Spiral

In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of patterns, a broad group that also includes concentric objects.

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Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius $r$ is a monotonic continuous function of angle $\backslash varphi$:

• $r=r(\backslash varphi)\backslash ;\; .$

The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant).

In $x$-$y$-coordinates the curve has the parametric representation:

• $x=r(\backslash varphi)\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=r(\backslash varphi)\backslash sin\backslash varphi\backslash ;\; .$

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

• The Archimedean spiral: $r=a\; \backslash varphi$
• The hyperbolic spiral: $r\; =\; a/\; \backslash varphi$
• Fermat's spiral: $r=\; a\backslash varphi^\{1/2\}$
• The lituus: $r\; =\; a\backslash varphi^\{-1/2\}$
• The logarithmic spiral: $r=ae^\{k\backslash varphi\}$
• 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

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 $\backslash varphi=\backslash tfrac\{1\}\{k\}\backslash cdot\; \backslash ln\; \backslash tfrac\{r\}\{a\}$. 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.

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

Polar slope angle

The angle $\backslash alpha$ between the spiral tangent and the corresponding polar circle (see diagram) is called angle of the polar slope and $\backslash tan\; \backslash alpha$ the polar slope.

From vector calculus in polar coordinates one gets the formula

$\backslash tan\backslash alpha=\backslash frac\{r\text{'}\}\{r\}\backslash \; .$

Hence the slope of the spiral $\backslash ;r=a\backslash varphi^n\; \backslash ;$ is

• $\backslash tan\backslash alpha=\backslash frac\{n\}\{\backslash varphi\}\backslash \; .$

In case of an Archimedean spiral ($n=1$) the polar slope is $\backslash ;\; \backslash tan\backslash alpha=\backslash tfrac\{1\}\{\backslash varphi\}\backslash \; .$

In a logarithmic spiral, $\backslash \; \backslash tan\backslash alpha=k\backslash$ is constant.

Curvature

The curvature $\backslash kappa$ of a curve with polar equation $r=r(\backslash varphi)$ is

$\backslash kappa\; =\; \backslash frac\{r^2\; +\; 2(r\text{'})^2\; -\; r\backslash ;\; r\text{'}\text{'}\}\{(r^2+(r\text{'})^2)^\{3/2\}\}\backslash \; .$

For a spiral with $r=a\backslash varphi^n$ one gets

• $\backslash kappa\; =\; \backslash dotsb\; =\; \backslash frac\{1\}\{a\backslash varphi^\{n-1\}\}\backslash frac\{\backslash varphi^2+n^2+n\}\{(\backslash varphi^2+n^2)^\{3/2\}\}\backslash \; .$

In case of $n=1$ (Archimedean spiral) $\backslash kappa=\backslash tfrac\{\backslash varphi^2+2\}\{a(\backslash varphi^2+1)^\{3/2\}\}$.
Only for inflection point.

The curvature of a logarithmic spiral $\backslash ;\; r=a\; e^\{k\backslash varphi\}\; \backslash ;$ is $\backslash ;\; \backslash kappa=\backslash tfrac\{1\}\{r\backslash sqrt\{1+k^2\}\}\; \backslash ;\; .$

Sector area

The area of a sector of a curve (see diagram) with polar equation $r=r(\backslash varphi)$ is

$A=\backslash frac\{1\}\{2\}\backslash int\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\; r(\backslash varphi)^2\backslash ;\; d\backslash varphi\backslash \; .$

For a spiral with equation $r=a\backslash varphi^n\backslash ;$ one gets

• $A=\backslash frac\{1\}\{2\}\backslash int\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\; a^2\backslash varphi^\{2n\}\backslash ;\; d\backslash varphi$

=\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},

$A=\backslash frac\{1\}\{2\}\backslash int\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\; \backslash frac\{a^2\}\{\backslash varphi\}\backslash ;\; d\backslash varphi$

=\frac{a^2}{2}(\ln\varphi_2-\ln\varphi_1)\ ,\quad \text{if} \quad n=-\frac{1}{2}\ .

