Abstract
The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.
Highlights
Quasicrystals, since their discovery [1], had been an active area of research in mathematics, physics, chemistry and mineralogy
In this paper we have reviewed the projection method and how it relates to the dualisation formalism of the Voronoi and Delaunay complexes of the lattice being projected
We have shown how empires can be computed within the context of the projection and dualisation methods
Summary
Quasicrystals, since their discovery [1], had been an active area of research in mathematics, physics, chemistry and mineralogy. A method investigated by Fang et al [12], is set in the context of the projection method and involves modifying [13] the cut-window in order to form a smaller empire window which serves as the acceptance domain for the empire of a specified patch of tiles. The latter construction is the more general of the three methods and has been used successfully to compute empires for several tilings generated as projections of cubic lattices Z N → En [12].
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