Dense quasicrystalline tilings by squares and equilateral triangles

Michael Okeeffe, Michael Treacy

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses.

Original languageEnglish (US)
Pages (from-to)5-9
Number of pages5
JournalActa Crystallographica Section A: Foundations of Crystallography
Volume66
Issue number1
DOIs
StatePublished - 2010

Fingerprint

Fractals
Cell Size
triangles
Diffraction patterns
apexes
Substitution reactions
diffraction patterns
substitutes
symmetry
cells
scattering
fractals
Scattering

Keywords

  • Fourier transforms
  • Quasicrystals
  • Tilings

ASJC Scopus subject areas

  • Structural Biology

Cite this

Dense quasicrystalline tilings by squares and equilateral triangles. / Okeeffe, Michael; Treacy, Michael.

In: Acta Crystallographica Section A: Foundations of Crystallography, Vol. 66, No. 1, 2010, p. 5-9.

Research output: Contribution to journalArticle

@article{a9ca3cecde4a49ea838ac3c463f514b7,
title = "Dense quasicrystalline tilings by squares and equilateral triangles",
abstract = "Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses.",
keywords = "Fourier transforms, Quasicrystals, Tilings",
author = "Michael Okeeffe and Michael Treacy",
year = "2010",
doi = "10.1107/S0108767309044183",
language = "English (US)",
volume = "66",
pages = "5--9",
journal = "Acta Crystallographica Section A: Foundations and Advances",
issn = "0108-7673",
publisher = "John Wiley and Sons Inc.",
number = "1",

}

TY - JOUR

T1 - Dense quasicrystalline tilings by squares and equilateral triangles

AU - Okeeffe, Michael

AU - Treacy, Michael

PY - 2010

Y1 - 2010

N2 - Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses.

AB - Dense square-symmetry tilings of the plane by equilateral triangles and squares are described. Repeated substitution of a vertex of a tiling by groups of vertices leads asymptotically to a limiting density that is independent of the starting pattern and to a family of quasicrystalline patterns with 12-fold symmetry. Diffraction patterns were computed by treating the vertices as point scatterers. As the number of substitutions increases, and as the unit-cell size increases, the diffraction patterns from a single unit cell develop a near-perfect 12-fold symmetry. In addition, the low-intensity background scattering in the diffraction patterns exhibits fractal-like self-similar properties, with motifs of local intensity recursively decorating the more intense features as the number of substitutions progresses.

KW - Fourier transforms

KW - Quasicrystals

KW - Tilings

UR - http://www.scopus.com/inward/record.url?scp=73449119914&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73449119914&partnerID=8YFLogxK

U2 - 10.1107/S0108767309044183

DO - 10.1107/S0108767309044183

M3 - Article

C2 - 20029127

AN - SCOPUS:73449119914

VL - 66

SP - 5

EP - 9

JO - Acta Crystallographica Section A: Foundations and Advances

JF - Acta Crystallographica Section A: Foundations and Advances

SN - 0108-7673

IS - 1

ER -