Some Properties of Lattice Substitution Systems

2003/11/05 by Dirk Frettlöh, Frettlöh, Dirk
Computer Science · Materials Science · Mathematics · #Cellular Automata and Applications #Computer science #FOS: Mathematics #Lattice (music) #Mathematics #Metric Geometry (math.MG) #Physics #Quasicrystal Structures and Properties #Statistical physics #Substitution (logic) #math.MG #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.math/0311061

18 pages, 4 figures This paper has been withdrawn by the author due to obscurity of the paper. All results are contained in the later paper by Bernd Sing and this author

openalex publication_date 2003/11/05 · arxiv created 2013/10/16 · arxiv updated 2013/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If a partition of a lattice in R^d is selfsimilar, it is called lattice substitution system (LSS). Such sets represent nonperiodic, but highly ordered structures. An important property of such structures is, whether they are model sets or not (equivalently, whether they are pure point diffractive or not). In this paper two conditions are given, which can be used to show that a given lattice substitution system does not consist of model sets.

Citations

• Tilings and Patterns.
• The Pinwheel Tilings of the Plane
• The undecidability of the domino problem

Related

• Lattice Substitution Systems and Model Sets
• Substitutions and their generalisations
• Bounded Displacement Non-Equivalence In Substitution Tilings
• Substitutions, tiling dynamical systems and minimal self-joinings
• Duality of Model Sets Generated by Substitutions