Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply c
onnected domain. We introduce the concept of crosscuts and investigate the rigidity of circle packings with respect to maximal crosscuts. The main result is a discrete version of an indentity theorem for analytic functions (in the spirit of Schwarz Lemma), which has implications to uniqueness statements for discrete conformal mappings.
We propose to visualize complex (meromorphic) functions $f$ by their phase $P_f:=f/|f|$. Color--coding the points on the unit circle converts the function $P_f$ to an image (the phase plot of $f$), which represents the function directly on its domain
. We discuss how special properties of $f$ are reflected by their phase plots and indicate several applications. In particular we reformulate a universality theorem for Riemanns Zeta function in the language of phase plots.
