Related papers: On the Penrose and Taylor-Socolar Hexagonal Tiling…
We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $\tau$-tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced…
This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive…
We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the…
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This…
In this paper we describe the pentagonal tiling of the plane defined in the article "A regular pentagonal tiling of the plane" by P. L. Bowers and K. Stephenson as a conformal substitution tiling and summarize many of its properties given…
In this paper, we try to give the appropriate class of languages to which belong various objects associated with tessellations in the hyperbolic plane.
We provide first a categorical exploration of, and then completion of the mapping of the relationships among, three fundamental perspectives on binary relations: as the incidence matrices of hypergraphs, as the formal contexts of concept…
This is a review (in Italian) on aperiodic tilings of the plane intended for a general audience. First, we recall some basic results about lattices and periodic tilings. Then, we move on to one-dimensional (domino) tilings and Wang tilings.…
The theory of fractal tilings of fractal blow-ups is extended to graph-directed iterated function systems, resulting in generalizations and extensions of some of the theory of Anderson and Putnam and of Bellisard et al. regarding…
As a non-perturbative and gauge invariant regularization the lattice provides a tool for deeper understanding of the celebrated Yang-Mills theory, QCD and chiral gauge theories. For illustration, I discuss some analytic developments on the…
One well studied way to construct quasicrystalline tilings is via inflate-and-subdivide (a.k.a. substitution) rules. These produce self-similar tilings--the Penrose, octagonal, and pinwheel tilings are famous examples. We present a…
Let $B_R$ denote the closed Euclidean ball of radius $R$ in the plane. In this paper we prove that, if $V$ is the set of vertices of any unit length rhombic Penrose tiling then, for $R\ge 2$, \[\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log…
We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with any irrational angle in degree: they are three $1$-parameter families of pentagonal subdivisions of the Platonic solids, with…
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice…
Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable MC Escher like image that works esthetically as well as functionally requires resolving incongruencies at a tile's…
In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.
We study Penrose tilings of the plane $\mathbb{R}^2$ and nonperiodic infinite frieze patterns from the point of view of Cohen--Macaulay representation theory: Triangulations of the completed infinity-gon correspond to subcategories of the…
Let X be a smooth elliptic fibration over a smooth base B. Under mild assumptions, we establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an O^* gerbe over a genus one fibration which is a…
The connection between symmetries and linearizations of discrete-time dynamical systems is being inverstigated. It is shown, that existence of semigroup structures related to the vector field and having linear representations enables…
Two-component mixtures in optical lattices reveal a rich variety of different phases. We employ an exact diagonalization method to obtain the relevant correlation functions in hexagonal optical lattices to characterize those phases. We…