  Our goal is to provide a novel method of representing 2D shapes, where each
shape will be assigned a unique fingerprint - a computable approximation to a
conformal map of the given shape to a canonical shape in 2D or 3D space (see
page 22 for a few examples). In this paper, we make the first significant step
in this program where we address the case of simply, and doubly-connected
planar domains. We prove uniform convergence of our approximation scheme to the
appropriate conformal mapping.
  Along the way, we affirm a conjecture raised by Ken Stephenson in the 90's
which predicts that the Riemann mapping can be approximated by a sequence of
electrical networks. In fact, we first treat a more general case. Consider a
planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a
sequence of triangulations exhausting it. We construct a corresponding sequence
of maps which converge uniformly on compact subsets of the domain, to a
conformal homeomorphism onto the interior of a Euclidean annulus bounded by two
concentric circles. The resolution of Stephenson's Conjecture then follows by a
limiting argument.
  Our methods involve harmonic mappings and boundary value problems: discrete
and analytic in the plane. The scheme we constructed is programmable.
