Circle packings are configurations of circles with specified patterns
of tangency, and lend themselves naturally to computer experimentation
and visualization. Maps between them display, with surprising
faithfulness, many of the geometric properties associated with
classical analytic functions. This paper introduces the fundamentals
of an emerging "discrete analytic function theory'' and investigates
connections with the classical theory. It then describes several experiments,
ranging from investigation of a conjectured
discrete Koebe $\sans{\quarter}$ theorem to a multigrid method for
computing discrete approximations of classical analytic functions.
These experiments were performed using CirclePack, a software package
described in the paper and available free of charge.