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Crystals 2018, 8(10), 370; https://doi.org/10.3390/cryst8100370

Quasicrystal Tilings in Three Dimensions and Their Empires

Quantum Gravity Research, Los Angeles, CA 90290, USA
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Received: 20 August 2018 / Revised: 15 September 2018 / Accepted: 19 September 2018 / Published: 20 September 2018
(This article belongs to the Collection Structure and Properties of Quasicrystals)
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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. View Full-Text
Keywords: quasicrystals; empires; forced tiles; cut-and-project quasicrystals; empires; forced tiles; cut-and-project
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Hammock, D.; Fang, F.; Irwin, K. Quasicrystal Tilings in Three Dimensions and Their Empires. Crystals 2018, 8, 370.

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