TY - JOUR

T1 - Self-similar one-dimensional quasilattices

AU - Boyle, Latham

AU - Steinhardt, Paul J.

N1 - Funding Information:
Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. PJS was supported in part by the Princeton University Innovation Fund for New Ideas in the Natural Sciences.
Publisher Copyright:
© 2022 American Physical Society.

PY - 2022/10/1

Y1 - 2022/10/1

N2 - We study 1D quasilattices, especially self-similar ones that can be used to generate two-, three-, and higher dimensional quasicrystalline tessellations that have matching rules and invertible self-similar substitution rules (also known as inflation rules) analogous to the rules for generating Penrose tilings. The lattice positions can be expressed in a closed-form expression we call floor form: xn=S(n-α)+(L-S)⌊κ(n-β)⌋, where L>S>0 and 0<κ<1 is an irrational number. We describe two equivalent geometric constructions of these quasilattices and show how they can be subdivided into various types of equivalence classes: (i) lattice equivalent, where any two quasilattices in the same lattice equivalence class may be derived from one another by a local decoration/gluing rule; (ii) self-similar, a proper subset of lattice equivalent where, in addition, the two quasilattices are locally isomorphic; and (iii) self-same, a proper subset of self-similar where, in addition, the two quasilattices are globally isomorphic (i.e., identical up to rescaling). For all three types of equivalence class, we obtain the explicit transformation law between the floor form expression for two quasilattices in the same class. We tabulate (in Table I and Fig. 5)the ten special self-similar 1D quasilattices relevant for constructing Ammann patterns and Penrose-like tilings in two dimensions and higher, and we explicitly construct and catalog the corresponding self-same quasilattices.

AB - We study 1D quasilattices, especially self-similar ones that can be used to generate two-, three-, and higher dimensional quasicrystalline tessellations that have matching rules and invertible self-similar substitution rules (also known as inflation rules) analogous to the rules for generating Penrose tilings. The lattice positions can be expressed in a closed-form expression we call floor form: xn=S(n-α)+(L-S)⌊κ(n-β)⌋, where L>S>0 and 0<κ<1 is an irrational number. We describe two equivalent geometric constructions of these quasilattices and show how they can be subdivided into various types of equivalence classes: (i) lattice equivalent, where any two quasilattices in the same lattice equivalence class may be derived from one another by a local decoration/gluing rule; (ii) self-similar, a proper subset of lattice equivalent where, in addition, the two quasilattices are locally isomorphic; and (iii) self-same, a proper subset of self-similar where, in addition, the two quasilattices are globally isomorphic (i.e., identical up to rescaling). For all three types of equivalence class, we obtain the explicit transformation law between the floor form expression for two quasilattices in the same class. We tabulate (in Table I and Fig. 5)the ten special self-similar 1D quasilattices relevant for constructing Ammann patterns and Penrose-like tilings in two dimensions and higher, and we explicitly construct and catalog the corresponding self-same quasilattices.

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U2 - 10.1103/PhysRevB.106.144112

DO - 10.1103/PhysRevB.106.144112

M3 - Article

AN - SCOPUS:85141480420

SN - 2469-9950

VL - 106

JO - Physical Review B

JF - Physical Review B

IS - 14

ER -