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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a spiral is a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
which emanates from a point, moving farther away as it revolves around the point.


Helices

Two major definitions of "spiral" in the
American Heritage Dictionary American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the "United States" or "America" ** Americans, citizens and nationals of the United States of America ** American ancestry, pe ...
are:Spiral
''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009.
# 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 A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
. The first definition describes a
planar Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), ...
curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a 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 Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
s. The second definition includes two kinds of 3-dimensional relatives of spirals: # a conical or
volute spring A volute spring, also known as a conical spring, is a compression spring in the form of a cone (somewhat like the classical volute decorative architectural ornament). Under compression, the coils slide past each other, thus enabling the spring to ...
(including the spring used to hold and make contact with the negative terminals of AA or AAA batteries in a battery box), and the
vortex In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
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 quite 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. In the side picture, the black curve at the bottom is an
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
, while the green curve is a helix. The curve shown in red is a conic helix.


Two-dimensional

A
two-dimensional In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
, or plane, spiral may be described most easily using
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
, where the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
r is a
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a
degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
case (the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; .


Examples

Some of the most important sorts of two-dimensional spirals include: * The
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
: r=a \varphi * The
hyperbolic spiral A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation :r=\frac of a hyperbola. Because it can be generated by a circle inversion of an Archimedean spiral, it is called Reciprocal spiral, too.. Pier ...
: r = a/ \varphi *
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...
: r= a\varphi^ * The
lituus The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments ...
: r = a\varphi^ * The
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
: r=ae^ * The
Cornu spiral An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals. Eu ...
or ''clothoid'' * The
Fibonacci spiral Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
and
golden spiral In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter turn it makes. Approximations of the golden spira ...
* The
Spiral of Theodorus In geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyre ...
: an approximation of the Archimedean spiral composed of contiguous right triangles * The
involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or ...
of a circle, used twice on each tooth of almost every modern
gear A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic pr ...
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). An ''Archimedean spiral'' is, for example, generated while coiling a carpet. 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). The name ''logarithmic spiral'' is due to the equation \varphi=\tfrac\cdot \ln \tfrac. 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.
The ''spiral of Theodorus'' is a polygon.
The ''Fibonacci Spiral'' consists of a sequence of circle arcs.
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 r=r(\varphi), especially for the cases r(\varphi)=a\varphi^n (Archimedean, hyperbolic, Fermat's, lituus spirals) and the logarithmic spiral r=ae^. ;Polar slope angle The angle \alpha between the spiral tangent and the corresponding polar circle (see diagram) is called ''angle of the polar slope and \tan \alpha the ''polar slope''. From vector calculus in polar coordinates one gets the formula :\tan\alpha=\frac\ . Hence the slope of the spiral \;r=a\varphi^n \; is * \tan\alpha=\frac\ . In case of an ''Archimedean spiral'' (n=1) the polar slope is \; \tan\alpha=\tfrac\ . The ''logarithmic spiral'' is a special case, because of \ \tan\alpha=k\ ''constant'' ! ;curvature The curvature \kappa of a curve with polar equation r=r(\varphi) is :\kappa = \frac\ . For a spiral with r=a\varphi^n one gets * \kappa = \dotsb = \frac\frac\ . In case of n=1 ''(Archimedean spiral)'' \kappa=\tfrac.
Only for -1 the spiral has an ''inflection point''. The curvature of a ''logarithmic spiral'' \; r=a e^ \; is \; \kappa=\tfrac \; . ;Sector area The area of a sector of a curve (see diagram) with polar equation r=r(\varphi) is :A=\frac\int_^ r(\varphi)^2\; d\varphi\ . For a spiral with equation r=a\varphi^n\; one gets * A=\frac\int_^ a^2\varphi^\; d\varphi =\frac\big(\varphi_2^- \varphi_1^\big)\ , \quad \text\quad n\ne-\frac, :A=\frac\int_^ \frac\; d\varphi =\frac(\ln\varphi_2-\ln\varphi_1)\ ,\quad \text \quad n=-\frac\ . The formula for a ''logarithmic spiral'' \; r=a e^ \; is \ A=\tfrac\ . ;Arc length The length of an arc of a curve with polar equation r=r(\varphi) is :L=\int\limits_^\sqrt\,\mathrm\varphi \ . For the spiral r=a\varphi^n\; the length is * L=\int_^ \sqrt\; d\varphi = a\int\limits_^\varphi^\sqrtd\varphi \ . 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 In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s only. The arc length of a ''logarithmic spiral'' \; r=a e^ \; is \ L=\tfrac\big(r(\varphi_2)-r(\varphi_1)\big) \ . ;Circle inversion The inversion at the unit circle has in polar coordinates the simple description: \ (r,\varphi) \mapsto (\tfrac,\varphi)\ . * The image of a spiral \ r= a\varphi^n\ under the inversion at the unit circle is the spiral with polar equation \ r= \tfrac\varphi^\ . For example: The inverse of an Archimedean spiral is a hyperbolic spiral. :A logarithmic spiral \; r=a e^ \; is mapped onto the logarithmic spiral \; r=\tfrac e^ \; .


