Editing Spiral

{{Short description|Curve that winds around a central point}}
{{Other uses}}
[[File:NautilusCutawayLogarithmicSpiral.jpg|right|thumb|Cutaway of a [[nautilus]] shell showing the chambers arranged in an approximately [[logarithmic spiral]]]]
In [[mathematics]], a '''spiral''' is a [[curve]] which emanates from a point, moving further away as it revolves around the point.<ref>{{Cite web|title=Spiral {{!}} mathematics|url=https://www.britannica.com/science/spiral-mathematics|access-date=2020-10-08|website=Encyclopedia Britannica|language=en}}</ref><ref>{{Cite web|title=Spiral Definition (Illustrated Mathematics Dictionary)|url=https://www.mathsisfun.com/definitions/spiral.html|access-date=2020-10-08|website=www.mathsisfun.com}}</ref><ref>{{Cite web|title=spiral.htm|url=https://www.math.tamu.edu/~dallen/digitalcam/spiral/spiral.htm|access-date=2020-10-08|website=www.math.tamu.edu}}</ref><ref>{{Cite web|date=2017-06-01|title=Math Patterns in Nature|url=https://www.fi.edu/math-patterns-nature|access-date=2020-10-08|website=The Franklin Institute|language=en}}</ref> It is a subtype of [[whorl]]ed patterns, a broad group that also includes [[concentric objects]].

==Two-dimensional==
{{main|List of spirals}}

[[File:Six types of spirals.png|thumb|Spirals generated by 6 mathematical relationships between radius and angle.]]

A [[two-dimensional]], or plane, spiral may be easily described using [[polar coordinates]], where the [[radius]] <math>r</math> is a [[monotonic]] [[continuous function]] of angle <math>\varphi</math>:
* <math>r=r(\varphi)\; .</math>
The circle would be regarded as a [[degenerate (mathematics)|degenerate]] case (the [[Function (mathematics)|function]] not being strictly monotonic, but rather [[Constant (mathematics)|constant]]).

In ''<math>x</math>-<math>y</math>-coordinates'' the curve has the parametric representation:
* <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; .</math>

=== Examples ===

Some of the most important sorts of two-dimensional spirals include:

* The [[Archimedean spiral]]: <math>r=a \varphi </math>
* The [[hyperbolic spiral]]: <math>r = a/ \varphi</math>
* [[Fermat's spiral]]: <math>r= a\varphi^{1/2}</math>
* The [[Lituus (mathematics)|lituus]]: <math>r = a\varphi^{-1/2}</math>
* The [[logarithmic spiral]]: <math>r=ae^{k\varphi}</math> 
* The [[Cornu spiral]] or ''clothoid''
* The [[Fibonacci spiral]] and [[golden spiral]]
* The [[Spiral of Theodorus]]: an approximation of the Archimedean spiral composed of contiguous right triangles
* The [[involute]] of a circle
<gallery>
Image:Archimedean spiral.svg|Archimedean spiral
Image:Hyperspiral.svg|hyperbolic spiral
Image:Fermat's spiral.svg|Fermat's spiral
Image:Lituus.svg|lituus
Image:Logarithmic Spiral Pylab.svg|logarithmic spiral
Image:Cornu Spiral.svg|Cornu spiral
Image:Spiral of Theodorus.svg|spiral of Theodorus
Image:Fibonacci_spiral.svg|Fibonacci Spiral (golden spiral)
Image:Archimedean-involute-circle-spirals-comparison.svg|The involute of a circle (black) is not identical to the Archimedean spiral (red).
</gallery>
[[File:Schraublinie-hyp-spirale.svg|thumb|upright=0.6|Hyperbolic spiral as central projection of a helix]]

An ''Archimedean spiral'' is, for example, generated while coiling a carpet.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Archimedean Spiral|url=https://mathworld.wolfram.com/ArchimedeanSpiral.html|access-date=2020-10-08|website=mathworld.wolfram.com|language=en}}</ref>

A ''hyperbolic spiral'' appears as image of a helix with a special central projection (see diagram). A hyperbolic spiral is some times called ''reciproke'' spiral, because it is the image of an Archimedean spiral with a circle-inversion (see below).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Hyperbolic Spiral|url=https://mathworld.wolfram.com/HyperbolicSpiral.html|access-date=2020-10-08|website=mathworld.wolfram.com|language=en}}</ref>

The name ''logarithmic spiral'' is due to the equation <math>\varphi=\tfrac{1}{k}\cdot \ln \tfrac{r}{a}</math>. Approximations of this are found in nature.

