Aperiodic Tilings by Right Triangles

2014 ◽  
pp. 29-41
Author(s):  
Nikolay Vereshchagin
Keyword(s):  
Aperiodic Tilings

scholarly journals Relating diffraction and spectral data of aperiodic tilings: Towards a Bloch theorem

2021 ◽  
pp. 104217
Author(s):  
Eric Akkermans ◽  
Yaroslav Don ◽  
Jonathan Rosenberg ◽  
Claude L. Schochet
Keyword(s):  
Spectral Data ◽  
Aperiodic Tilings ◽  
Bloch Theorem

Generalized matching rules for aperiodic tilings

2001 ◽  
Vol 316 (1-2) ◽  
pp. 39-45 ◽  
Author(s):  
Paolo Bellingeri ◽  
Paolo M. Ossi
Keyword(s):  
Aperiodic Tilings ◽  
Matching Rules

Aperiodic tilings of the hyperbolic plane by convex polygons

1998 ◽  
Vol 107 (1) ◽  
pp. 319-325 ◽  
Author(s):  
G. A. Margulis ◽  
S. Mozes
Keyword(s):  
Hyperbolic Plane ◽  
Convex Polygons ◽  
Aperiodic Tilings

Percolation on Penrose Tilings

1998 ◽  
Vol 41 (2) ◽  
pp. 166-177 ◽  
Author(s):  
A. Hof
Keyword(s):  
Large Class ◽  
Arbitrary Dimension ◽  
Bond Percolation ◽  
Critical Probability ◽  
Site Percolation ◽  
Aperiodic Tilings ◽  
Infinite Clusters ◽  
Almost All ◽  
Percolation Processes ◽  
Penrose Tilings

AbstractIn Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ℤd, and to other percolation processes, including Bernoulli bond percolation.

Radiation properties of planar antenna arrays based on certain categories of aperiodic tilings

2005 ◽  
Vol 53 (2) ◽  
pp. 635-644 ◽  
Author(s):  
V. Pierro ◽  
V. Galdi ◽  
G. Castaldi ◽  
I.M. Pinto ◽  
L.B. Felsen
Keyword(s):  
Antenna Arrays ◽  
Planar Antenna ◽  
Aperiodic Tilings ◽  
Radiation Properties

LATTICE GAS MODELS ON SELF-SIMILAR APERIODIC TILINGS

1991 ◽  
Vol 03 (02) ◽  
pp. 163-221 ◽  
Author(s):  
C. P. M. GEERSE ◽  
A. HOF
Keyword(s):  
Lattice Gas ◽  
Arbitrary Dimension ◽  
Quantum Systems ◽  
Translation Invariance ◽  
Aperiodic Tilings ◽  
Mean Pressure ◽  
Lattice Gas Models ◽  
Self Similar ◽  
The Mean ◽  
Penrose Tilings

We discuss lattice gas models on the vertices of tilings in arbitrary dimension that are self-similar in the way Penrose tilings of the plane are self-similar. Among these, there are systems that fundamentally lack translation invariance. Under natural hypotheses on the interactions and the states, we prove the existence of thermodynaraic functions — the mean pressure, the mean energy and the mean entropy — and derive the variational principle. The relation between Gibbs states and tangent functionals to the mean pressure is investigated. Generalizations to quantum systems are also discussed. Our work extends results known for lattice gas models on periodic lattices.

scholarly journals Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesBn

2015 ◽  
Vol 71 (2) ◽  
pp. 175-185 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Ramazan Koc
Keyword(s):  
Three Dimensional ◽  
Lattice Points ◽  
Icosahedral Symmetry ◽  
The Novel ◽  
Voronoi Cells ◽  
Aperiodic Tilings ◽  
Coxeter Number ◽  
Hypercubic Lattice ◽  
Higher Dimensional ◽  
Coxeter Plane

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter–Weyl groupWa(Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroupDhofW(Bn) withh= 2nrepresenting the Coxeter number describes theh-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed forn= 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroupsW(A3),W(H2) ×W(A1) andW(H3) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the latticeB4onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. TheB5lattice is used to describe both fivefold and tenfold symmetries. The latticeB6can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

scholarly journals Pattern Equivariant Mass Transport in Aperiodic Tilings and Cohomology

2020 ◽  
Author(s):  
Michael Kelly ◽  
Lorenzo Sadun
Keyword(s):  
Mass Transport ◽  
Aperiodic Tilings ◽  
Mass Distributions ◽  
Aperiodic Tiling ◽  
Cech Cohomology

Abstract Suppose that we have a repetitive and aperiodic tiling ${\textbf{T}}$ of ${\mathbb{R}}^n$ and two mass distributions $f_1$ and $f_2$ on ${\mathbb{R}}^n$, each pattern equivariant (PE) with respect to ${\textbf{T}}$. Under what circumstances is it possible to do a bounded transport from $f_1$ to $f_2$? When is it possible to do this transport in a strongly or weakly PE way? We reduce these questions to properties of the Čech cohomology of the hull of ${\textbf{T}}$, properties that in most common examples are already well understood.

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