Bounded spirals

The function r(\varphi) of a spiral is usually strictly monotonic, continuous and un bounded. For the standard spirals r(\varphi) is either a power function or an exponential function. If one chooses for r(\varphi) a ''bounded'' function the spiral is bounded, too. A suitable bounded function is the
arctan In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
function: ;Example 1 Setting \;r=a \arctan(k\varphi)\; and the choice \;k=0.1, a=4, \;\varphi\ge 0\; gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius \;r=a\pi/2\; (diagram, left). ;Example 2 For \;r=a (\arctan(k\varphi)+\pi/2)\; and \;k=0.2, a=2,\; -\infty<\varphi<\infty\; one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius \;r=a\pi\; (diagram, right).


Three-dimensional

Two well-known spiral
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s are ''conic spirals'' and ''spherical spirals'', defined below. Another instance of space spirals is the ''toroidal spiral''. A "a spiral wound around a helix", also known as ''double-twisted helix'', represents objects such as coiled coil filaments or the
Slinky The Slinky is a helical spring toy invented by Richard James in the early 1940s. It can perform a number of tricks, including travelling down a flight of steps end-over-end as it stretches and re-forms itself with the aid of gravity and its ow ...
spring toy.


Conical spirals

If in the x-y-plane a spiral with parametric representation :x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi is given, then there can be added a third coordinate z(\varphi), such that the now space curve lies on the
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
with equation \;m(x^2+y^2)=(z-z_0)^2\ ,\ m>0\;: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\ , \qquad \color \ . Spirals based on this procedure are called conical spirals. ;Example Starting with an ''archimedean spiral'' \;r(\varphi)=a\varphi\; one gets the conical spiral (see diagram) :x=a\varphi\cos\varphi \ ,\qquad y=a\varphi\sin\varphi\ , \qquad z=z_0 + ma\varphi \ ,\quad \varphi \ge 0 \ .


Spherical spirals

If one represents a sphere of radius r by: : \begin x &=& r \cdot \sin \theta \cdot \cos \varphi \\ y &=& r \cdot \sin \theta \cdot \sin \varphi \\ z &=& r \cdot \cos \theta \end and sets the linear dependency \; \varphi=c\theta , \; c> 2\; , for the angle coordinates, one gets a
spherical curve A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
called spherical spiral with the parametric representation (with c equal to twice the number of turns) * \begin x &=& r \cdot \sin \theta \cdot \cos \\ y &=& r \cdot \sin \theta \cdot \sin \\ z &=& r \cdot \cos \theta\qquad \qquad 0\le\theta\le \pi \ . \end Spherical spirals were known to Pappus, too. Remark: a
rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb li ...
is ''not'' a spherical spiral in this sense. KUGSPI-5 Archimedische Kugelspirale.gif, Spherical spiral KUGSPI-9_Loxodrome.gif, Loxodrome A
rhumb line In navigation, a rhumb line, rhumb (), or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north. Introduction The effect of following a rhumb li ...
(also known as a loxodrome or "spherical spiral") is the curve on a sphere traced by a ship with constant bearing (e.g., travelling from one
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
to the other while keeping a fixed
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
with respect to the meridians). The loxodrome has an
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
number of
revolution In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
s, with the separation between them decreasing as the curve approaches either of the poles, unlike an
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
which maintains uniform line-spacing regardless of radius.