Spirals which do not fit into this scheme of the first 5 examples:

A ''Cornu spiral'' has two asymptotic points.<br>
The ''spiral of Theodorus'' is a polygon.<br>
The ''Fibonacci Spiral'' consists of a sequence of circle arcs.<br>
The ''involute of a circle'' looks like an Archimedean, but is not: see [[Involute#Examples]].

=== Geometric properties ===
The following considerations are dealing with spirals, which can be described by a polar equation <math>r=r(\varphi)</math>, especially for the cases <math>r(\varphi)=a\varphi^n</math> (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral <math>r=ae^{k\varphi}</math>.

[[File:Sektor-steigung-pk-def.svg|thumb|Definition of sector (light blue) and polar slope angle <math>\alpha</math>]]

;Polar slope angle
The angle <math>\alpha</math> between the spiral tangent and the corresponding polar circle (see diagram) is called ''angle of the polar slope'' and <math>\tan \alpha</math> the ''polar slope''.

From [[polar coordinate system#Vector calculus|vector calculus in polar coordinates]] one gets the formula
:<math>\tan\alpha=\frac{r'}{r}\ .</math>

Hence the slope of the spiral <math>\;r=a\varphi^n \;</math>  is
* <math>\tan\alpha=\frac{n}{\varphi}\ .</math>

In case of an ''Archimedean spiral'' (<math>n=1</math>) the polar slope is <math>\; \tan\alpha=\tfrac{1}{\varphi}\ .</math>

In a ''logarithmic spiral'', <math>\ \tan\alpha=k\ </math> is constant.

;Curvature
The curvature <math>\kappa</math> of a curve with polar equation <math>r=r(\varphi)</math> is

:<math>\kappa = \frac{r^2 + 2(r')^2 - r\; r''}{(r^2+(r')^2)^{3/2}}\ .</math>

For a spiral with <math>r=a\varphi^n</math> one gets
* <math>\kappa = \dotsb = \frac{1}{a\varphi^{n-1}}\frac{\varphi^2+n^2+n}{(\varphi^2+n^2)^{3/2}}\ .</math>

In case of <math>n=1</math> ''(Archimedean spiral)''
<math>\kappa=\tfrac{\varphi^2+2}{a(\varphi^2+1)^{3/2}}</math>.<br> 
Only for <math>-1<n<0 </math> the spiral has an ''inflection point''.

The curvature of a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is <math>\; \kappa=\tfrac{1}{r\sqrt{1+k^2}} \; .</math>

;Sector area
The area of a sector of a curve (see diagram) with polar equation <math>r=r(\varphi)</math> is
:<math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} r(\varphi)^2\; d\varphi\ .</math>

For a spiral with equation <math>r=a\varphi^n\; </math> one gets

* <math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} a^2\varphi^{2n}\; d\varphi
=\frac{a^2}{2(2n+1)}\big(\varphi_2^{2n+1}- \varphi_1^{2n+1}\big)\ , \quad \text{if}\quad n\ne-\frac{1}{2},
</math>
:<math>A=\frac{1}{2}\int_{\varphi_1}^{\varphi_2} \frac{a^2}{\varphi}\; d\varphi
=\frac{a^2}{2}(\ln\varphi_2-\ln\varphi_1)\ ,\quad \text{if} \quad n=-\frac{1}{2}\ .</math>

The formula for a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is
<math>\ A=\tfrac{r(\varphi_2)^2-r(\varphi_1)^2)}{4k}\ .</math>

;Arc length
The length of an arc of a curve with polar equation <math>r=r(\varphi)</math> is
:<math>L=\int\limits_{\varphi_1}^{\varphi_2}\sqrt{\left(r^\prime(\varphi)\right)^2+r^2(\varphi)}\,\mathrm{d}\varphi \ .</math>

For the spiral <math>r=a\varphi^n\; </math> the length is

* <math>L=\int_{\varphi_1}^{\varphi_2} \sqrt{\frac{n^2r^2}{\varphi^2} +r^2}\; d\varphi
= a\int\limits_{\varphi_1}^{\varphi_2}\varphi^{n-1}\sqrt{n^2+\varphi^2}d\varphi
\ .</math>
Not all these integrals can be solved by a suitable table. In case of a Fermat's spiral, the integral can be expressed by [[elliptic integral]]s only.

