TOPICS

# Archimedes' Spiral

Archimedes' spiral is an Archimedean spiral with polar equation

 (1)

This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral.

The curvature of Archimedes' spiral is

 (2)

and the arc length is

 (3) (4)

This has the series expansion

 (5) (6)

(OEIS A091154 and A002595), where is a Legendre polynomial.

Archimedes' spiral can be used for compass and straightedge division of an angle into parts (including angle trisection) and can also be used for circle squaring. In addition, the curve can be used as a cam to convert uniform circular motion into uniform linear motion (Brown 1923; Steinhaus 1999, p. 137). The cam consists of one arch of the spiral above the x-axis together with its reflection in the x-axis. Rotating this with uniform angular velocity about its center will result in uniform linear motion of the point where it crosses the y-axis.

Archimedean Spiral, Hyperbolic Spiral

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## References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 225, 1987.Brown, H. T. 507 Mouvements mécaniques. Liège, Belgium: Desoer, p. 28, 1923.Gardner, M. The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, pp. 106-107, 1991.Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 90-92, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 186-187, 1972.Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 173-164, 1967.Sloane, N. J. A. Sequences A002595/M4233 and A091154 in "The On-Line Encyclopedia of Integer Sequences."Smith, D. E. History of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New York: Dover, pp. 329 and 330, 1958.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, p. 137, 1999.

## Cite this as:

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