  We introduce orthogonal ring patterns consisting of pairs of concentric
circles generalizing circle patterns. We show that orthogonal ring patterns are
governed by the same equation as circle patterns. For every ring pattern there
exists a one parameter family of patterns that interpolates between a circle
pattern and its dual. We construct ring pattern analogues of the Doyle spiral,
Erf and $z^{\alpha}$ functions. We also derive a variational principle and
compute ring patterns based on Dirichlet and Neumann boundary conditions.