The arc length of a ''logarithmic spiral'' <math>\; r=a e^{k\varphi} \;</math> is <math>\ L=\tfrac{\sqrt{k^2+1}}{k}\big(r(\varphi_2)-r(\varphi_1)\big) \ .</math>

;Circle inversion
The [[Circle inversion|inversion at the unit circle]] has in polar coordinates the simple description: <math>\ (r,\varphi) \mapsto (\tfrac{1}{r},\varphi)\ </math>.

* The image of a spiral <math>\ r= a\varphi^n\ </math> under the inversion at the unit circle is the spiral with polar equation <math>\ r= \tfrac{1}{a}\varphi^{-n}\ </math>. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.
:A logarithmic spiral <math>\; r=a e^{k\varphi} \;</math> is mapped onto the logarithmic spiral <math>\; r=\tfrac{1}{a} e^{-k\varphi} \; .</math>

=== Bounded spirals ===
[[File:Spiral-arctan-1-2.svg|thumb|upright=1.4|Bounded spirals:<br>
<math>r=a \arctan(k\varphi)</math> (left), <br>
<math>r=a (\arctan(k\varphi)+\pi/2) </math> (right)]]
The function <math>r(\varphi)</math> of a spiral is usually strictly monotonic, continuous
and un[[Bounded function|bounded]]. For the standard spirals  <math>r(\varphi)</math> is either a power function or an exponential function. If one chooses for <math>r(\varphi)</math> a ''bounded'' function, the spiral is bounded, too. A suitable bounded function is the [[arctan]] function:

;Example 1
Setting <math>\;r=a \arctan(k\varphi)\;</math> and the choice <math>\;k=0.1, a=4, \;\varphi\ge 0\;</math> gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius <math>\;r=a\pi/2\;</math> (diagram, left).
;Example 2
For <math>\;r=a (\arctan(k\varphi)+\pi/2)\;</math> and <math>\;k=0.2, a=2,\; -\infty<\varphi<\infty\;</math> one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius <math>\;r=a\pi\;</math> (diagram, right).

== Three-dimensional ==
=== Helices {{anchor|Spiral or helix}} ===
[[File:Schraube und archimedische Spirale.png|right|thumb|An Archimedean spiral (black), a helix (green), and a conical spiral (red)]]
Two major definitions of "spiral" in the [[American Heritage Dictionary]] are:<ref name=free>"[http://www.thefreedictionary.com/spiral Spiral], ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009.</ref>
# a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
# a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a [[helix]].

The first definition describes a [[Plane (mathematics)|planar]] curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a [[gramophone record]] closely approximates a plane spiral (and it is by the finite width and depth of the groove, but ''not'' by the wider spacing between than within tracks, that it falls short of being a perfect example); note that successive loops ''differ'' in diameter. In another example, the "center lines" of the arms of a [[spiral galaxy]] trace [[logarithmic spiral]]s.

The second definition includes two kinds of 3-dimensional relatives of spirals:
* A conical or [[volute spring]] (including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a [[Battery (electricity)|battery box]]), and the [[vortex]] that is created when water is draining in a sink is often described as a spiral, or as a [[conical helix]].
* Quite explicitly, definition 2 also includes a cylindrical coil spring and a strand of [[DNA]], both of which are fairly helical, so that "helix" is a more ''useful'' description than "spiral" for each of them. In general, "spiral" is seldom applied if successive "loops" of a curve have the same diameter.<ref name=free/>

In the side picture, the black curve at the bottom is an [[Archimedean spiral]], while the green curve is a helix. The curve shown in red is a conical spiral.

{{redirect|Space spiral|the building|Space Spiral}}

Two well-known spiral [[space curve]]s are ''conical spirals'' and ''spherical spirals'', defined below.
Another instance of space spirals is the ''toroidal spiral''.<ref name="von Seggern 1994 p. 241">{{cite book | last=von Seggern | first=D.H. | title=Practical Handbook of Curve Design and Generation | publisher=Taylor & Francis | year=1994 | isbn=978-0-8493-8916-0 | url=https://books.google.com/books?id=PVKXqob2dhAC&pg=PA241 | access-date=2022-03-03 | page=241}}</ref> A spiral wound around a helix,<ref name="Wolfram MathWorld 2002">{{cite web |date=2002-09-13 |title=Slinky -- from Wolfram MathWorld |url=https://mathworld.wolfram.com/Slinky.html |access-date=2022-03-03 |website=Wolfram MathWorld}}</ref> also known as ''double-twisted helix'',<ref name="Ugajin Ishimoto Kuroki Hirata 2001 pp. 437–451">{{cite journal | last1=Ugajin | first1=R. | last2=Ishimoto | first2=C. | last3=Kuroki | first3=Y. | last4=Hirata | first4=S. | last5=Watanabe | first5=S. | title=Statistical analysis of a multiply-twisted helix | journal=Physica A: Statistical Mechanics and Its Applications | publisher=Elsevier BV | volume=292 | issue=1–4 | year=2001 | issn=0378-4371 | doi=10.1016/s0378-4371(00)00572-0 | pages=437–451| bibcode=2001PhyA..292..437U }}</ref> represents objects such as [[coiled coil filament]]s.

