Discrete analytic functions, also known as circle-packing maps, are mappings between circle packings whose properties faithfully reflect the properties that are characteristic of classical analytic functions. A circle packing is nothing more than a collection of circles with prescribed tangencies that lie on a surface. The tangency information may be encoded in a graph in which each circle is represented by a vertex, and each point of tangency at which two circles meet is represented by an edge connecting the vertices that correspond to those two circles.

A discrete analytic function then may be represented by a mapping between two circle-packing graphs of this type. Though discrete analytic functions may be used to build discrete approximations of classical analytic functions of both planar domains and surfaces, their real importance derives from the discovery that their properties mark them as true analogs in the discrete setting of the familiar analytic mappings of classical complex analysis. As such, discrete analytic functions offer both the rigidity and versatility of their classical counterparts and, in particular, offer a range of manipulations of planar domains that makes them valuable for flat two-dimensional representations of surfaces in three dimensions.

The theory of discrete analytic functions has been developed to the point that it stands alone as a part of a general movement in the recent development of mathematics—that of the discretization of geometry. Before presenting a more detailed description of these functions, it will be instructive to place the subject in its proper context within the recent development of mathematics.