In nature

The study of spirals in
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
has a long history.
Christopher Wren Sir Christopher Wren PRS FRS (; – ) was one of the most highly acclaimed English architects in history, as well as an anatomist, astronomer, geometer, and mathematician-physicist. He was accorded responsibility for rebuilding 52 churches ...
observed that many shells form a
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
;
Jan Swammerdam Jan Swammerdam (February 12, 1637 – February 17, 1680) was a Dutch biologist and microscopist. His work on insects demonstrated that the various phases during the life of an insect—egg, larva, pupa, and adult—are different forms of the ...
observed the common mathematical characteristics of a wide range of shells from ''
Helix A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, ...
'' to ''
Spirula ''Spirula spirula'' is a species of deep-water squid-like cephalopod mollusk. It is the only extant member of the genus ''Spirula'', the family Spirulidae, and the order Spirulida. Because of the shape of its internal shell, it is commonly known ...
''; and
Henry Nottidge Moseley Henry Nottidge Moseley FRS (14 November 1844 – 10 November 1891) was a British naturalist who sailed on the global scientific expedition of HMS ''Challenger'' in 1872 through 1876. Life Moseley was born in Wandsworth, London, the son of Hen ...
described the mathematics of
univalve The gastropods (), commonly known as snails and slugs, belong to a large taxonomic class of invertebrates within the phylum Mollusca called Gastropoda (). This class comprises snails and slugs from saltwater, from freshwater, and from land. ...
shells.
D’Arcy Wentworth Thompson Sir D'Arcy Wentworth Thompson CB FRS FRSE (2 May 1860 – 21 June 1948) was a Scottish biologist, mathematician and classics scholar. He was a pioneer of mathematical and theoretical biology, travelled on expeditions to the Bering Strait an ...
's ''
On Growth and Form ''On Growth and Form'' is a book by the Scottish mathematical biologist D'Arcy Wentworth Thompson (1860–1948). The book is long – 793 pages in the first edition of 1917, 1116 pages in the second edition of 1942. The book covers many top ...
'' gives extensive treatment to these spirals. He describes how shells are formed by rotating a closed curve around a fixed axis: the
shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...
of the curve remains fixed but its size grows in a
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
. In some shells, such as ''
Nautilus The nautilus (, ) is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina. It comprises six living species in ...
'' and
ammonite Ammonoids are a group of extinct marine mollusc animals in the subclass Ammonoidea of the class Cephalopoda. These molluscs, commonly referred to as ammonites, are more closely related to living coleoids (i.e., octopuses, squid and cuttlefish) ...
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 helico-spiral pattern. Thompson also studied spirals occurring in
horn Horn most often refers to: *Horn (acoustic), a conical or bell shaped aperture used to guide sound ** Horn (instrument), collective name for tube-shaped wind musical instruments *Horn (anatomy), a pointed, bony projection on the head of various ...
s,
teeth A tooth ( : teeth) is a hard, calcified structure found in the jaws (or mouths) of many vertebrates and used to break down food. Some animals, particularly carnivores and omnivores, also use teeth to help with capturing or wounding prey, tear ...
,
claw A claw is a curved, pointed appendage found at the end of a toe or finger in most amniotes (mammals, reptiles, birds). Some invertebrates such as beetles and spiders have somewhat similar fine, hooked structures at the end of the leg or tarsus ...
s and
plant Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae exclud ...
s. A model for the pattern of
floret This glossary of botanical terms is a list of definitions of terms and concepts relevant to botany and plants in general. Terms of plant morphology are included here as well as at the more specific Glossary of plant morphology and Glossary o ...
s in the head of a
sunflower The common sunflower (''Helianthus annuus'') is a large annual forb of the genus ''Helianthus'' grown as a crop for its edible oily seeds. Apart from cooking oil production, it is also used as livestock forage (as a meal or a silage plant), as ...
was proposed by H. Vogel. This has the form :\theta = n \times 137.5^,\ r = c \sqrt where ''n'' is the index number of the floret and ''c'' is a constant scaling factor, and is a form of
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance ...
. The angle 137.5° is the golden angle which is related to the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
and gives a close packing of florets. Spirals in plants and animals are frequently described as
whorls A whorl ( or ) is an individual circle, oval, volution or equivalent in a whorled pattern, which consists of a spiral or multiple concentric objects (including circles, ovals and arcs). Whorls in nature File:Photograph and axial plane floral ...
. This is also the name given to spiral shaped
fingerprint A fingerprint is an impression left by the friction ridges of a human finger. The recovery of partial fingerprints from a crime scene is an important method of forensic science. Moisture and grease on a finger result in fingerprints on surfac ...
s. 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.