=== Conical spirals ===
[[File:Spiral-cone-arch-s.svg|thumb|upright=0.8|Conical spiral with Archimedean spiral as floor plan]]
{{main|conical spiral}}
If in the <math>x</math>-<math>y</math>-plane a spiral with parametric representation 
:<math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi</math> 
is given, then there can be added a third coordinate <math>z(\varphi)</math>, such that the now space curve lies on the [[cone]] with equation <math>\;m(x^2+y^2)=(z-z_0)^2\ ,\  m>0\;</math>:
* <math>x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color{red}{z=z_0 + mr(\varphi)} \ .</math>

Spirals based on this procedure are called '''conical spirals'''.

;Example
Starting with an ''archimedean spiral'' <math>\;r(\varphi)=a\varphi\;</math> one gets the conical spiral (see diagram)
:<math>x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .</math>

=== Spherical spirals ===
[[File:Kugel-spirale-1-2.svg|thumb|upright=1.2|Clelia curve with <math>c=8</math>]]

Any [[cylindrical map projection]] can be used as the basis for a '''spherical spiral''': draw a straight line on the map and find its inverse projection on the sphere, a kind of [[spherical curve]].

One of the most basic families of spherical spirals is the [[Clelia curve]]s, which project to straight lines on an [[equirectangular projection]]. These are curves for which [[longitude]] and [[colatitude]] are in a linear relationship, analogous to Archimedean spirals in the plane; under the [[azimuthal equidistant projection]] a Clelia curve projects to a planar Archimedean spiral.

If one represents a unit sphere by [[spherical coordinates]]

: <math>
x = \sin \theta \, \cos \varphi, \quad
y = \sin \theta \, \sin \varphi, \quad
z = \cos \theta,
</math>

then setting the linear dependency <math> \varphi=c\theta</math> for the angle coordinates gives a [[parametric curve]] in terms of parameter {{tmath|\theta}},<ref>Kuno Fladt: ''Analytische Geometrie spezieller Flächen und Raumkurven'', Springer-Verlag, 2013, {{ISBN|3322853659}}, 9783322853653, S. 132</ref>

:<math>
\bigl( \sin \theta\, \cos c\theta,\, \sin \theta\, \sin c\theta,\, \cos \theta \,\bigr).
</math>

<gallery>
KUGSPI-5 Archimedische Kugelspirale.gif|Clelia curve
KUGSPI-9_Loxodrome.gif|Loxodrome
</gallery>

Another family of spherical spirals is the [[rhumb line]]s or loxodromes, that project to straight lines on the [[Mercator projection]]. These are the trajectories traced by a ship traveling with constant [[bearing (navigation)|bearing]]. Any loxodrome (except for the meridians and parallels) spirals infinitely around either pole, closer and closer each time, unlike a Clelia curve which maintains uniform spacing in colatitude. Under [[stereographic projection]], a loxodrome projects to a logarithmic spiral in the plane.

==In nature==
The study of spirals in [[nature]] has a long history. [[Christopher Wren]] observed that many [[Exoskeleton|shells]] form a [[logarithmic spiral]]; [[Jan Swammerdam]] observed the common mathematical characteristics of a wide range of shells from ''[[Helix (genus)|Helix]]'' to ''[[Spirula]]''; and [[Henry Nottidge Moseley]] described the mathematics of [[univalve]] shells. [[D'Arcy Wentworth Thompson|D’Arcy Wentworth Thompson]]'s ''[[On Growth and Form]]'' gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis: the [[shape]] of the curve remains fixed, but its size grows in a [[geometric progression]]. In some shells, such as ''[[Nautilus]]'' and [[ammonite]]s, the generating curve revolves in a plane perpendicular to the axis and the shell will form a planar discoid shape. In others it follows a skew path forming a [[helix|helico]]-spiral pattern. Thompson also studied spirals occurring in [[Horn (anatomy)|horn]]s, [[teeth]], [[claw]]s and [[plant]]s.<ref>{{Cite book|first=D'Arcy|last=Thompson|title=On Growth and Form |year=1942 |orig-year=1917 |url=https://archive.org/details/ongrowthform00thom |publisher=Cambridge : University Press ; New York : Macmillan| pages=748–933}}</ref>

