Conformally Symmetric Circle Packings: A Generalization of Doyle's Spirals

Abstract

From the geometric study of the elementary cell of hexagonal circle packings---a flower of 7 circles---the class of conformally symmetric circle packings is defined. Up to Möbius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are given explicitly. It is shown that these circle packings can be viewed as descretization s of the quotient of two Airy functions. The online version of this paper contains Java applets that let you experiment with the circle packings directly. The applets are found at http://www-sfb288.math.tu-berlin.de/Publications/online/cscpOnline/Applets.html