2019/05/13 by Max Leppmeier, Leppmeier, Max
Materials Science · Mathematics · #FOS: Mathematics #History and Overview (math.HO) #Mathematics and Applications #Metric Geometry (math.MG) #Point processes and geometric inequalities #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1905.05837
openalex publication_date 2019/05/13 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying Jensen's inequality on convex functions to receive a local estimate which is globally valid. Based on the concept of Voronoi cells, we will introduce a new tessellation into so-called L-triangles which can be related to fundamental parallelograms of lattice circle packings. Therefore a globally disordered circle packing can be reduced to locally ordered configurations: We will show how the theorem of Lagrange on lattice circle packings can be applied to non-lattice circle packings. Thus we receive a new proof of Thue's theorem.