A model for the pattern of [[floret]]s in the head of a [[sunflower]]<ref>{{cite web|url=https://www.geogebra.org/m/B4C9bbuy|title=Geogebra: Sunflowers are Irrationally Pretty|author=Ben Sparks}}</ref> was proposed by H. Vogel. This has the form
:<math>\theta = n \times 137.5^{\circ},\ r = c \sqrt{n}</math>
where ''n'' is the index number of the floret and ''c'' is a constant scaling factor, and is a form of [[Fermat's spiral]]. The angle 137.5° is the [[golden angle]] which is related to the [[golden ratio]] and gives a close packing of florets.<ref>{{cite book | last =Prusinkiewicz | first =Przemyslaw | author-link =Przemyslaw Prusinkiewicz | author2 =Lindenmayer, Aristid | author-link2 =Aristid Lindenmayer | title =The Algorithmic Beauty of Plants | publisher =Springer-Verlag | year =1990 | pages =[https://archive.org/details/algorithmicbeaut0000prus/page/101 101–107] | url =https://archive.org/details/algorithmicbeaut0000prus/page/101 | isbn =978-0-387-97297-8 }}</ref>

Spirals in plants and animals are frequently described as [[whorl (botany)|whorls]]. This is also the name given to spiral shaped [[fingerprint]]s.

<gallery widths="220" heights="160">
The center Galaxy of Cat's Eye.jpg|An artist's rendering of a spiral galaxy.
Helianthus whorl.jpg|Sunflower head displaying florets in spirals of 34 and 55 around the outside.
</gallery>

==As a symbol==
The [[Celts|Celtic]] triple-spiral is in fact a pre-Celtic symbol.<ref>Anthony Murphy and Richard Moore, ''Island of the Setting Sun: In Search of Ireland's Ancient Astronomers,'' 2nd ed., Dublin: The Liffey Press, 2008, pp. 168-169</ref> It is carved into the rock of a stone lozenge near the main entrance of the prehistoric [[Newgrange]] monument in [[County Meath]], [[Republic of Ireland|Ireland]]. Newgrange was built around 3200 [[BCE]], predating the Celts; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture.<ref name="knowth.com">{{cite web |url= http://knowth.com/newgrange.htm |title= Newgrange Ireland - Megalithic Passage Tomb - World Heritage Site |publisher= Knowth.com |date= 2007-12-21 |access-date= 2013-08-16 |url-status= live |archive-url= https://web.archive.org/web/20130726102318/http://www.knowth.com/newgrange.htm |archive-date=2013-07-26 }}</ref> The [[triskelion]] symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include [[Mycenaean Greece|Mycenaean]] vessels, coinage from [[Lycia]], [[stater]]s of [[Pamphylia]] (at [[Aspendos]], 370–333 BC) and [[Pisidia]], as well as the [[heraldic]] emblem on warriors' shields depicted on Greek pottery.<ref>For example, the trislele on [[Achilles]]' round shield on an Attic late sixth-century ''[[hydria]]'' at the [[Boston Museum of Fine Arts]], illustrated in John Boardman, Jasper Griffin and Oswyn Murray, ''Greece and the Hellenistic World'' (Oxford History of the Classical World) vol. I (1988), p. 50.</ref>

