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Acta Crystallographica Section A, ISSN 2053-2733, 01/2019, Volume 75, Issue 1, pp. 94 - 106
In this article, a framework is presented that allows the systematic derivation of planar edge‐to‐edge k‐isocoronal tilings from tile‐s‐transitive tilings, s ≤... 
vertex‐k‐transitive tilings | k‐isocoronal tilings | k‐uniform tilings | isocoronal tilings | uniform tilings | Isocoronal tilings | k-isocoronal tilings | Uniform tilings | k-uniform tilings | Vertex-k-transitive tilings | Euclidean geometry | Tiling | Apexes | Symmetry
Journal Article
Acta Crystallographica Section A, ISSN 2053-2733, 01/2019, Volume 75, Issue 1, pp. 3 - 13
Journal Article
Discrete and Computational Geometry, ISSN 0179-5376, 2019, Volume 63, Issue 3, pp. 577 - 606
We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in... 
Non-periodic tilings | Robust tilings | Ammann tilings | Self-similar tilings | Substitution tilings | MATHEMATICS | COMPUTER SCIENCE, THEORY & METHODS | Theorems | Tiling | Self-similarity | Mathematics | Computational Geometry | Combinatorics | General Mathematics | Computer Science | Discrete Mathematics
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 03/2020, Volume 373, Issue 3, pp. 2173 - 2190
As part of our ongoing work on the enumeration of symmetry classes of lozenge tilings of hexagons with certain four-lobed structures removed from their center,... 
MATHEMATICS | SYMMETRIES | GAPS | TILINGS
Journal Article
Acta Crystallographica Section A, ISSN 2053-2733, 07/2018, Volume 74, Issue 4, pp. 388 - 398
This work introduces the idea of symmetry order, which describes the rotational symmetry types of tilings in the hull of a given substitution. Definitions are... 
aperiodic tilings | rotation‐invariant tilings | symmetry order | dense tile orientations | substitution tilings | Symmetry order | Aperiodic tilings | Substitution tilings | Dense tile orientations | Rotation-invariant tilings | rotation-invariant tilings | CRYSTALLOGRAPHY | CHEMISTRY, MULTIDISCIPLINARY | Substitutes | Hulls (structures) | Hulls | Symmetry
Journal Article
European Journal of Combinatorics, ISSN 0195-6698, 01/2020, Volume 83, p. 103022
The Golomb–Welch conjecture (1968) states that there are no e-perfect Lee codes in Zn for n≥3 and e≥2. This conjecture remains open even for linear codes. A... 
MATHEMATICS | THEOREMS | TILINGS
Journal Article
Physical Review Letters, ISSN 0031-9007, 06/2018, Volume 120, Issue 24, p. 247401
We study a continuum of photonic quasicrystal heterostructures derived from local isomorphism (LI) classes of pentagonal quasicrystal tilings. These tilings... 
PHYSICS, MULTIDISCIPLINARY | RULES | TILINGS
Journal Article
Mathematica Scandinavica, ISSN 0025-5521, 2015, Volume 117, Issue 2, pp. 186 - 202
We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C* crossed products to describe the K-theory of C-r*(S) when S is an... 
MATHEMATICS | TILINGS
Journal Article
Discrete Mathematics, ISSN 0012-365X, 04/2018, Volume 341, Issue 4, pp. 957 - 964
Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in... 
Integer partitions | Tilings | Gap sequence | MATHEMATICS | ONE-DIMENSIONAL TILINGS
Journal Article
Physical Review Letters, ISSN 0031-9007, 09/2005, Volume 95, Issue 13, p. 133901
We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase matching of... 
SPACE-GROUPS | GENERATION | PHYSICS, MULTIDISCIPLINARY | TILINGS
Journal Article
Acta Crystallographica Section A, ISSN 2053-2733, 05/2017, Volume 73, Issue 3, pp. 246 - 256
The intimate relationship between the Penrose and the Taylor–Socolar tilings is studied, within both the context of double hexagon tiles and the algebraic... 
inverse sequences of hierarchical lattices | nested triangularizations | Taylor–Socolar tiling | Penrose tiling | double hexagon tiling | Taylor-Socolar tiling | CRYSTALLOGRAPHY | CHEMISTRY, MULTIDISCIPLINARY | Foundations | Algebra | Tiles | Hexagons | Lattices | Tiling | Inverse | Crystallography
Journal Article
Chemical Society Reviews, ISSN 0306-0012, 10/2012, Volume 41, Issue 20, pp. 6821 - 6843
Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The... 
