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# Archimedean Spiral

An Archimedean spiral is a spiral with polar equation

 (1)

where is the radial distance, is the polar angle, and is a constant which determines how tightly the spiral is "wrapped."

Values of corresponding to particular special named spirals are summarized in the following table, together with the colors with which they are depicted in the plot above.

 spiral color lituus red hyperbolic spiral orange Archimedes' spiral green 1 Fermat's spiral blue 2

The curvature of an Archimedean spiral is given by

 (2)

and the arc length for by

 (3)

where is a hypergeometric function.

If a fly crawls radially outward along a uniformly spinning disk, the curve it traces with respect to a reference frame in which the disk is at rest is an Archimedean spiral (Steinhaus 1999, p. 137). Furthermore, a heart-shaped frame composed of two arcs of an Archimedean spiral which is fixed to a rotating disk converts uniform rotational motion to uniform back-and-forth motion (Steinhaus 1999, pp. 136-137).

## See also

Archimedean Spiral Inverse Curve, Archimedes' Spiral, Daisy, Fermat's Spiral, Hyperbolic Spiral, Lituus, Spiral

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## References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90-92, 1997.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 59-60, 1991.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186 and 189, 1972.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 175, 1967.MacTutor History of Mathematics Archive. "Spiral of Archimedes." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Spiral.html.Pappas, T. "The Spiral of Archimedes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 149, 1989.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 136-137, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 8-9, 1991.

## Cite this as:

Weisstein, Eric W. "Archimedean Spiral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ArchimedeanSpiral.html