Archimedean spiral

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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

$\, r=a+b\theta$

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

$r=a+b\theta^{1\!/\!x}.$

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 St. Catherine's wheel) are Archimedean.