Abstract
Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (<em>L</em>,π<sub>1</sub>, π<sub>2</sub>) is given by a lattice <em>L</em> in R<sup>s</sup> and projections π<sub>1</sub>, π<sub>2</sub> to suitable subspaces V<sub>1</sub>, V<sub>2</sub>. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice <em>L</em> and transformations of its projections π<sub>1</sub>(<em>L</em>), π<sub>2</sub>(<em>L</em>). For a self-similarity of a set Σ we take any linear mapping A such that AΣ ⊂ Σ, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project scheme such that π<sub>1</sub>(<em>L</em>) ⊂ R<sup>2</sup> is invariant under an isometry of order 5. We describe all linear self-similarities of the scheme thus constructed and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.
Highlights
Quasicrystals, their mathematical models and their physical properties stand in the front row of interest of scientists since 1984 when Shechtmann and his collegues [18] published his 1982 discovery of non-crystallographic materials with long-range order
While crystals are modeled by periodic lattices, as a suitable mathematical model of atomic positions in quasicrystals one recognizes the cut-and-project method that stems in projecting lattice points from a higherdimensional space to suitable subspaces
We derive what is the necessary form of the cut-and-project scheme (L ⊂ R4, π1, π2) if one aims to obtain a quasicrystal model with 10-fold symmetry
Summary
Quasicrystals, their mathematical models and their physical properties stand in the front row of interest of scientists since 1984 when Shechtmann and his collegues [18] published his 1982 discovery of non-crystallographic materials with long-range order. We may want to fix the cut-and-project scheme and ask about all linear self-similarities allowed by this specific scheme To this aim we present a matrix formalism for the study of cut-and-project schemes and derive several general statements (Theorem 3.1 and Proposition 3.3). We derive what is the necessary form of the cut-and-project scheme (L ⊂ R4, π1, π2) if one aims to obtain a quasicrystal model with 10-fold symmetry. We show that these mappings form an 8-dimensional Z-algebra Z and we provide explicit description of its elements. We provide an example illustrating that a cut-and-project set may have a linear self-similarity which is not a scaled rotation
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