Spirals occur commonly in pre-Columbian art in Latin and Central America. The more than 1,400 [[petroglyphs]] (rock engravings) in [[Las Plazuelas]], [[Guanajuato]] [[Mexico]], dating 750-1200 AD, predominantly depict spirals, dot figures and scale models.<ref>{{cite web | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites | date = June 2006 | url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | page = 5 | access-date = 4 January 2014 | url-status = live | archive-url = https://web.archive.org/web/20140105032613/http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | archive-date = 5 January 2014}}</ref> In Colombia, monkeys, frog and lizard-like figures depicted in petroglyphs or as gold offering-figures frequently include spirals, for example on the palms of hands.<ref>{{cite web | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites | date = June 2006 | url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | page = 99 | access-date = 4 January 2014 | url-status = live | archive-url = https://web.archive.org/web/20140105032613/http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | archive-date = 5 January 2014}}</ref> In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters.<ref>{{cite web | title = Rock Art Of Latin America & The Caribbean | publisher = International Council on Monuments & Sites | date = June 2006 | url = http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | page = 17 | access-date = 4 January 2014 | url-status = live | archive-url = https://web.archive.org/web/20140105032613/http://www.icomos.org/studies/rockart-latinamerica/fulltext.pdf | archive-date = 5 January 2014}}</ref> Spirals also appear among the [[Nazca Lines]] in the coastal desert of Peru, dating from 200 BC to 500 AD. The [[geoglyphs]] number in the thousands and depict animals, plants and geometric motifs, including spirals.<ref>{{cite web | last = Jarus | first = Owen | title = Nazca Lines: Mysterious Geoglyphs in Peru | publisher = LiveScience | date = 14 August 2012 | url = http://www.livescience.com/22370-nazca-lines.html | access-date = 4 January 2014 | url-status = live | archive-url = https://web.archive.org/web/20140104122842/http://www.livescience.com/22370-nazca-lines.html | archive-date = 4 January 2014}}</ref>

Spirals are also a symbol of [[hypnosis]], stemming from the [[cliché]] of people and cartoon characters being hypnotized by staring into a spinning spiral (one example being [[Kaa]] in Disney's [[The Jungle Book (1967 film)| ''The Jungle Book'']]). They are also used as a symbol of [[dizziness]], where the eyes of a cartoon character, especially in [[anime]] and [[manga]], will turn into spirals to suggest that they are dizzy or dazed. The spiral is also found in structures as small as the [[double helix]] of [[DNA]] and as large as a [[spiral galaxy|galaxy]]. Due to this frequent natural occurrence, the spiral is the official symbol of the [[World Pantheist Movement]].<ref name=WPM>{{cite web|last= Harrison |first= Paul|title= Pantheist Art|url= http://www.pantheism.net/pan/free/pan9.pdf|publisher= World Pantheist Movement |access-date= 7 June 2012}}</ref>
The spiral is also a symbol of the [[dialectic]] process and of [[Dialectical monism]].

<blockquote>
The spiral is a frequent symbol for [[spiritual experience | spiritual]] purification, both within [[Christianity]] and beyond (one thinks of the spiral as the [[neoplatonism | neo-Platonist]] symbol for prayer and contemplation, circling around a subject and ascending at the same time, and as a [[Buddhist]] symbol for the gradual process on the Path to [[Enlightenment in Buddhism | Enlightenment]]). [...] while a helix is repetitive, a spiral expands and thus epitomizes [[Exponential growth | growth]] - conceptually ''ad infinitum''.<ref>
{{cite book
 |last1                = Bruhn
 |first1               = Siglind
 |author-link1         = Siglind Bruhn
 |year                 = 1997
 |chapter              = The Exchange of Natures and the Nature(s) of Time and Silence
 |title                = Images and Ideas in Modern French Piano Music: The Extra-musical Subtext in Piano Works by Ravel, Debussy, and Messiaen
 |url                  = https://books.google.com/books?id=_2V4i07PNzkC
 |series               = Aesthetics in music, ISSN 1062-404X, number 6
 |publication-place    = Stuyvesant, New York
 |publisher            = Pendragon Press
 |page                 = 353
 |isbn = 978-0-945193-95-1
 |access-date          = 30 June 2024
}} 
</ref>
</blockquote>

<gallery mode="packed" heights="150px">
File:库库特尼陶碗陶罐.JPG|[[Cucuteni Culture]] spirals on a bowl on stand, a vessel on stand, and an amphora, 4300-4000 BCE, ceramic, [[Palace of Culture (Iași)|Palace of Culture]], [[Iași]], [[Romania]]
Newgrange Entrance Stone.jpg|[[Neolithic Europe|Neolithic]] spirals on the [[Newgrange]] entrance slab, unknown sculptor or architect, 3rd millennium BC

File:Mycenaean funerary stele at the National Archaeological Museum of Athens on October 6, 2021.jpg|[[Mycenaean Greece|Mycenaean]] spirals on a burial stela, Grave Circle A, {{circa}}1550 BC, stone, [[National Archaeological Museum, Athens|National Archaeological Museum]], [[Athens]], Greece

