The Archimedean spiral (also known as the arithmetic spiral) is a

spiral In mathematics, a spiral is a curve 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:Greek

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

. 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 In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

. Equivalently, in

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 ...

it can be described by the equation

r = a + b\cdot\theta

with

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...

and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book ''

On Spirals ''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle. Contents Preface Archimedes begins ''O ...

''.

Conon of Samos Conon of Samos ( el, Κόνων ὁ Σάμιος, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices. Life and work Conon was born on Samos ...

was a friend of his and Pappus states that this spiral was discovered by Conon.

Derivation of general equation of spiral

A physical approach is used below to understand the notion of Archimedean spirals. Suppose a point object moves in the Cartesian system with a constant

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

directed parallel to the -axis, with respect to the -plane. Let at time , the object was at an arbitrary point . If the plane rotates with a constant

about the -axis, then the velocity of the point with respect to -axis may be written as: Spiral derivation

\begin
, v_0, &=\sqrt \\
v_x&=v \cos \omega t - \omega (vt+c) \sin \omega t \\
v_y&=v \sin \omega t + \omega (vt+c) \cos \omega t
\end

Here is the modulus of the

position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or s ...

of the particle at any time , is the velocity component along the -axis and is the component along the -axis. The figure shown alongside explains this.

\begin
\int v_x \,dt &=x \\
\int v_y \,dt &=y
\end

The above equations can be integrated by applying

integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...

, leading to the following parametric equations:

\begin
x&=(vt + c) \cos \omega t \\
y&=(vt+c) \sin \omega t
\end

Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation

\sqrt=\frac\cdot \arctan \frac +c

(using the fact that and ) or

\tan \left(\left(\sqrt-c\right)\cdot\frac\right) = \frac

Its polar form is

r= \frac\cdot \theta +c.

Arc length and curvature

Osculating circles of the Archimedean spiral

Given the parametrization in cartesian coordinates

f\colon\theta\mapsto (r\,\cos \theta, r\,\sin \theta) = (b\, \theta\,\cos \theta,b\, \theta\,\sin\theta)

the

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

from

\theta_1

\theta_2

\frac\left theta\,\sqrt+\ln\left(\theta+\sqrt\right)\right^

or, equivalently:

\frac\left theta\,\sqrt+\operatorname\theta\right^.

The total length from

\theta_1=0

\theta_2=\theta

is therefore

\frac\left theta\,\sqrt+\ln \left(\theta+\sqrt \right)\right

The curvature is given by

\kappa=\frac

Characteristics

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to if is measured in

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s), hence the name "arithmetic spiral". In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. The Archimedean spiral has two arms, one for and one for . The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the -axis will yield the other arm. For large a point moves with well-approximated uniform acceleration along the Archimedean spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity (see contribution from Mikhail Gaichenkov). As the Archimedean spiral grows, its evolute asymptotically approaches a circle with radius .

General Archimedean spiral

Sometimes the term ''Archimedean spiral'' is used for the more general group of spirals

r = a + b\cdot\theta^\frac.

The normal Archimedean spiral occurs when . Other spirals falling into this group include 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.. Pierre ...

(),

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 f ...

(), and the lituus ().

Applications

One method of

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty ...

, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to

trisect an angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge a ...

. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs. Two moving spirals scroll pump

The Archimedean spiral has a variety of real-world applications.

Scroll compressor A scroll compressor (also called ''spiral compressor'', scroll pump and scroll vacuum pump) is a device for compressing air or refrigerant. It is used in air conditioning equipment, as an automobile supercharger (where it is known as a scroll- ...

s, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, involutes of a circle of the same size that almost resemble Archimedean spirals, or hybrid curves. Archimedean spirals can be found in

spiral antenna In microwave systems, a spiral antenna is a type of RF antenna. It is shaped as a two-arm spiral, or more arms may be used. Spiral antennas were first described in 1956. ''Logarithmic'' spiral antennas belong to the class of frequency independ ...

, which can be operated over a wide range of frequencies. The coils of

watch A watch is a portable timepiece intended to be carried or worn by a person. It is designed to keep a consistent movement despite the motions caused by the person's activities. A wristwatch is designed to be worn around the wrist, attached by ...

balance springs and the grooves of very early

gramophone record A phonograph record (also known as a gramophone record, especially in British English), or simply a record, is an analog sound storage medium in the form of a flat disc with an inscribed, modulated spiral groove. The groove usually starts nea ...

s form Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record). Asking for a patient to draw an Archimedean spiral is a way of quantifying human

tremor A tremor is an involuntary, somewhat rhythmic, muscle contraction and relaxation involving oscillations or twitching movements of one or more body parts. It is the most common of all involuntary movements and can affect the hands, arms, eyes, fa ...

; this information helps in diagnosing neurological diseases. Archimedean spirals are also used in digital light processing (DLP) projection systems to minimize the " rainbow effect", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly. Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter. Celestial spiral with a twist

They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine's wheel) are Archimedean. For instance, the star

LL Pegasi LL Pegasi (AFGL 3068) is a Mira variable star surrounded by a pinwheel-shaped nebula, IRAS 23166+1655, thought to be a Protoplanetary nebula, preplanetary nebula. It is a binary system (astronomy), binary system that includes an extreme ...

shows an approximate Archimedean spiral in the dust clouds surrounding it, thought to be ejected matter from the star that has been shepherded into a spiral by another companion star as part of a double star system.

References

External links

Jonathan Matt making the Archimedean spiral interesting - Video : The surprising beauty of Mathematics
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TedX Talks TED Conferences, LLC (Technology, Entertainment, Design) is an American-Canadian non-profit media organization that posts international talks online for free distribution under the slogan "ideas worth spreading". TED was founded by Richard Sau ...

Green Farms Green's Farms is the oldest neighborhood in the town of Westport, Connecticut, Westport in Fairfield County, Connecticut, Fairfield County, Connecticut, United States. It was listed as a census-designated place prior to the 2020 United States cens ...

* *
Page with Java application to interactively explore the Archimedean spiral and its related curves

Online exploration using JSXGraph (JavaScript)

Archimedean spiral at "mathcurve"
{{DEFAULTSORT:Archimedean Spiral Squaring the circle Spirals

Spiral In mathematics, a spiral is a curve 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:Articles with example R code