  We study a new discrete-time dynamical system on circle patterns with the
combinatorics of the square grid. This dynamics, called Miquel dynamics, relies
on Miquel's six circles theorem. We provide a coordinatization of the
appropriate space of circle patterns on which the dynamics acts and use it to
derive local recurrence formulas. Isoradial circle patterns arise as periodic
points of Miquel dynamics. Furthermore, we prove that certain signed sums of
intersection angles are preserved by the dynamics. Finally, when the initial
circle pattern is spatially biperiodic with a fundamental domain of size two by
two, we show that the appropriately normalized motion of intersection points of
circles takes place along an explicit quartic curve.
