The approximation of conformal structures via circle packing

. This is a pictorial tour and survey of circle packing techniques in the approximation of classical conformal objects. It begins with numerical conformal mapping and the conjecture of Thurston which launched this topic, moves to approximation of more general analytic functions, and ends with recent work on the approximation of conformal tilings and conformal structures. x1 Introduction A circle packing is a configuration of circles with a specified pattern of tangencies. The regular hexagonal or "penny" packing in the plane --- every circle tangent to six others --- is certainly familiar to everyone, and the literature contains a smattering of other examples stretching back to the ancient Greeks. But I offer this as a first illustration of the type of packings we will discuss. S 2 Circle Packing 1 Circle packing has its beginning as a distinct topic with applications to 3-manifolds in Thurston's Notes [20]. Its connections to analytic functions, the subject of this survey, can b...