Miquel dynamics for circle patterns

TL;DR

This paper introduces Miquel dynamics, a new discrete-time system on square grid circle patterns, revealing its properties, invariants, and special periodic points, with implications for geometric and combinatorial structures.

Contribution

It develops a coordinatization of circle patterns under Miquel dynamics, derives recurrence formulas, and characterizes periodic and invariant configurations.

Findings

Isoradial patterns are periodic points of the dynamics

Certain signed sums of intersection angles are conserved

Intersection points follow quartic curves in biperiodic cases

Abstract

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.