SQUARE-TRIANGLE TILINGS | NON-PERIODIC TILINGS | DODECAGONAL QUASI-CRYSTALS | MATCHING RULES | PENROSE TILINGS | PURE POINT DIFFRACTION | REAL-SPACE STRUCTURE | LOCAL DERIVABILITY | DIRAC COMBS | CHEMISTRY, MULTIDISCIPLINARY | DIFFUSE-SCATTERING
Journal Article
Acta Crystallographica Section A, ISSN 2053-2733, 05/2017, Volume 73, Issue 3, pp. 227 - 230
All trinodal, edge‐2‐transitive polyhedra and 2‐periodic tilings are enumerated and described. These are of special interest for the design and synthesis of... 
nets | minimal transitivity | tilings | DESIGN | METAL-ORGANIC FRAMEWORKS | CRYSTALLOGRAPHY | CHEMISTRY, MULTIDISCIPLINARY | TRANSITIVE TILINGS | Polyhedrons | Foundations | Tiling | Polyhedra | Synthesis | Crystallography
Journal Article
Journal of Combinatorial Theory, Series A, ISSN 0097-3165, 11/2018, Volume 160, pp. 409 - 445
Aperiodic tiling — a form of complex global geometric structure arising through locally checkable, constant-time matching rules — has long been closely tied to... 
Discrete geometry | Substitution tiling | Aperiodic tiling | MATHEMATICS | SUBSTITUTION | DYNAMICAL-SYSTEMS | TILINGS | NONPERIODICITY
Journal Article
Acta Crystallographica Section A, ISSN 2053-2733, 01/2019, Volume 75, Issue 1, pp. 121 - 134
An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the... 
dual tiling | uniform tiling | tetravalent vertices | Cairo tiling | coordination sequences | trivalent vertices | Trivalent vertices | Tetravalent vertices | Uniform tiling | Coordination sequences | Dual tiling | CRYSTAL-STRUCTURES | CRYSTALLOGRAPHY | CHEMISTRY, MULTIDISCIPLINARY | NETS | LATTICES | GROWTH SERIES | TOPOLOGICAL DENSITY | Coloring | Graph theory | Tiling | Apexes | Mathematics - Combinatorics
Journal Article
2016, Encyclopedia of mathematics and its applications, ISBN 9781107077027, Volume 159, xviii, 475 page
"Internationally recognised researchers look at developing trends in combinatorics with applications in the study of words and in symbolic dynamics. They... 
Symbolic dynamics | Computer science | MATHEMATICS / Discrete Mathematics | Combinatorial analysis | symbolic dynamics | tilings | combinatorics
Book
Advances in Mathematics, ISSN 0001-8708, 2010, Volume 225, Issue 6, pp. 3134 - 3165
The construction of friezes is motivated by the theory of cluster algebras. It gives, for each acyclic quiver, a family of integer sequences, one for each... 
Dynkin diagrams | Laurent phenomenon | [formula omitted] tilings | Rationality | Periodicity | Cluster algebras | Positivity | Euclidean diagrams | Frieze patterns | tilings | MATHEMATICS | SL2 tilings
Journal Article
Michigan Mathematical Journal, ISSN 0026-2285, 12/2016, Volume 65, Issue 4, pp. 715 - 747
We define spectral triples for stationary Bratteli diagrams and study associated zeta functions, traces of heat kernels, and their spectral states. We observe... 
MATHEMATICS | TILINGS | GEOMETRY
Journal Article
Physical Review Letters, ISSN 0031-9007, 08/2018, Volume 121, Issue 9
While the enhancement of spin-space symmetry from the usual SU(2) to SU(N) is promising for finding nontrivial quantum spin liquids, its realization in... 
3-PERIODIC NETS | SYSTEMS | PHYSICS, MULTIDISCIPLINARY | TILINGS | TOPOLOGICAL ORDER | GAP
Journal Article
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