Saturday, February 6, 2010

Mathematical term in circle packing: Apollonian gasket

Apollonian packings of circles are named after the Greek geometer Apollonius of Perga (ca. 262-190 B.C.) [1]. In such packings, groups of circles are mutually tangent. A key generation step in the construction of Apollonian packings is the placement of a new circle into an interstice, which is formed by mutually tangent circles, in such a way that the new circle is tangent with those interstice-defining circles [2-5]. The successive repetition of this generation step with the newly forming interstices is building up an Apollonian gasket. Apollonian gaskets and similar constructions based on polygons instead of circles are of interest in physics for investigating the “architecture” of foams and powders by computer simulations. Also, fascinating relations between Apollonian gaskets and number theory have recently been studied. Finally, Apollonian gaskets are always good in the design of geometrically inspiring screensavers and op-art images.

Keywords: mathematics, geometry, fractals

[1] Biography: Apollonius of Perga.
[2] Kenneth Stephenson: Introduction to Circle PackingThe Theory of Discrete Analytic Functions. Cambridge University Press, Cambridge and New York, 2005; pp. 213-214.
[3] Edward Kasner and Fred Supnick: The Apollonian Packing of Circles. Proc. Natl. Acad. Sci USA December 1943, 29 (11), pp.378-384. Access.
[4] Dana Mackenzie: A Tisket, a Tasket, an Apollonian GasketFractals made of circles do funny things to mathematicians. American Scientist January-February 2010, Volume 98, Number 1, pp. 10-14.
[5] Wolfram MathWorld: Apollonian Gasket.

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