Conformally symmetric circle packings: A generalization of Doyle's spirals

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 discretization 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.