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

A. I. Bobenko·T. Hoffmann·Yu. B. Suris
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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 $\log z$.

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A. I. Bobenko, T. Hoffmann, Yu. B. Suris. 2001-04-25. Hexagonal circle patterns and integrable systems: Patterns with the multi-ratio property and Lax equations on the regular triangular lattice. https://arxiv.org/abs/math/0104244

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