# 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 of points corresponding 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. Equivalently, in polar coordinates ("r", "θ") it can be described by the equation

:$, r=a+b heta$

with real numbers "a" and "b". Changing the parameter "a" will turn the spiral, while "b" controls the distance between successive turnings.

Archimedes described such a spiral in his book "On Spirals".

Characteristics

This Archimedean spiral is distinguished from the logarithmic spiral by the fact that successive turnings of the spiral have a constant separation distance (equal to 2π"b" if &theta; is measured in radians), while in a logarithmic spiral these distances form a geometric progression.

Note that the Archimedean spiral has two arms, one for &theta; > 0 and one for &theta; < 0. 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 y-axis will yield the other arm.

One method of squaring the circle, by relaxing the strict limitations on the use of straightedge and compass in ancient Greek geometric proofs, makes use of an Archimedean spiral.

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

:$r=a+b heta^\left\{1!/!x\right\}.$

The normal Archimedean spiral occurs when "x" = 1. Other spirals falling into this group include the hyperbolic spiral, Fermat's spiral, and the lituus. Virtually all static spirals appearing in nature are logarithmic spirals, not Archimedean ones. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine's wheel) are Archimedean.

Applications

The Archimedean spiral has a plethora of real-world applications. Scroll compressors, made from two interleaved Archimedean spirals of the same size, are used for compressing liquids and gases. [cite web|url=http://www.freepatentsonline.com/5603614.html|title=Fluid compressing device having coaxial spiral members|accessdate=2006-11-25|author=Sakata, Hirotsugu and Masayuki Okuda] The coils of watch balance springs and the grooves of very early gramophone records form Archimedean spirals, making the grooves evenly spaced and maximizing the amount of music that could be fit onto the record (although this was later changed to allow better sound quality). [cite web|url=http://ronpenndorf.com/journalofrecordedmusic5.html|title=Early Development of the LP|author=Penndorf, Ron|accessdate=2005-11-25] Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor; this information helps in diagnosing neurological diseases. Archimedean spirals are also used in 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 fast. [cite web|url=http://electronics.howstuffworks.com/dlp1.htm|title=Adding Color and the Reliability of DLP|author=Wilson, Tracy V.|accessdate=2005-11-25]

* Clackson scroll formula
* Hyperbolic spiral
* Fermat's spiral
* Logarithmic spiral
* Triple spiral symbol

References

*
*
* [http://www-groups.dcs.st-and.ac.uk/~history/Java/Spiral.html Page with Java application to interactively explore the Archimedean spiral and its related curves]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• archimedean spiral — noun Usage: usually capitalized A : a plane curve generated by a point moving away from or toward a fixed point at a constant rate while the radius vector from the fixed point rotates at a constant rate * * * Archimedean spiral noun The curve… …   Useful english dictionary

• Archimedean spiral — Archimedo spiralė statusas T sritis fizika atitikmenys: angl. Archimedean spiral; Archimedes spiral vok. Archimedische Spirale, f rus. архимедова спираль, f; спираль Архимеда, f pranc. spirale d’Archimède, f …   Fizikos terminų žodynas

• Archimedean spiral — noun A spiral that increases in distance from the point of origin at a constant rate. Syn: arithmetic spiral, spiral of Archimedes …   Wiktionary

• Archimedean — means of or pertaining to or named in honor of the Greek mathematician Archimedes. These are most commonly:* Archimedean property * Archimedean absolute value * Archimedean solid * Archimedean point * Archimedean tiling * Archimedean spiral *… …   Wikipedia

• Spiral — In mathematics, a spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.piral or helixA spiral and a helix are two terms that are easily confused, but represent different objects …   Wikipedia

• Spiral of Theodorus — In geometry, the spiral of Theodorus (also called square root spiral or Einstein spiral ) is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene.ConstructionThe spiral is started with an isosceles… …   Wikipedia

• spiral of Archimedes — noun A spiral that increases in distance from the point of origin at a constant rate. Syn: Archimedean spiral, arithmetic spiral …   Wiktionary

• spiral of archimedes — Usage: usually capitalized A : archimedean spiral * * * Geom. a curve that is the locus of a point that moves outward with uniform speed along a vector, beginning at the origin, while the vector rotates about the origin with uniform angular… …   Useful english dictionary

• Archimedean screw — sraigtinis konvejeris statusas T sritis automatika atitikmenys: angl. Archimedean screw; helical conveyor; helical screw conveyor; helicoid conveyor; screw conveyer; screw conveyor; spiral conveyor vok. Archimedesche Schraube, f; Förderschnecke,… …   Automatikos terminų žodynas

• archimedean — adj. of or associated with the Greek mathematician Archimedes (d. 212 BC). Phrases and idioms: Archimedean screw a device of ancient origin for raising water by means of a spiral tube …   Useful english dictionary