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Quasicrystal Tilings in Three Dimensions and Their Empires

Quantum Gravity Research, Los Angeles, CA 90290, USA
Author to whom correspondence should be addressed.
Crystals 2018, 8(10), 370;
Received: 20 August 2018 / Revised: 15 September 2018 / Accepted: 19 September 2018 / Published: 20 September 2018
(This article belongs to the Special Issue Structure and Properties of Quasicrystals)
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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MDPI and ACS Style

Hammock, D.; Fang, F.; Irwin, K. Quasicrystal Tilings in Three Dimensions and Their Empires. Crystals 2018, 8, 370.

AMA Style

Hammock D, Fang F, Irwin K. Quasicrystal Tilings in Three Dimensions and Their Empires. Crystals. 2018; 8(10):370.

Chicago/Turabian Style

Hammock, Dugan, Fang Fang, and Klee Irwin. 2018. "Quasicrystal Tilings in Three Dimensions and Their Empires" Crystals 8, no. 10: 370.

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