Rigidity of quasicrystallic and Z^\gamma-circle patterns

TL;DR

This paper proves the rigidity and uniqueness of certain quasicrystallic and Z^gamma-circle patterns, showing they are determined by combinatorics and intersection angles, with implications for their geometric structure.

Contribution

It establishes the rigidity and uniqueness of quasicrystallic and Z^gamma-circle patterns based on combinatorics and intersection angles, extending previous results.

Findings

Uniqueness of orthogonal Z^gamma-circle patterns under boundary conditions.

Rigidity of large classes of quasicrystallic circle patterns covering the plane.

Determination of these patterns up to affine transformations.

Abstract

The uniqueness of the orthogonal Z^\gamma-circle patterns as studied by Bobenko and Agafonov is shown, given the combinatorics and some boundary conditions. Furthermore we study (infinite) rhombic embeddings in the plane which are quasicrystallic, that is they have only finitely many different edge directions. Bicoloring the vertices of the rhombi and adding circles with centers at vertices of one of the colors and radius equal to the edge length leads to isoradial quasicrystallic circle patterns. We prove for a large class of such circle patterns which cover the whole plane that they are uniquely determined up to affine transformations by the combinatorics and the intersection angles. Combining these two results, we obtain the rigidity of large classes of quasicrystallic Z^\gamma-circle patterns.