As a symbol

A spiral like form has been found in
Mezine Mezine is a place within the modern country of Ukraine which has the most artifact finds of Paleolithic culture origin. The Epigravettian site is located on a bank of the Desna River in Novhorod-Siverskyi Raion of Chernihiv Oblast, northern Ukrain ...
,
Ukraine Ukraine ( uk, Україна, Ukraïna, ) is a country in Eastern Europe. It is the second-largest European country after Russia, which it borders to the east and northeast. Ukraine covers approximately . Prior to the ongoing Russian inv ...
, as part of a decorative object dated to 10,000 BCE. The spiral and
triple spiral A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of ...
motif is a
Neolithic The Neolithic period, or New Stone Age, is an Old World archaeological period and the final division of the Stone Age. It saw the Neolithic Revolution, a wide-ranging set of developments that appear to have arisen independently in several parts ...
symbol in Europe (
Megalithic Temples of Malta The Megalithic Temples of Malta ( mt, It-Tempji Megalitiċi ta' Malta) are several prehistoric temples, some of which are UNESCO World Heritage Sites, built during three distinct periods approximately between 3600 BC and 2500 BC on the island coun ...
). The
Celtic Celtic, Celtics or Keltic may refer to: Language and ethnicity *pertaining to Celts, a collection of Indo-European peoples in Europe and Anatolia **Celts (modern) *Celtic languages **Proto-Celtic language * Celtic music *Celtic nations Sports Fo ...
symbol the triple spiral is in fact a pre-Celtic symbol. It is carved into the rock of a stone lozenge near the main entrance of the prehistoric
Newgrange Newgrange ( ga, Sí an Bhrú) is a prehistoric monument in County Meath in Ireland, located on a rise overlooking the River Boyne, west of Drogheda. It is an exceptionally grand passage tomb built during the Neolithic Period, around 3200 BC, ...
monument in
County Meath County Meath (; gle, Contae na Mí or simply ) is a county in the Eastern and Midland Region of Ireland, within the province of Leinster. It is bordered by Dublin to the southeast, Louth to the northeast, Kildare to the south, Offaly to the sou ...
,
Ireland Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic Ocean, in Northwestern Europe, north-western Europe. It is separated from Great Britain to its east by the North Channel (Grea ...
. Newgrange was built around 3200 BCE predating the Celts and the triple spirals were carved at least 2,500 years before the Celts reached Ireland but has long since been incorporated into Celtic culture. The
triskelion A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts o ...
symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures, including Mycenaean vessels, on coinage in
Lycia Lycia (Lycian language, Lycian: 𐊗𐊕𐊐𐊎𐊆𐊖 ''Trm̃mis''; el, Λυκία, ; tr, Likya) was a state or nationality that flourished in Anatolia from 15–14th centuries BC (as Lukka) to 546 BC. It bordered the Mediterranean ...
, on
stater The stater (; grc, , , statḗr, weight) was an ancient coin used in various regions of Ancient Greece, Greece. The term is also used for similar coins, imitating Greek staters, minted elsewhere in ancient Europe. History The stater, as a Gr ...
s of
Pamphylia Pamphylia (; grc, Παμφυλία, ''Pamphylía'') was a region in the south of Asia Minor, between Lycia and Cilicia, extending from the Mediterranean to Mount Taurus (all in modern-day Antalya province, Turkey). It was bounded on the north by ...
(at
Aspendos Aspendos or Aspendus ( Pamphylian: ΕΣΤϜΕΔΥΣ; Attic: Ἄσπενδος) was an ancient Greco-Roman city in Antalya province of Turkey. The site is located 40 km east of the modern city of Antalya. It was situated on the Eurymedon ...
, 370–333 BC) and
Pisidia Pisidia (; grc-gre, Πισιδία, ; tr, Pisidya) was a region of ancient Asia Minor located north of Pamphylia, northeast of Lycia, west of Isauria and Cilicia, and south of Phrygia, corresponding roughly to the modern-day province of An ...
, as well as on the
heraldic Heraldry is a discipline relating to the design, display and study of armorial bearings (known as armory), as well as related disciplines, such as vexillology, together with the study of ceremony, rank and pedigree. Armory, the best-known branc ...
emblem on warriors' shields depicted on Greek pottery. Spirals can be found throughout pre-Columbian art in Latin and Central America. The more than 1,400
petroglyphs A petroglyph is an image created by removing part of a rock surface by incising, picking, carving, or abrading, as a form of rock art. Outside North America, scholars often use terms such as "carving", "engraving", or other descriptions ...
(rock engravings) in Las Plazuelas,
Guanajuato Guanajuato (), officially the Free and Sovereign State of Guanajuato ( es, Estado Libre y Soberano de Guanajuato), is one of the 32 states that make up the Federal Entities of Mexico. It is divided into 46 municipalities and its capital city i ...
Mexico Mexico (Spanish: México), officially the United Mexican States, is a country in the southern portion of North America. It is bordered to the north by the United States; to the south and west by the Pacific Ocean; to the southeast by Guatema ...
, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models. In Colombia monkeys, frog and lizard like figures depicted in petroglyphs or as gold offering figures frequently includes spirals, for example on the palms of hands. In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters. Spirals can also be found among the
Nazca Lines The Nazca Lines are a group of geoglyphs made in the soil of the Nazca Desert in southern Peru. They were created between 500 BCE and 500 CE by people making depressions or shallow incisions in the desert floor, removing pebbles and l ...
in the coastal desert of Peru, dating from 200 BC to 500 AD. The
geoglyphs A geoglyph is a large design or motif (generally longer than 4 metres) produced on the ground by durable elements of the landscape, such as stones, stone fragments, gravel, or earth. A positive geoglyph is formed by the arrangement and alignmen ...
number in the thousands and depict animals, plants and geometric motifs, including spirals. Spiral shapes, including the
swastika The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. It ...
,
triskele A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts of ...
, etc., have often been interpreted as
solar symbol A solar symbol is a symbol representing the Sun. Common solar symbols include circles (with or without rays), crosses, and spirals. In religious iconography, personifications of the Sun or solar attributes are often indicated by means of a hal ...
s. Roof tiles dating back to the
Tang Dynasty The Tang dynasty (, ; zh, t= ), or Tang Empire, was an Dynasties in Chinese history, imperial dynasty of China that ruled from 618 to 907 AD, with an Zhou dynasty (690–705), interregnum between 690 and 705. It was preceded by the Sui dyn ...
with this symbol have been found west of the ancient city of
Chang'an Chang'an (; ) is the traditional name of Xi'an. The site had been settled since Neolithic times, during which the Yangshao culture was established in Banpo, in the city's suburbs. Furthermore, in the northern vicinity of modern Xi'an, Qin Shi ...
(modern-day Xi'an). Spirals are also a symbol of
hypnosis Hypnosis is a human condition involving focused attention (the selective attention/selective inattention hypothesis, SASI), reduced peripheral awareness, and an enhanced capacity to respond to suggestion.In 2015, the American Psychologica ...
, stemming from the
cliché A cliché ( or ) is an element of an artistic work, saying, or idea that has become overused to the point of losing its original meaning or effect, even to the point of being weird or irritating, especially when at some earlier time it was consi ...
of people and cartoon characters being hypnotized by staring into a spinning spiral (one example being
Kaa Kaa is a fictional character from ''The Jungle Book'' stories written by Rudyard Kipling. He is a giant snake who is 30 feet long. In the books and many of the screen adaptations, Kaa is an ally of main protagonist Mowgli, acting as a friend ...
in Disney's ''
The Jungle Book ''The Jungle Book'' (1894) is a collection of stories by the English author Rudyard Kipling. Most of the characters are animals such as Shere Khan the tiger and Baloo the bear, though a principal character is the boy or "man-cub" Mowgli, ...
''). They are also used as a symbol of
dizziness Dizziness is an imprecise term that can refer to a sense of disorientation in space, vertigo, or lightheadedness. It can also refer to disequilibrium or a non-specific feeling, such as giddiness or foolishness. Dizziness is a common medical c ...
, where the eyes of a cartoon character, especially in
anime is Traditional animation, hand-drawn and computer animation, computer-generated animation originating from Japan. Outside of Japan and in English, ''anime'' refers specifically to animation produced in Japan. However, in Japan and in Japane ...
and
manga Manga (Japanese: 漫画 ) are comics or graphic novels originating from Japan. Most manga conform to a style developed in Japan in the late 19th century, and the form has a long prehistory in earlier Japanese art. The term ''manga'' is u ...
, will turn into spirals to show they are dizzy or dazed. The spiral is also found in structures as small as the
double helix A double is a look-alike or doppelgänger; one person or being that resembles another. Double, The Double or Dubble may also refer to: Film and television * Double (filmmaking), someone who substitutes for the credited actor of a character * ...
of DNA and as large as a
galaxy A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
. Because of this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement. The spiral is also a symbol of the
dialectic Dialectic ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing ...
process and
Dialectical monism Dialectical monism, also known as dualistic monism or monistic dualism, is an ontological position that holds that reality is ultimately a unified whole, distinguishing itself from monism by asserting that this whole necessarily expresses itself in ...
.