The formula for a logarithmic spiral $\backslash ;\; r=a\; e^\{k\backslash varphi\}\; \backslash ;$ is $\backslash \; A=\backslash tfrac\{r(\backslash varphi\_2)^2-r(\backslash varphi\_1)^2)\}\{4k\}\backslash \; .$

Arc length

The length of an arc of a curve with polar equation $r=r(\backslash varphi)$ is

$L=\backslash int\backslash limits\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\backslash sqrt\{\backslash left(r^\backslash prime(\backslash varphi)\backslash right)^2+r^2(\backslash varphi)\}\backslash ,\backslash mathrm\{d\}\backslash varphi\; \backslash \; .$

For the spiral $r=a\backslash varphi^n\backslash ;$ the length is

• $L=\backslash int\_\{\backslash varphi\_1\}^\{\backslash varphi\_2\}\; \backslash sqrt\{\backslash frac\{n^2r^2\}\{\backslash varphi^2\}\; +r^2\}\backslash ;\; d\backslash varphi$

= a\int\limits_{\varphi_1}^{\varphi_2}\varphi^{n-1}\sqrt{n^2+\varphi^2}d\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 integrals only.

The arc length of a logarithmic spiral $\backslash ;\; r=a\; e^\{k\backslash varphi\}\; \backslash ;$ is $\backslash \; L=\backslash tfrac\{\backslash sqrt\{k^2+1\}\}\{k\}\backslash big(r(\backslash varphi\_2)-r(\backslash varphi\_1)\backslash big)\; \backslash \; .$

Circle inversion

The Circle inversion has in polar coordinates the simple description: $\backslash \; (r,\backslash varphi)\; \backslash mapsto\; (\backslash tfrac\{1\}\{r\},\backslash varphi)\backslash$.

• The image of a spiral $\backslash \; r=\; a\backslash varphi^n\backslash$ under the inversion at the unit circle is the spiral with polar equation $\backslash \; r=\; \backslash tfrac\{1\}\{a\}\backslash varphi^\{-n\}\backslash$. For example: The inverse of an Archimedean spiral is a hyperbolic spiral.

A logarithmic spiral $\backslash ;\; r=a\; e^\{k\backslash varphi\}\; \backslash ;$ is mapped onto the logarithmic spiral $\backslash ;\; r=\backslash tfrac\{1\}\{a\}\; e^\{-k\backslash varphi\}\; \backslash ;\; .$

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

Example 1

Setting $\backslash ;r=a\; \backslash arctan(k\backslash varphi)\backslash ;$ and the choice $\backslash ;k=0.1,\; a=4,\; \backslash ;\backslash varphi\backslash ge\; 0\backslash ;$ gives a spiral, that starts at the origin (like an Archimedean spiral) and approaches the circle with radius $\backslash ;r=a\backslash pi/2\backslash ;$ (diagram, left).

Example 2

For $\backslash ;r=a\; (\backslash arctan(k\backslash varphi)+\backslash pi/2)\backslash ;$ and one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius $\backslash ;r=a\backslash pi\backslash ;$ (diagram, right).

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Three-dimensional

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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.

1. a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point.
2. 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 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 spirals.

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 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.

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.

Two well-known spiral are conical spirals and spherical spirals, defined below. Another instance of space spirals is the toroidal spiral.

A spiral wound around a helix, also known as double-twisted helix, represents objects such as coiled coil filaments.

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Conical spirals If in the $x$-$y$-plane a spiral with parametric representation

$x=r(\backslash varphi)\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=r(\backslash varphi)\backslash sin\backslash varphi$

is given, then there can be added a third coordinate $z(\backslash varphi)$, such that the now space curve lies on the cone with equation $\backslash ;m(x^2+y^2)=(z-z\_0)^2\backslash \; ,\backslash \; m>0\backslash ;$:

• $x=r(\backslash varphi)\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=r(\backslash varphi)\backslash sin\backslash varphi\backslash \; ,\; \backslash qquad\; \backslash color\{red\}\{z=z\_0\; +\; mr(\backslash varphi)\}\; \backslash \; .$

Spirals based on this procedure are called conical spirals.

