The Archimedean spiral (also known as the arithmetic spiral) is a
spiral named after the 3rd-century BC
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
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. It is the
locus
Locus (plural loci) is Latin for "place". It may refer to:
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* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
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** ''Locus Award' ...
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
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 th ...
it can be described by the equation
with
real numbers 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''.
Conon of 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 i ...
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
angular velocity about the -axis, then the velocity of the point with respect to -axis may be written as:
Here is the modulus of the
position vector 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.
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:
Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation
(using the fact that and ) or
Its polar form is
Arc length and curvature
Given the parametrization in cartesian coordinates
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
to
is
or, equivalently:
The total length from
to
is therefore
The curvature is given by
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 tha ...
s), hence the name "arithmetic spiral". In contrast to this, in a
logarithmic spiral
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" ("ewige Linie"). More ...
these distances, as well as the distances of the intersection points measured from the origin, form a
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero 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 ex ...
.
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
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...
asymptotically approaches a circle with radius .
General Archimedean spiral
Sometimes the term ''Archimedean spiral'' is used for the more general group of spirals
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..
Pier ...
(),
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 ...
(), and the
lituus
The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments ...
().
Applications
One method of
squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to
trisect an angle. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs.
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, 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 b ...
balance spring
A balance spring, or hairspring, is a spring attached to the balance wheel in mechanical timepieces. It causes the balance wheel to oscillate with a resonant frequency when the timepiece is running, which controls the speed at which the wheels of ...
s 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 ne ...
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; this information helps in diagnosing neurological diseases.
Archimedean spirals are also used in
digital light processing
Digital Light Processing (DLP) is a set of chipsets based on optical micro-electro-mechanical technology that uses a digital micromirror device. It was originally developed in 1987 by Larry Hornbeck of Texas Instruments. While the DLP imagin ...
(DLP) projection systems to minimize the "
rainbow effect
Digital Light Processing (DLP) is a set of chipsets based on optical micro-electro-mechanical technology that uses a digital micromirror device. It was originally developed in 1987 by Larry Hornbeck of Texas Instruments. While the DLP imagin ...
", 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.
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
The heliospheric current sheet, or interplanetary current sheet, is a surface separating regions of the heliosphere where the interplanetary magnetic field points toward and away from the Sun. A small electrical current with a current density of ...
of the
solar wind
The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between . The composition of the sol ...
, or the pattern made by a
Catherine's wheel) are Archimedean. For instance, the star
LL Pegasi 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.
See also
*
Archimedes' screw
The Archimedes screw, also known as the Archimedean screw, hydrodynamic screw, water screw or Egyptian screw, is one of the earliest hydraulic machines. Using Archimedes screws as water pumps (Archimedes screw pump (ASP) or screw pump) dates back ...
*
Euler spiral
An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros, clothoids, or Cornu spirals.
E ...
*
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 ...
*
Golden spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter turn it makes.
Approximations of the golden spira ...
*
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..
Pier ...
*
List of spirals
This list of spirals includes named spirals that have been described mathematically.
See also
* Catherine wheel (firework)
* List of spiral galaxies
* Parker spiral
* Spirangle
* Spirograph
Spirograph is a geometric drawing device that ...
*
Logarithmic spiral
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" ("ewige Linie"). More ...
*
Spiral of Theodorus
In geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyre ...
*
Triple spiral symbol
References
External links
Jonathan Matt making the Archimedean spiral interesting - Video : The surprising beauty of Mathematics-
TedX Talks,
Green Farms
*
*
Page with Java application to interactively explore the Archimedean spiral and its related curvesOnline exploration using JSXGraph (JavaScript)Archimedean spiral at "mathcurve"
{{DEFAULTSORT:Archimedean Spiral
Squaring the circle
Spirals
Spiral
Articles with example R code