In art

The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is
Robert Smithson Robert Smithson (January 2, 1938 – July 20, 1973) was an American artist known for sculpture and land art who often used drawing and photography in relation to the spatial arts. His work has been internationally exhibited in galleries and mu ...
's earthwork, "
Spiral Jetty ''Spiral Jetty'' is an earthwork sculpture constructed in April 1970 that is considered to be the most important work of American sculptor Robert Smithson. Smithson documented the construction of the sculpture in a 32-minute color film also tit ...
", at the
Great Salt Lake The Great Salt Lake is the largest saltwater lake in the Western Hemisphere and the eighth-largest terminal lake in the world. It lies in the northern part of the U.S. state of Utah and has a substantial impact upon the local climate, particula ...
in Utah. The spiral theme is also present in David Wood's Spiral Resonance Field at the Balloon Museum in Albuquerque, as well as in the critically acclaimed
Nine Inch Nails Nine Inch Nails, commonly abbreviated as NIN and stylized as NIИ, is an American industrial rock band formed in Cleveland in 1988. Singer, songwriter, multi-instrumentalist, and producer Trent Reznor was the only permanent member of the band ...
1994 concept album ''
The Downward Spiral ''The Downward Spiral'' is the second studio album by American industrial rock band Nine Inch Nails, released on March 8, 1994 by Nothing Records in the United States and Island Records in Europe. It is a concept album detailing the self-destru ...
''. The Spiral is also a prominent theme in the anime
Gurren Lagann ''Gurren Lagann'', known in Japan as , is a Japanese mecha anime, mecha anime television series animated by Gainax and co-produced by Aniplex and Konami. It ran for 27 episodes on TV Tokyo between April and September 2007. It was directe ...
, 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 is a Japanese horror manga series written and illustrated by Junji Ito. Appearing as a serial in the weekly ''seinen'' manga magazine ''Big Comic Spirits'' from 1998 to 1999, the chapters were compiled into three bound volumes by Shoga ...
'' by
Junji Ito is a Japanese horror manga artist. Some of his most notable works include ''Tomie'', a series chronicling an immortal girl who drives her stricken admirers to madness; '' Uzumaki'', a three-volume series about a town obsessed with spirals; and ...
, where a small coastal town is afflicted by a curse involving spirals. ''2012 A Piece of Mind By Wayne A Beale'' also depicts a large spiral in this book of dreams and images. The coiled spiral is a central image in Australian artist Tanja Stark's
Suburban Gothic Suburban Gothic is a subgenre of Gothic fiction, art, film and television, focused on anxieties associated with the creation of suburban communities, particularly in the United States and the West, from the 1950s and 1960s onwards. Criteria It o ...
iconography, that incorporates spiral electric stove top elements as symbols of domestic alchemy and spirituality.


