Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice

2001/04/25 by A. I. Bobenko, T. Hoffmann, Yu. B. Suris
Mathematics · Physics and Astronomy · #math.CV #nlin.SI

paper · pdf

published as Intern. Math. Research Notices, 2002, No 3, p.111-164 · 43 pages, 13 figures

arxiv created 2001/04/25 · arxiv updated 2009/11/30

Abstract

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a B"acklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogons to the analytic functions z^α and log z.

Related

• Hexagonal Circle Patterns and Integrable Systems. Patterns with Constant Angles
• Producing circle patterns via configurations
• Circle patterns with obtuse exterior intersection angles
• Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
• Spherical and hyperbolic orthogonal ring patterns: integrability and variational principles