Fractal Penrose tiles II: Tiles with fractal boundary as duals of Penrose triangles.
Fractal representation of the attractive lamination of an automorphism of the free group
In this paper, we extend to automorphisms of free groups some results and constructions that classically hold for morphisms of the free monoid, i.e., the so-called substitutions. A geometric representation of the attractive lamination of a class of automorphisms of the free group (irreducible with irreducible powers (iwip) automorphisms) is given in the case where the dilation coefficient of the automorphism is a unit Pisot number. The shift map associated with the attractive symbolic lamination...
Gromov hyperbolicity of planar graphs
We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version...
Indecomposable tilings of the integers with exponentially long periods.
Inflationary tilings with a similarity structure.
Integer partitions, tilings of -gons and lattices
In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of -gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a -gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.
Integer Partitions, Tilings of 2D-gons and Lattices
In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2D-gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a 2D-gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.
Isohedral tilings of the plane by polygons.
Isonemality and mononemality of woven fabrics
The paper studies the diagrams of woven fabrics consisting of white and black squares as geometrical objects and described their symmetries. The concepts of isonemality and mononemality due to B. Grünbaum and G. C. Shephard are used. A conjecture of these authors is proved in a particular case.
Locally toroidal regular polytopes of rank 4.
Moments of inertia associated with the lozenge tilings of a hexagon.
Monomer-dimer tatami tilings of rectangular regions.
...-Morphic Sets of Prototiles.
Multiple tilings of with long periods, and tiles with many-generated level semigroups.
Nestandardní zápisy čísel
Non-integer lattice tilings by (k, n) truncated crosses.
Non-intersecting, simple, symmetric random walks and the extended Hahn kernel
We show using non-intersecting paths, that a random rhombus tiling of a hexagon, or a boxed planar partition, is described by a determinantal point process given by an extended Hahn kernel.
Non-repetitive tilings.
Note on certain partitions of points in
Notes on a class of tiling problems