Hexagonal circle patterns and integrable systems: Patterns with constant angles

DOI10.1215/S0012-7094-03-11635-XzbMath1037.52018arXivmath/0109018MaRDI QIDQ1395923

Tim Hoffmann, Alexander Ivanovich Bobenko

Publication date: 2003

Published in: Duke Mathematical Journal (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0109018

zbMATH Keywords

integrable systemshexagonal circle patternsSchramm's circle patterns

Mathematics Subject Classification ID

Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions (37K20) Lattice dynamics; integrable lattice equations (37K60) Circle packings and discrete conformal geometry (52C26)

Related Items (11)

Integrable cluster dynamics of directed networks and pentagram maps ⋮ Explicit solutions for a nonlinear vector model on the triangular lattice ⋮ Hexagonal circle patterns with constant intersection angles and discrete Painlevé and Riccati equations ⋮ Approximation of quasiconformal mappings by SG circle patterns ⋮ Integrability and geometry of the Wynn recurrence ⋮ Rigidity of quasicrystallic and \(Z ^{\gamma }\)-circle patterns ⋮ Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices ⋮ Discrete time Toda systems ⋮ Explicit solutions for a nonlinear model on the honeycomb and triangular lattices ⋮ Circle patterns and critical Ising models ⋮ Discrete power functions on a hexagonal lattice I: derivation of defining equations from the symmetry of the Garnier system in two variables

Cites Work

• Circle patterns with the combinatorics of the square grid
• The convergence of circle packings to the Riemann mapping
• Schwarz's lemma for circle packings
• The \(C^\infty\)-convergence of hexagonal disk packings to the Riemann map
• Rigidity of infinite disk patterns
• Spiral hexagonal circle packings in the plane
• Imbedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations
• Legendre transforms on a triangular lattice
• The discrete Korteweg-de Vries equation
• Conformally Symmetric Circle Packings: A Generalization of Doyle's Spirals
• "ISOMONODROMY" SOLUTIONS OF EQUATIONS OF ZERO CURVATURE
• Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds
• On some integrable systems related to the Toda lattice
• Circle packing immersions form regularly exhaustible surfaces
• Menelaus$apos$ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
• Trivalent graphs and solitons
• Discrete isothermic surfaces.
• Circle Packing: Experiments In Discrete Analytic Function Theory
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