Substitution Tilings
نویسنده
چکیده
Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all substitution tilings showing tiles in infinitely many orientations. Dedicated to my teacher Ludwig Danzer on the occasion of his 80th birthday
منابع مشابه
A Fractal Fundamental Domain with 12-fold Symmetry
Square triangle tilings are relevant models for quasicrystals. We introduce a new self-similar tile-substitution which yields the well-known nonperiodic square triangle tilings of Schlottmann. It is shown that the new tilings are locally derivable from Schlottmann’s, but not vice versa, and that they are mutually locally derivable with the undecorated square triangle tilings. Furthermore, the r...
متن کاملFractal Dual Substitution Tilings
Starting with a substitution tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles have fractal boundary. We show that each of the new tilings is mutually locally derivable to the original tiling. Thus, at the tiling space level, the new substitution rules are expressing...
متن کاملCyclotomic Aperiodic Substitution Tilings
The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmet...
متن کاملA Primer of Substitution Tilings of the Euclidean Plane
This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutio...
متن کاملOne-dimensional substitution tilings with an interval projection structure
We study nonperiodic tilings of the line obtained by a projection method with an interval projection structure. We obtain a geometric characterisation of all interval projection tilings that admit substitution rules and describe the set of substitution rules for each such a tiling. We show that each substitution tiling admits a countably infinite number of nonequivalent substitution rules. We a...
متن کاملNon-Periodic Rhomb Substitution Tilings that Admit Order n Rotational Symmetry
For each n ∈ N we construct a substitution rule using the set of rhombs with angles (πk)/n. These substitution rules generate a local isomorphism class of tilings closed under rotation of order 2n, and also admit singular tilings fixed under a rotation of order n. The scaling factors for this set of substitution rules includes algebraic numbers of every rank.
متن کامل