Conformally symmetric circle packings. A generalization of Doyle 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 Moebius 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 discretizations of the quotient of two Airy functions. There is an interactive version of this paper that contains JAVA applets that let you experiment with the circle packings directly. It is available at http://www-sfb288.math.tu-berlin.de/Publications/online/