Some Generalizations of the Pinwheel Tiling

Scripta Metallurgica, 1986

2005

Acta Crystallographica Section A Foundations of Crystallography, 1988

We generalize the grid-projection method for the construction of quasiperiodic tilings. A rather general fundamental domain of the associated higher dimensional lattice is used for the construction of the acceptance region. The arbitrariness of the fundamental domain allows for a choice which obeys all the symmetries of the lattice, which is important for the construction of tilings with a given non-trivial point group symmetry in Fourier space. As an illustration, the construction of a 2d quasiperiodic tiling with twelvefold orientational symmetry is described.

Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01, 2001

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity at least n. This adds a quantitative angle to classical results on non-recursivity of tilings -that we also develop in terms of Turing degrees of unsolvability.

arXiv: Metric Geometry, 2015

General substitution rules for non-periodic rhomb tilings are derived. From the requirement that all substitution tiles consist of a discrete number of prototiles, it follows that a substitution tile with angle s*pi/n must be built out of pairs of prototiles with angles (s+t)*pi/n and (s-t)*pi/n, except if t=0. In addition, we require that a discrete number of prototile edges must fit between the beginning and endpoint of the substitution tile edge. By comparing the total area of the discrete number of prototiles constituting the substitution tile and the total area derived from the assumed edge shape, a set of substitution rules is derived. The generalization involves the introduction of prototiles with a negative area or subtraction tiles.

The American Mathematical Monthly

The purpose of this paper is to give the flavor of the subject of self-similar tilings in a relatively elementary setting, and to provide a novel method for the construction of such polygonal tilings.

Journal of Integer Sequences, 2009

We consider a weighted square-and-domino tiling model obtained by assigning real number weights to the cells and boundaries of an n-board. An important special case apparently arises when these weights form periodic sequences. When the weights of an nm-tiling form sequences having period m, it is shown that such a tiling may be regarded as a meta-tiling of length n whose weights have period 1 except for the first cell (i.e., are constant). We term such a contraction of the period in going from the longer to the shorter tiling as "period compression." It turns out that period compression allows one to provide combinatorial interpretations for certain identities involving continued fractions as well as for several identities involving Fibonacci and Lucas numbers (and their generalizations).

Труды математического института им. Стеклова, 2015

A new method for constructing aperiodic tilings is presented. The method is illustrated by constructing a particular tiling and its hull. The properties of this tiling and the hull are studied. In particular it is shown that these tilings have a substitution rule, that they are nonperiodic, aperiodic, limitperiodic and pure point diffractive. Dedicated to Nikolai Petrovich Dolbilin on the occasion of his 70th birthday.

SCIENCE AND TECHNOLOGY, 2012

Abstract. Parametric Tile Pasting P System (PTPPS) is a new computational model, based on the structure and functioning of living cells, to generate tiling patterns, using simple rules of gluing tiles (pasting rule) at their edges and geometric operations on the tiles. In this ...

2000

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 locations Λ, where Λ is a multiset in the plane. The pairing property of K makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for K to tile with a given lattice Λ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them) we move on to prove that a large class of such polygons tiles only quasi-periodically, which for us means that Λ must be a finite union of translated 2-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile necessarily quasi-periodically, if at all.