# Spiral

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In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. [1] [2] [3] [4]

## Helices

Two major definitions of "spiral" in the American Heritage Dictionary are: [5]

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 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:

1. 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.
2. 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. [5]

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 conic helix.

## Two-dimensional

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

• ${\displaystyle r=r(\varphi )\;.}$

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

In ${\displaystyle x}$-${\displaystyle y}$-coordinates the curve has the parametric representation:

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

### Examples

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

An Archimedean spiral is, for example, generated while coiling a carpet. [6]

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). [7]

The name logarithmic spiral is due to the equation ${\displaystyle \varphi ={\tfrac {1}{k}}\cdot \ln {\tfrac {r}{a}}}$. 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.

### Geometric properties

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

Polar slope angle

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

From vector calculus in polar coordinates one gets the formula

${\displaystyle \tan \alpha ={\frac {r'}{r}}\ .}$

Hence the slope of the spiral ${\displaystyle \;r=a\varphi ^{n}\;}$ is

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

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

The logarithmic spiral is a special case, because of ${\displaystyle \ \tan \alpha =k\ }$constant !

curvature

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

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

For a spiral with ${\displaystyle r=a\varphi ^{n}}$ one gets

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

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

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

Sector area

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

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

For a spiral with equation ${\displaystyle r=a\varphi ^{n}\;}$ one gets

• ${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}a^{2}\varphi ^{2n}\;d\varphi ={\frac {a^{2}}{2(2n+1)}}{\big (}\varphi _{2}^{2n+1}-\varphi _{1}^{2n+1}{\big )}\ ,\quad {\text{if}}\quad n\neq -{\frac {1}{2}},}$
${\displaystyle A={\frac {1}{2}}\int _{\varphi _{1}}^{\varphi _{2}}{\frac {a^{2}}{\varphi }}\;d\varphi ={\frac {a^{2}}{2}}(\ln \varphi _{2}-\ln \varphi _{1})\ ,\quad {\text{if}}\quad n=-{\frac {1}{2}}\ .}$

The formula for a logarithmic spiral${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ A={\tfrac {r(\varphi _{2})^{2}-r(\varphi _{1})^{2})}{4k}}\ .}$

Arc length

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

${\displaystyle L=\int \limits _{\varphi _{1}}^{\varphi _{2}}{\sqrt {\left(r^{\prime }(\varphi )\right)^{2}+r^{2}(\varphi )}}\,\mathrm {d} \varphi \ .}$

For the spiral ${\displaystyle r=a\varphi ^{n}\;}$ the length is

• ${\displaystyle L=\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {{\frac {n^{2}r^{2}}{\varphi ^{2}}}+r^{2}}}\;d\varphi =a\int \limits _{\varphi _{1}}^{\varphi _{2}}\varphi ^{n-1}{\sqrt {n^{2}+\varphi ^{2}}}d\varphi \ .}$

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${\displaystyle \;r=ae^{k\varphi }\;}$ is ${\displaystyle \ L={\tfrac {\sqrt {k^{2}+1}}{k}}{\big (}r(\varphi _{2})-r(\varphi _{1}){\big )}\ .}$

Circle inversion

The inversion at the unit circle has in polar coordinates the simple description: ${\displaystyle \ (r,\varphi )\mapsto ({\tfrac {1}{r}},\varphi )\ }$.

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

### Bounded spirals

The function ${\displaystyle r(\varphi )}$ of a spiral is usually strictly monotonic, continuous and unbounded. For the standard spirals ${\displaystyle r(\varphi )}$ is either a power function or an exponential function. If one chooses for ${\displaystyle r(\varphi )}$ a bounded function the spiral is bounded, too. A suitable bounded function is the arctan function:

Example 1

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

Example 2

For ${\displaystyle \;r=a(\arctan(k\varphi )+\pi /2)\;}$ and ${\displaystyle \;k=0.2,a=2,\;-\infty <\varphi <\infty \;}$ one gets a spiral, that approaches the origin (like a hyperbolic spiral) and approaches the circle with radius ${\displaystyle \;r=a\pi \;}$ (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. [8] A "a spiral wound around a helix", [9] also known as double-twisted helix, [10] represents objects such as coiled coil filaments or the Slinky spring toy.

### Conical spirals

If in the ${\displaystyle x}$-${\displaystyle y}$-plane a spiral with parametric representation

${\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi }$

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

• ${\displaystyle x=r(\varphi )\cos \varphi \ ,\qquad y=r(\varphi )\sin \varphi \ ,\qquad \color {red}{z=z_{0}+mr(\varphi )}\ .}$

Spirals based on this procedure are called conical spirals.

Example

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

${\displaystyle x=a\varphi \cos \varphi \ ,\qquad y=a\varphi \sin \varphi \ ,\qquad z=z_{0}+ma\varphi \ ,\quad \varphi \geq 0\ .}$

### Spherical spirals

If one represents a sphere of radius ${\displaystyle r}$ by:

${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos \varphi \\y&=&r\cdot \sin \theta \cdot \sin \varphi \\z&=&r\cdot \cos \theta \end{array}}}$

and sets the linear dependency ${\displaystyle \;\varphi =c\theta ,\;c>2\;,}$ for the angle coordinates, one gets a spherical curve called spherical spiral [11] with the parametric representation (with ${\displaystyle c}$ equal to twice the number of turns)

• ${\displaystyle {\begin{array}{cll}x&=&r\cdot \sin \theta \cdot \cos {\color {red}c\theta }\\y&=&r\cdot \sin \theta \cdot \sin {\color {red}c\theta }\\z&=&r\cdot \cos \theta \qquad \qquad 0\leq \theta \leq \pi \ .\end{array}}}$

Spherical spirals were known to Pappus, too.

Remark: a rhumb line is not a spherical spiral in this sense.

A rhumb line (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 Archimedean spiral which maintains uniform line-spacing regardless of radius.

## In nature

The study of spirals in nature has a long history. Christopher Wren 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 ammonites, 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. [12] [ page needed ]

A model for the pattern of florets in the head of a sunflower [13] was proposed by H. Vogel. This has the form

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

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. [14]

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

## As a symbol

A spiral like form has been found in Mezine, Ukraine, as part of a decorative object dated to 10,000 BCE.[ citation needed ]

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. [15] 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. [16] 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. [17]

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 Mexico, dating 750-1200 AD, predominantly depict spirals, dot figures and scale models. [18] 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. [19] In Lower Central America spirals along with circles, wavy lines, crosses and points are universal petroglyphs characters. [20] 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. [21]

Spiral shapes, including the swastika, triskele, etc., have often been interpreted as solar symbols.[ citation needed ] 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).[ citation needed ][ year needed ]

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 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. Because of this frequent natural occurrence, the spiral is the official symbol of the World Pantheist Movement. [22] The spiral is also a symbol of the dialectic 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 in Utah. [23] 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. 2012 A Piece of Mind By Wayne A Beale also depicts a large spiral in this book of dreams and images. [24] [ full citation needed ] [25] [ verification needed ] The coiled spiral is a central image in Australian artist Tanja Stark's Suburban Gothic iconography, that incorporates spiral electric stove top elements as symbols of domestic alchemy and spirituality. [26] [27]

## Related Research Articles

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

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 is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

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". More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

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The Archimedean 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 constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation

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 from the spiral center, contrasting with the Archimedean spiral and the logarithmic spiral. Fermat spirals are named after Pierre de Fermat.

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by Carl Gustav Jakob Jacobi (1829). Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later.

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In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates

In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin, which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them.

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal even though it is not an independent quantity and it relates to as .

In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called conchospiral.

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