Editing Logarithmic spiral (section)

==In nature==
{{further|Patterns in nature#Spirals}}
{{Multiple image|total_width=480
|image1=Low pressure system over Iceland.jpg|caption1=An [[extratropical cyclone]] over [[Iceland]] shows an approximately logarithmic spiral pattern
|image2=Messier51 sRGB.jpg|caption2=The arms of [[Spiral galaxy|spiral galaxies]] often have the shape of a logarithmic spiral, here the [[Whirlpool Galaxy]]}}
[[File:Nautilus Cutaway with Logarithmic Spiral.png|thumb|right|Cutaway of a [[nautilus]] shell showing the chambers arranged in an approximately logarithmic spiral.  The plotted spiral (dashed blue curve) is based on growth rate parameter <math>b = 0.1759</math>, resulting in a pitch of <math>\arctan b \approx 10^\circ</math>.]]

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follow some examples and reasons:
*The approach of a [[hawk]] to its prey in [[classical pursuit]], 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.<ref>{{cite journal |first=Gilbert J. |last=Chin |date=8 December 2000 |title=Organismal Biology: Flying Along a Logarithmic Spiral |journal=[[Science (journal)|Science]] |volume=290 |issue=5498 |page=1857 |doi=10.1126/science.290.5498.1857c|s2cid=180484583 }}</ref>
*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 line.<ref>
{{cite book
 | title = Discovering Moths: Nighttime Jewels in Your Own Backyard
 | author = John Himmelman
 | publisher = Down East Enterprise Inc
 | year = 2002
 | isbn = 978-0-89272-528-1
 | page = 63
 | url = https://books.google.com/books?id=iGn6ohfKhbAC&pg=PA63
 }}</ref>  In the same token, a [[rhumb line]] approximates a logarithmic spiral close to a pole.
*The arms of spiral [[galaxy|galaxies]].<ref>
{{cite book
 | title = Spiral structure in galaxies: a density wave theory
 | author = G. Bertin and C. C. Lin
 | publisher = MIT Press
 | year = 1996
 | isbn = 978-0-262-02396-2
 | page = 78
 | url = https://books.google.com/books?id=06yfwrdpTk4C&pg=PA78
 }}</ref> The [[Milky Way]] galaxy has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees.<ref>
{{cite book
 | title = The universal book of mathematics: from Abracadabra to Zeno's paradoxes
 | author = David J. Darling
 | publisher = John Wiley and Sons
 | year = 2004
 | isbn = 978-0-471-27047-8
 | page = 188
 | url = https://books.google.com/books?id=nnpChqstvg0C&pg=PA188
 }}</ref> However, although spiral galaxies have often been modeled as logarithmic spirals, [[Archimedean spiral]]s, or [[hyperbolic spiral]]s, 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.<ref>{{cite journal
 | last1 = Savchenko | first1 = S. S.
 | last2 = Reshetnikov | first2 = V. P.
 | date = September 2013
 | doi = 10.1093/mnras/stt1627
 | issue = 2
 | journal = Monthly Notices of the Royal Astronomical Society
 | pages = 1074–1083
 | title = Pitch angle variations in spiral galaxies
 | volume = 436| doi-access = free
 | arxiv = 1309.4308
 }}</ref>
*The nerves of the [[cornea]] (this is, corneal nerves of the subepithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).<ref name="Yu">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.</ref>
*The [[rainband|bands]] of [[tropical cyclone]]s, such as hurricanes.<ref>
{{cite book
 | title = Treatise on physics, Volume 1
 | author = Andrew Gray
 | publisher = Churchill
 | year = 1901
 | pages = [https://archive.org/details/atreatiseonphys02graygoog/page/n386 356]–357
 | url = https://archive.org/details/atreatiseonphys02graygoog
 }}</ref>
*Many [[Biology|biological]] structures including the shells of [[Mollusca|mollusk]]s.<ref>
{{cite book
 | title = Spiral symmetry
 | chapter = The form, function, and synthesis of the molluscan shell
 | author = Michael Cortie
 | editor = István Hargittai and Clifford A. Pickover
 | publisher = World Scientific
 | year = 1992
 | isbn = 978-981-02-0615-4
 | page = 370
 | chapter-url = https://books.google.com/books?id=Ga8aoiIUx1gC&pg=PA370
 }}</ref>  In these cases, the reason may be construction from expanding similar shapes, as is the case for [[polygon]]al figures.
*[[Logarithmic spiral beaches]] can form as the result of wave refraction and diffraction by the coast. [[Half Moon Bay (California)]] is an example of such a type of beach.<ref>
{{cite book
 | title = Beach management: principles and practice
 | author = Allan Thomas Williams and Anton Micallef
 | publisher = Earthscan
 | year = 2009
 | isbn = 978-1-84407-435-8
 | page = 14
 | url = https://books.google.com/books?id=z_vKEMeJXKYC&pg=PA14
 }}</ref>
{{clear}}