See also

* Celtic maze (straight-line spiral) *
Concentric circles In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...
* DNA *
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
*
Hypogeum of Ħal-Saflieni A hypogeum or hypogaeum (plural hypogea or hypogaea, pronounced ; literally meaning "underground", from Greek ''hypo'' (under) and ''ghê'' (earth)) is an underground temple or tomb. Hypogea will often contain niches for cremated human rema ...
*
Megalithic Temples of Malta The Megalithic Temples of Malta ( mt, It-Tempji Megalitiċi ta' Malta) are several prehistoric temples, some of which are UNESCO World Heritage Sites, built during three distinct periods approximately between 3600 BC and 2500 BC on the island coun ...
*
Patterns in nature Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foa ...
*
Seashell surface In mathematics, a seashell surface is a surface made by a circle which spirals up the ''z''-axis while decreasing its own radius and distance from the ''z''-axis. Not all seashell surfaces describe actual seashells found in nature. Parametriza ...
*
Spirangle In geometry, a spirangle is a figure related to a spiral. Spirangles are similar to spirals in that they expand from a center point as they grow larger, but they are made out of straight line segments, instead of curves. Spirangle vectographs ...
*
Spiral vegetable slicer Spiral vegetable slicers (also known as spiralizers) are kitchen appliances used for cutting vegetables, such as zucchinis (to make zoodles), potatoes, cucumbers, carrots, apples, parsnips, and beetroots, into linguine-like strands which can be ...
* Spiral stairs *
Triskelion A triskelion or triskeles is an ancient motif consisting of a triple spiral exhibiting rotational symmetry. The spiral design can be based on interlocking Archimedean spirals, or represent three bent human legs. It is found in artefacts o ...


References


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. * * Harary, G., Tal, A., 2011. ''The natural 3D spiral''. Computer Graphics Forum 30 (2), 237 – 24

* 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 Associatio

* * * 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), 457–46

* 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. 166 – 17

* * * * 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), 896–90

* Yoshida, N., Saito, T., 2007. ''Quasi-aesthetic curves in rational cubic Bézier forms''. Computer-Aided Design and Applications 4 (9–10), 477–48

* 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—14

* Ziatdinov, R., Yoshida, N., Kim, T., 2012. ''Fitting G2 multispiral transition curve joining two straight lines'', Computer-Aided Design 44(6), 591—59

* 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, 201

* 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—123


External links


Jamitzer_Jamnitzer, Jamnitzer
-Galerie:_3D-Spirals.html" ;"title="Jamitzer Jamnitzer, Jamnitzer">Jamitzer Jamnitzer, Jamnitzer
-Galerie: 3D-Spirals">Jamitzer Jamnitzer, Jamnitzer">Jamitzer Jamnitzer, Jamnitzer
-Galerie: 3D-Spiralsbr>Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS
{{Authority control Spirals,