Related papers: 6-Fold Quasiperiodic Tilings With Two Diamond Shap…
It is shown that there are primitive substitution tilings with dense tile orientations invariant under n-fold rotation for n=2,3,4,5,6,8. The proof for dense tile orientations uses a general result about irrationality of angles in certain…
We classify the dihedral edge-to-edge tilings of the sphere by regular polygons with gonality at least 5 and rhombi.
A systematic, decoration-based technique to discover the atomic structure of a decagonal quasicrystal, given pair potentials and experimentally measured lattice constants, is applied to the ``basic'' cobalt-rich decagonal Al-Co-Ni…
What do you get when you cross a crystal with a quasicrystal? The surprising answer stretches from Fibonacci to Kepler, who nearly 400 years ago showed how the ancient tiles of Archimedes form periodic patterns.
The study of tilings is a major problem in many mathematical instances, which is studied in two main different approaches: when considering the existence (or obstructions to the existence) of a tiling with a given tile and the other…
Motivated by a question of Erd\"{o}s and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$)…
A \emph{semi-equivelar gem} of a PL $d$-manifold is a regular colored graph that represents the manifold and admits a regular embedding on a surface, such that the cyclic sequence of face degrees around each vertex is identical. In [1,4],…
We describe how an ensemble of four-level atoms in the diamond-type configuration can be applied to create a fully controllable effective coupling between two cavity modes. The diamond-type configuration allows one to use a bimodal cavity…
We briefly review the standard methods used to construct quasiperiodic tilings, such as the projection, the inflation, and the grid method. A number of sample Mathematica programs, implementing the different approaches for one- and…
Quasicrystals allow for symmetries that are impossible in crystalline materials, such as eight-fold rotational symmetry, enabling the existence of novel higher-order topological insulators in two dimensions without crystalline counterparts.…
In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For…
We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a3b). Parallel to a complete classification by Cheung, Luk and Yan, the method implemented…
We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$, then there are at most three nonsimilar triangles $T$ such that the angles of $T$…
Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles…
The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory…
Graphite and diamond are two well-known allotropes of carbon with distinct physical properties due to different atomic connectivity. Graphite has a layered structure in which the honeycomb carbon sheets can easily glide, while atoms in…
Motivated by problems on the \'etale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the \'etale…
The Ammann A2 tiling is a simple aperiodically ordered tiling of the plane. We consider the graph derived from this tiling, by treating each corner of each tile as a vertex and each side of each tile as an edge. We present a closed-form…
Tilings of a surface of negative Euler characteristic by n-gons with n\ge 7 is a finite problem. We develop the algorithm for finding all the tilings for fixed number of tiles and present the calculation for tilings of surfaces of small…
We show that convex pentagons that can generate edge-to-edge monohedral tilings of the plane can be classified into exactly eight types. Using these results, it is also proved that no single convex polygon can be an aperiodic prototile…