What Type of Apollonian Circle Packing Will Appear?

Abstract

An Apollonian circle packing is created by starting with three pairwise tangent circles, adding the two circles—or circle and line—tangent to the first three, then repeating the process forever by successively adding new circles and lines tangent to every new tangent threesome of circles and/or lines. The resulting packing is one of four types: bounded (enclosed by one of the circles), half-plane (with one line), strip (with two lines), or full-plane. Given three starting circles, what type of Apollonian circle packing will appear? This article gives an answer in the form of a picture, i.e., a plot in the parameter space of relative sizes of the starting circles, indicating the types of packings. The result is a fractal, which is then further explored.

Keywords:

• MSC: Primary 52C26
• Secondary 14H50
• 28A80

Acknowledgments

The author thanks Gerardo Lafferriere of the Fariborz Maseeh Department of Mathematics and Statistics at Portland State University for the welcome to PSU as a Visiting Scholar during a portion of this project, and thanks anonymous reviewers for helpful comments among which was the information about reference [2]. This work received support from the Colby College Research Grant Program.

ORCID

Jan E. Holly http://orcid.org/0000-0002-6419-3038

Jan E. Holly is a mathematician who continues to be intrigued by interesting problems in pure mathematics while also doing research in applied mathematics. This interest in exploration apparently extends to geography as well, with an education and career that have included University of Colorado, University of New Mexico, University of Illinois (Ph.D. in mathematics with specialization in logic and algebra), Robert S. Dow Neurological Sciences Institute in Oregon (postdoc in mathematical neuroscience), Colby College in Maine, as well as stints at The Hebrew University of Jerusalem, Center for Computational Biology in Montana, and Santa Fe Institute in New Mexico.

Department of Mathematics and Statistics, Colby College, Waterville, ME 04901 [email protected]