File:Temple of Amun alley of rams (4) (34143965175).jpg|[[Meroë|Meroitic]] spirals on a ram of the alley of the [[Amun]] Temple of [[Naqa]], unknown sculptor, 1st century AD, stone, [[in situ]]

File:Samarra, Iraq (25270211056) edited.jpg|[[Islamic architecture|Islamic]] spiral design of the [[Great Mosque of Samarra]], [[Samarra]], [[Iraq]], unknown architect, {{circa}} 851

File:Nantes Maison compagnonnage Clocher tors.jpg|[[Gothic Revival]] spiralling bell-tower of the Maison des compagnons du tour de France, [[Nantes]], unknown architect, {{circa}} 1910
</gallery>

==In art==
The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is [[Robert Smithson]]'s [[earthworks (art)|earthwork]], "[[Spiral Jetty]]", at the [[Great Salt Lake]] in Utah.<ref>{{cite book | author1=Israel, Nico | title=Spirals : the whirled image in twentieth-century literature and art | date=2015 | publisher=New York Columbia University Press | pages=161–186 | isbn=978-0-231-15302-7 }}</ref> The spiral theme is also present in David Wood's Spiral Resonance Field at the [[Anderson-Abruzzo Albuquerque International Balloon Museum|Balloon Museum]] in Albuquerque, as well as in the critically acclaimed [[Nine Inch Nails]] 1994 concept album ''[[The Downward Spiral]]''. The Spiral is also a prominent theme in the anime ''[[Gurren Lagann]]'', where it represents a philosophy and way of life. It also central in Mario Merz and Andy Goldsworthy's work. The spiral is the central theme of the horror manga ''[[Uzumaki]]'' by [[Junji Ito]], where a small coastal town is afflicted by a curse involving spirals.

==See also==
*[[Celtic maze]] (straight-line spiral)
*[[Concentric circles]]
*[[DNA]]
*[[Fibonacci number]]
*[[Hypogeum of Ħal-Saflieni]]
*[[Megalithic Temples of Malta]]
*[[Patterns in nature]]
*[[Seashell surface]]
*[[Spirangle]]
*[[Spiral vegetable slicer]]
*[[Spiral stairs]]
*[[Triskelion]]

