Consider three mutually tangent circles, and draw their inner Soddy circle. Then draw the inner Soddy circles
of this circle with each pair of the original three, and continue iteratively. The
steps in the process are illustrated above (Trott 2004, pp. 34-35).
An animation illustrating the construction of the gasket is shown above.
The points which are never inside a circle form a set of measure 0 having fractal
dimension approximately 1.3058 (Mandelbrot 1983, p. 172). The Apollonian
gasket corresponds to a limit set that is invariant
under a Kleinian group (Wolfram 2002, p. 986).
The Apollonian gasket can also be generalized to three dimensions (Boyd 1973, Andrade et al. 2005), as illustrated above. A graph obtained by connecting the centers
of touching spheres in a three-dimensional Apollonian gasket by edges is known as
an Apollonian network.
Andrade, J. S. Jr.; Herrmann, H. J.; Andrade, R. F. S.; 2 and da Silva, L. R. "Apollonian Networks: Simultaneously
Scale-Free, Small World, Euclidean, Space Filling, and with Matching Graphs."
Phys. Rev. Lett.94, 01870-1-4, 2005.Boyd, D. W.
"Improved Bounds for the Disk Packing Constants." Aeq. Math.9,
99-106, 1973.Boyd, D. W. "The Residual Set Dimension of the
Apollonian Packing." Mathematika20, 170-174, 1973.Boyd,
D. W. "The Osculatory Packing of a Three Dimensional Sphere." Canad.
J. Math.25, 303-322, 1973.Kasner, E. and Supnick, F. "The
Apollonian Packing of Circles." Proc. Nat. Acad. Sci. USA29,
378-384, 1943.Mandelbrot, B. B. The
Fractal Geometry of Nature. New York: W. H. Freeman, pp. 169-172,
1983.Trott, M. The
Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 3-4, 1991.Wolfram, S. A
New Kind of Science. Champaign, IL: Wolfram Media, p. 986,
2002.
