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 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. The first definition describes a planar curve, that extends in both of the perpendicular directions within its plane; the groove on one side of a

record A record, recording or records may refer to: An item or collection of data Computing * Record (computer science), a data structure ** Record, or row (database), a set of fields in a database related to one entity ** Boot sector or boot 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 spirals. 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 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, or plane, spiral may be described most easily using polar coordinates, where the radius

r

is a monotonic

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), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...

case (the function 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

r=a \varphi

* The hyperbolic spiral:

r = a/ \varphi

* Fermat's spiral:

r= a\varphi^

* The lituus:

r = a\varphi^

* The logarithmic spiral:

r=ae^

* 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, used twice on each tooth of almost every modern gear 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). Schraublinie-hyp-spirale

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 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 Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...

. 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 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 curves 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 filament An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a wire filament heated until it glows. The filament is enclosed in a glass bulb with a vacuum or inert gas to protect the filament from oxidat ...

s 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 own ...

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 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 th ...

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

(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 to the other while keeping a fixed angle with respect to the meridians). The loxodrome has an infinite number of revolutions, with the separation between them decreasing as the curve approaches either of the poles, unlike an

which maintains uniform line-spacing regardless of radius.

In nature

The study of spirals in nature 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; Jan Swammerdam observed the common mathematical characteristics of a wide range of shells from '' Helix'' to '' Spirula''; and Henry Nottidge Moseley described the mathematics of univalve shells. 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 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 horns, teeth, claws and plants. A model for the pattern of florets 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. The angle 137.5° is the

golden angle In geometry, the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the l ...

which is related to the golden ratio and gives a close packing of florets. Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped fingerprints. 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, Ukraine, as part of a decorative object dated to 10,000 BCE. The spiral and triple spiral motif is a Neolithic symbol in Europe ( Megalithic Temples of Malta). The Celtic 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 monument in County Meath, Ireland. 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 symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures, including Mycenaean vessels, on coinage in Lycia, on staters of Pamphylia (at Aspendos, 370–333 BC) and Pisidia, as well as on the heraldic 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 (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, 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 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. 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, etc., have often been interpreted as solar symbols. Roof tiles dating back to the Tang Dynasty with this symbol have been found west of the ancient city of Chang'an (modern-day Xi'an). Spirals are also a symbol of hypnosis, 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 in Disney's '' The Jungle Book''). They are also used as a symbol of dizziness, where the eyes of a cartoon character, especially in anime 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 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.

In art

The spiral has inspired artists throughout the ages. Among the most famous of spiral-inspired art is Robert Smithson's earthwork, " Spiral Jetty", 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 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. ''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.

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