Example

Starting with an archimedean spiral $\backslash ;r(\backslash varphi)=a\backslash varphi\backslash ;$ one gets the conical spiral (see diagram)

$x=a\backslash varphi\backslash cos\backslash varphi\; \backslash \; ,\backslash qquad\; y=a\backslash varphi\backslash sin\backslash varphi\backslash \; ,\; \backslash qquad\; z=z\_0\; +\; ma\backslash varphi\; \backslash \; ,\backslash quad\; \backslash varphi\; \backslash ge\; 0\; \backslash \; .$

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Spherical spirals 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 , 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

$$

x = \sin \theta \, \cos \varphi, \quad y = \sin \theta \, \sin \varphi, \quad z = \cos \theta,

then setting the linear dependency $\backslash varphi=c\backslash theta$ for the angle coordinates gives a parametric curve in terms of parameter ,Kuno Fladt: Analytische Geometrie spezieller Flächen und Raumkurven, Springer-Verlag, 2013, , 9783322853653, S. 132

$$

\bigl( \sin \theta\, \cos c\theta,\, \sin \theta\, \sin c\theta,\, \cos \theta \,\bigr).

Another family of spherical spirals is the or loxodromes, that project to straight lines on the Mercator projection. These are the trajectories traced by a ship traveling with constant 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.

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In nature The study of spirals in nature has a long history. Christopher Wren observed that many Exoskeleton 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 , 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-spiral pattern. Thompson also studied spirals occurring in horns, teeth, and .

A model for the pattern of in the head of a sunflower was proposed by H. Vogel. This has the form

$\backslash theta\; =\; n\; \backslash times\; 137.5^\{\backslash circ\},\backslash \; r\; =\; c\; \backslash sqrt\{n\}$

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.

Przemyslaw Prusinkiewicz (1990). 9780387972978, Springer-Verlag. . ISBN 9780387972978

Spirals in plants and animals are frequently described as whorls. This is also the name given to spiral shaped .

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.

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As a symbol The Celts triple-spiral is in fact a pre-Celtic symbol.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 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; triple spirals were carved at least 2,500 years before the Celts reached Ireland, but have long since become part of Celtic culture. The triskelion symbol, consisting of three interlocked spirals or three bent human legs, appears in many early cultures: examples include Mycenaean Greece vessels, coinage from Lycia, of Pamphylia (at Aspendos, 370–333 BC) and Pisidia, as well as the heraldic emblem on warriors' shields depicted on Greek pottery.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.

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. 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. In Lower Central America, spirals along with circles, wavy lines, crosses and points are universal petroglyph characters. 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.

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). 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. Due to this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement. The spiral is also a symbol of the dialectic process and of Dialectical monism.

The spiral is a frequent symbol for spiritual purification, both within Christianity and beyond (one thinks of the spiral as the neoplatonism 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). ... while a helix is repetitive, a spiral expands and thus epitomizes growth - conceptually ad infinitum.

(1997). 9780945193951, Pendragon Press. . ISBN 9780945193951

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, Romania Newgrange Entrance Stone.jpg|Neolithic Europe 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 spirals on a burial stela, Grave Circle A, 1550 BC, stone, National Archaeological Museum, Athens, Greece

File:Temple of Amun alley of rams (4) (34143965175).jpg|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 spiral design of the Great Mosque of Samarra, Samarra, Iraq, unknown architect, 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, 1910

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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 in Utah.

(2025). 9780231153027, New York Columbia University Press. ISBN 9780231153027

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 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.

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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

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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 – 246 [2] .
• 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 [3].
• 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 [4] .
• 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 – 171 [5] .
• 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 [6] .
• Yoshida, N., Saito, T., 2007. Quasi-aesthetic curves in rational cubic Bézier forms. Computer-Aided Design and Applications 4 (9–10), 477–486 [7] .
• 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 [8].
• Ziatdinov, R., Yoshida, N., Kim, T., 2012. Fitting G2 multispiral transition curve joining two straight lines, Computer-Aided Design 44(6), 591—596 [9].
• 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 [10].
• 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 [11].

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External links

• [12]
• Archimedes' spiral transforms into Galileo's spiral. Mikhail Gaichenkov, OEIS

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