==References==
{{Reflist}}

== Related publications ==
* Cook, T., 1903. ''Spirals in nature and art''. Nature 68 (1761), 296.
* Cook, T., 1979. ''The curves of life''. Dover, New York.
* Habib, Z., Sakai, M., 2005. ''Spiral transition curves and their applications''. Scientiae Mathematicae Japonicae 61 (2), 195 – 206.
* {{cite journal|doi=10.1007/s11075-008-9252-1|title=Fair cubic transition between two circles with one circle inside or tangent to the other|journal=Numerical Algorithms|volume=51|issue=4|pages=461–476|year=2009|last1=Dimulyo|first1=Sarpono|last2=Habib|first2=Zulfiqar|last3=Sakai|first3=Manabu|bibcode=2009NuAlg..51..461D|s2cid=22532724}}
* Harary, G., Tal, A., 2011. ''The natural 3D spiral''. Computer Graphics Forum 30 (2), 237 – 246 [http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf] {{Webarchive|url=https://web.archive.org/web/20151122013249/http://webee.technion.ac.il/~ayellet/Ps/11-HararyTal.pdf |date=2015-11-22 }}.
* Xu, L., Mould, D., 2009. ''Magnetic curves: curvature-controlled aesthetic curves using magnetic fields''. In: Deussen, O., Hall, P. (Eds.), Computational Aesthetics in Graphics, Visualization, and Imaging. The Eurographics Association [http://gigl.scs.carleton.ca/sites/default/files/ling_xu/artn-cae.pdf].
* {{cite journal|doi=10.1016/j.cagd.2004.04.001|title=Designing fair curves using monotone curvature pieces|journal=Computer Aided Geometric Design|volume=21|issue=5|pages=515–527|year=2004|last1=Wang|first1=Yulin|last2=Zhao|first2=Bingyan|last3=Zhang|first3=Luzou|last4=Xu|first4=Jiachuan|last5=Wang|first5=Kanchang|last6=Wang|first6=Shuchun}}
* {{cite journal|doi=10.1016/j.cagd.2009.12.004|title=Applying inversion to construct planar, rational spirals that satisfy two-point G2 Hermite data|journal=Computer Aided Geometric Design|volume=27|issue=3|pages=262–280|year=2010|last1=Kurnosenko|first1=A.|arxiv=0902.4834|s2cid=14476206 }}
* A. Kurnosenko. ''Two-point G2 Hermite interpolation with spirals by inversion of hyperbola''. Computer Aided Geometric Design, 27(6), 474–481, 2010.
* Miura, K.T., 2006. ''A general equation of aesthetic curves and its self-affinity''. Computer-Aided Design and Applications 3 (1–4), 457–464 [http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/KTMiura-CAD06Final.pdf] {{Webarchive|url=https://web.archive.org/web/20130628000547/http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/KTMiura-CAD06Final.pdf |date=2013-06-28 }}.
* Miura, K., Sone, J., Yamashita, A., Kaneko, T., 2005. ''Derivation of a general formula of aesthetic curves''. In: 8th International Conference on Humans and Computers (HC2005). Aizu-Wakamutsu, Japan, pp.&nbsp;166 – 171 [http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/acurveHC0.pdf] {{Webarchive|url=https://web.archive.org/web/20130628051506/http://ktm11.eng.shizuoka.ac.jp/profile/ktmiura/pdf/acurveHC0.pdf |date=2013-06-28 }}.
* {{cite journal|doi=10.1016/0377-0427(89)90076-9|title=The use of Cornu spirals in drawing planar curves of controlled curvature|journal=Journal of Computational and Applied Mathematics|volume=25|pages=69–78|year=1989|last1=Meek|first1=D.S.|last2=Walton|first2=D.J.|doi-access=free}}
* {{cite journal|doi=10.1134/S1990793117010328|title=Potassium sulfate forms a spiral structure when dissolved in solution|journal=Russian Journal of Physical Chemistry B|volume=11|pages=195–198|year=2017|last1=Thomas|first1=Sunil|issue=1|bibcode=2017RJPCB..11..195T|s2cid=99162341}}
* {{cite journal|doi=10.1016/j.cagd.2006.03.004|title=Class a Bézier curves|journal=Computer Aided Geometric Design|volume=23|issue=7|pages=573–581|year=2006|last1=Farin|first1=Gerald}}
* Farouki, R.T., 1997. ''Pythagorean-hodograph quintic transition curves of monotone curvature''. Computer-Aided Design 29 (9), 601–606.
* Yoshida, N., Saito, T., 2006. ''Interactive aesthetic curve segments''. The Visual Computer 22 (9), 896–905 [http://www.yoshida-lab.net/aesthetic/ias2006pg.pdf] {{Webarchive|url=https://web.archive.org/web/20160304064701/http://www.yoshida-lab.net/aesthetic/ias2006pg.pdf |date=2016-03-04 }}.
* Yoshida, N., Saito, T., 2007. ''Quasi-aesthetic curves in rational cubic Bézier forms''. Computer-Aided Design and Applications 4 (9–10), 477–486 [http://www.yoshida-lab.net/aesthetic/cad07yoshida.pdf] {{Webarchive|url=https://web.archive.org/web/20160303205632/http://www.yoshida-lab.net/aesthetic/cad07yoshida.pdf |date=2016-03-03 }}.
* Ziatdinov, R., Yoshida, N., Kim, T., 2012. ''Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions''. Computer Aided Geometric Design 29 (2), 129—140 [https://www.sciencedirect.com/science/article/abs/pii/S0167839611001452].
* Ziatdinov, R., Yoshida, N., Kim, T., 2012. ''Fitting G2 multispiral transition curve joining two straight lines'', Computer-Aided Design 44(6), 591—596 [https://www.sciencedirect.com/science/article/pii/S001044851200019X].
* Ziatdinov, R., 2012. ''Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function''. Computer Aided Geometric Design 29(7): 510–518, 2012 [https://www.sciencedirect.com/science/article/abs/pii/S0167839612000325].
* Ziatdinov, R., Miura K.T., 2012. ''On the Variety of Planar Spirals and Their Applications in Computer Aided Design''. European Researcher 27(8–2), 1227—1232 [http://www.erjournal.ru/pdf.html?n=1345307278.pdf].

==External links==
{{Commons category|Spirals|Spiral}}
* [http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html] {{Webarchive|url=https://web.archive.org/web/20210702004420/http://www.mathe.tu-freiberg.de/~hebisch/cafe/jamnitzer/galerie7g.html |date=2021-07-02 }}
*[http://oeis.org/A202407 Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS]

{{Spirals}}
{{Authority control}}

[[Category:Spirals| ]]