Hexagonal circle patterns with constant intersection angles and discrete Painleve and Riccati equations
TL;DR: In this paper, the Riccati equation is expressed in terms of the hypergeometric function and the solution of the Painleve equation is described by a special separatrix solution.
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Abstract: Hexagonal circle patterns with constant intersection angles mimicking holomorphic maps z^c and log(z) are studied. It is shown that the corresponding circle patterns are immersed and described by special separatrix solutions of discrete Painleve and Riccati equations. The general solution of the Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete z^c and log(z), are established.
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Discrete differential geometry. Consistency as integrability
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Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
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Asymptotic behavior of discrete holomorphic maps z c and log(z)
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Hypergeometric tau Functions of the q-Painlevé Systems of Types A_4^{(1)} and (A_1+A_1')^{(1)}
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References
•Book
Quasicrystals and geometry
Marjorie Senechal
- 01 Jan 1995
TL;DR: An atlas of tiling transforms is given in this article, along with a mathematical toolbag for tiling transform analysis and a diagram of the Penrose tilings of the plane.
•Book
Discrete integrable geometry and physics
Alexander I. Bobenko,Ruedi Seiler +1 more
- 01 Jan 1999
TL;DR: The Pinkall Discretization of Surfaces and Integrable Systems as mentioned in this paper is a generalization of the Darboux Transform for Isothermic Surfaces (DHT).
On Thurston's formulation and proof of Andreev's theorem
Albert Marden,Burt Rodin +1 more
- 01 Jan 1990
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