2012/01/04 by Piotr W. Nowak, Shmuel Weinberger, Nowak, Piotr W. +1 · 1 citation
Computer Science · Materials Science · Mathematics · #Aperiodic graph #Cellular Automata and Applications #Combinatorics #Computer science #Conjecture #Construct (python library) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometry #Group (periodic table) #Group Theory (math.GR) #Homology (biology) #Mathematical Dynamics and Fractals #Mathematics #Metric Geometry (math.MG) #Penrose tiling #Physics #Pure mathematics #Quantum mechanics #Quasicrystal #Quasicrystal Structures and Properties #Substitution tiling #math.DS #math.GR #math.GT #math.MG
paper · pdf · doi:10.48550/arxiv.1201.0919
published in arXiv (Cornell University) (Cornell University) · Withdrawn by the authors
openalex publication_date 2012/01/04 · arxiv created 2012/05/16 · arxiv updated 2012/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture.