2536:, or
2521:line.
2398:over
705:Latin
272:with
83:curve
77:is a
73:, or
47:pitch
4042:Ulam
3998:List
3899:Beta
3860:Hemi
3815:List
3784:and
3497:Wave
3435:Foam
3430:Dune
3285:ISBN
3254:ISSN
3209:ISBN
3109:ISBN
3082:ISBN
2980:ISBN
2953:ISBN
2926:ISBN
2862:ISBN
2822:ISBN
2793:ISBN
2768:ISBN
2636:kerf
2550:The
2511:hawk
2357:The
2180:line
1938:and
1887:same
1358:>
309:>
159:the
89:was
49:10°)
3277:doi
3246:doi
3201:doi
3174:doi
3059:356
3023:doi
3009:436
2893:doi
2889:290
2760:doi
2677:In
2554:of
2394:An
2295:sin
2280:cos
2048:253
2032:218
2016:173
2000:109
1309:sin
1222:sin
1021:cos
837:tan
820:):
613:sin
598:cos
525:sin
490:cos
460:by
439:sin
417:cos
220:or
124:In
4059::
3909:Pi
3888:10
3780:,
3331:,
3310:.
3283:.
3252:.
3242:42
3240:.
3217:.
3207:.
3195:.
3172:.
3162:80
3160:.
3156:.
3131:.
3021:.
3007:.
3003:.
2899:.
2887:.
2766:.
2681:,
2667:A
2483:10
2369:.
2118::
2080:,
2070:).
1920:20
1050::
969::
963:.)
829::
703:,
634::
324:,
244:ln
69:,
65:A
3886:3
3770:e
3763:t
3756:v
3403:e
3396:t
3389:v
3316:.
3293:.
3279::
3260:.
3248::
3225:.
3203::
3180:.
3176::
3168::
3141:.
3117:.
3090:.
3031:.
3025::
3015::
2988:.
2961:.
2934:.
2907:.
2895::
2870:.
2830:.
2801:.
2776:.
2762::
2743:.
2501:.
2476:b
2447:=
2444:b
2304:)
2301:t
2292:i
2289:+
2286:t
2277:(
2272:t
2269:k
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2259:b
2255:e
2247:=
2242:t
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2235:e
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2220:t
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2209:=
2204:)
2201:t
2198:(
2195:z
2191:e
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2164:+
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2157:b
2154:+
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2148:k
2145:(
2138:=
2135:)
2132:t
2129:(
2126:z
2106:0
2041:,
2025:,
2009:,
1976:5
1973:,
1970:4
1967:,
1964:3
1961:,
1958:2
1955:,
1952:1
1949:=
1946:a
1916:=
1872:,
1869:.
1866:.
1863:.
1860:,
1857:2
1851:,
1848:1
1842:=
1839:n
1836:,
1827:2
1824:n
1821:k
1817:e
1791:0
1783:k
1776:e
1750:k
1746:e
1738:0
1730:k
1723:e
1719:a
1716:=
1713:r
1691:0
1659:5
1656:,
1653:4
1650:,
1647:3
1644:,
1641:2
1638:,
1635:1
1632:=
1629:a
1602:.
1593:k
1586:e
1579:a
1576:1
1570:=
1567:r
1542:k
1538:e
1534:a
1531:=
1528:r
1508:r
1504:/
1500:1
1494:r
1486:(
1463:k
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1443:1
1435:(
1432:r
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1420:)
1414:2
1406:(
1403:r
1397:=
1394:A
1361:0
1355:k
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1322:(
1304:)
1299:2
1291:(
1288:r
1282:=
1279:)
1274:2
1266:,
1257:(
1254:L
1217:)
1212:1
1204:(
1201:r
1195:)
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1182:(
1179:r
1173:=
1168:)
1163:)
1158:1
1150:(
1147:r
1141:)
1136:2
1128:(
1125:r
1120:(
1113:k
1109:1
1106:+
1101:2
1097:k
1090:=
1087:)
1082:2
1074:,
1069:1
1061:(
1058:L
1031:r
1015:=
1005:2
1001:k
997:+
994:1
989:r
985:1
980:=
951:a
911:0
908:=
905:k
865:)
853:(
849:k
846:=
804:,
801:0
795:k
791:,
782:k
778:e
774:a
771:=
768:r
676:.
668:)
665:i
662:+
659:k
656:(
652:e
648:a
645:=
642:z
622:)
610:i
607:+
595:=
587:i
583:e
578:,
575:y
572:i
569:+
566:x
563:=
560:z
557:(
534:.
517:k
513:e
509:a
506:=
503:y
499:,
482:k
478:e
474:a
471:=
468:x
448:)
436:r
433:=
430:y
426:,
414:r
411:=
408:x
405:(
380:k
376:e
372:a
369:=
366:r
338:0
332:k
312:0
306:a
281:e
260:,
255:a
252:r
239:k
236:1
231:=
208:,
204:R
193:,
185:k
181:e
177:a
174:=
171:r
147:)
141:,
138:r
135:(
38:.
20:)
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