2019/11/11 by Leonardo Di Giosia, Jahangir Habib, Di Giosia, Leonardo +15 · 1 citation
Computer Science · Materials Science · #Cellular Automata and Applications #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.1911.04476
openalex publication_date 2019/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The hexagon is the least-perimeter tile in the Euclidean plane for any given area. On hyperbolic surfaces, this "isoperimetric" problem differs for every given area, as solutions do not scale. Cox conjectured that a regular k-gonal tile with 120-degree angles is isoperimetric. For area π/3, the regular heptagon has 120-degree angles and therefore tiles many hyperbolic surfaces. For other areas, we show the existence of many tiles but provide no conjectured optima. On closed hyperbolic surfaces, we verify via a reduction argument using cutting and pasting transformations and convex hulls that the regular 7-gon is the optimal n-gonal tile of area π/3 for 3≤ n ≤ 10. However, for n>10, it is difficult to rule out non-convex n-gons that tile irregularly.