iPedia Free Information Center

Information about Archimedean Spiral

Enlarge picture
Three 360° turnings of one arm of an Archimedean spiral


An Archimedean spiral (also 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



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.

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 2πb if θ 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 θ > 0 and one for θ < 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



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

Mechanism of a scroll pump
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.[1] The grooves of very early gramophone records form an Archimedean spiral, 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).[2] Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor, this information helps in diagnosis 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.[3]

See also

References

1. ^ Sakata, Hirotsugu and Masayuki Okuda. Fluid compressing device having coaxial spiral members. Retrieved on 2006-11-25.
2. ^ Penndorf, Ron. Early Development of the LP. Retrieved on 2005-11-25.
3. ^ Wilson, Tracy V.. Adding Color and the Reliability of DLP. Retrieved on 2005-11-25.

External links

spiral is a curve which emanates from a central point, getting progressively farther away as it revolves around the point.

Spiral or helix

A "spiral" and a "helix" are two terms that are easily confused, but represent different objects.
..... Click the link for more information.
Archimedes of Syracuse (Greek: Άρχιμήδης c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer.
..... Click the link for more information.
locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve.
..... Click the link for more information.
angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed at which an object is rotating along with the direction in which it is rotating.
..... Click the link for more information.
For a more advanced and general treatment, see .
Coordinates are numbers which describe the location of points in a plane or in space. For example, the height above sea level
..... Click the link for more information.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
Editing of this page by unregistered or newly registered users is currently disabled due to vandalism.
If you are prevented from editing this page, and you wish to make a change, please discuss changes on the talk page, request unprotection, log in, or .
..... Click the link for more information.
radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
..... Click the link for more information.
geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ...
..... Click the link for more information.
Squaring the circle is a problem proposed by ancient geometers. It is the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.
..... Click the link for more information.
A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. It has the polar equation rθ = a, and is the inverse to the Archimedean spiral.
..... Click the link for more information.
Fermat's spiral (also known as a parabolic spiral) follows the equation



in polar coordinates (the more general Fermat's spiral follows .) It is a type of Archimedean spiral.
..... Click the link for more information.
lituus is a spiral in which the angle is inversely proportional to the square of the radius (as expressed in polar coordinates). This spiral is asymptotic to the line .

Roman ritual



The lituus
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
Parker spiral is the shape of the Sun's extended magnetic field as it extends through the solar system. Unlike the familiar shape of the field from a bar magnet, the Sun's extended field is twisted into an arithmetic spiral by the influence of the solar wind and
..... Click the link for more information.
solar wind is a stream of charged particles (i.e., a plasma) which are ejected from the upper atmosphere of the sun. It consists mostly of high-energy electrons and protons (about 1 keV) that are able to escape the sun's gravity in part because of the high temperature of the corona
..... Click the link for more information.
Catherine wheel may refer to:
  • Breaking wheel, an instrument of torturous execution, associated with Saint Catherine of Alexandria.
  • Catherine wheel (firework), type of spinning firework.
  • Catherine wheel (window), type of spoked circular window.

..... Click the link for more information.
scroll compressor, also known as scroll pump and scroll vacuum pump, uses two interleaved spiral-like vanes to pump or compress fluids such as liquids and gases. The vane geometry may be involute, archimedean spiral, or hybrid curves.
..... Click the link for more information.
gramophone record (also phonograph record, or simply record) is an analogue sound storage medium consisting of a flat disc with an inscribed modulated spiral groove starting near the periphery and ending near the center of the disc.
..... Click the link for more information.
For , see .


Digital Light Processing (DLP) is a technology used in projectors and video projectors. It was originally developed at Texas Instruments, in 1987 by Dr. Larry Hornbeck.
..... Click the link for more information.
The Clackson scroll formula [1],
is used in blacksmithing for estimating the length of stock required to produce a scroll of turns with a spacing between the turns.
..... Click the link for more information.
A hyperbolic spiral is a transcendental plane curve also known as a reciprocal spiral. It has the polar equation rθ = a, and is the inverse to the Archimedean spiral.
..... Click the link for more information.
Fermat's spiral (also known as a parabolic spiral) follows the equation



in polar coordinates (the more general Fermat's spiral follows .) It is a type of Archimedean spiral.
..... Click the link for more information.
A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jakob Bernoulli, who called it
..... Click the link for more information.
triple spiral or triskele is a Celtic and pre-Celtic symbol found on a number of Irish Megalithic and Neolithic sites, most notably inside the Newgrange passage tomb, on the entrance stone, and on some of the curbstones surrounding the mound.
..... Click the link for more information.
Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He currently works for Wolfram Research, Inc.
..... Click the link for more information.
MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc., the creators of the Mathematica computer algebra system. It is also partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at
..... Click the link for more information.
PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigour, openness, pedagogy, real-time content, interlinked content, and community.
..... Click the link for more information.


This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.
Herod_Archelaus


page counter