Quasiperiodicity and Non-computability in Tilings

Discrete & Computational Geometry, 1998

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite.

Scripta Metallurgica, 1986

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.

Journal of Physics A: Mathematical and General, 2003

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen initial vertex. We compare different ways of counting and demonstrate that suitable averaging improves convergence to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-avoiding walks on the square lattice. For polygons, the obtained enumeration data does not allow us to draw decisive conclusions about the exponent.

2009

We propose a new simple construction of an aperiodic tile set based on self-referential (fixed point) argument.

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).

Geometriae Dedicata, 1987

The paper discusses homeomorphic types of (periodic) filings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set ~ on which three involutions ao,al and tr2 act from the right such that troa ~ = a2ao and there are two maps mol ,m12 :~ ~ ~ satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Griinbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique,, are stated.

Transactions of the American Mathematical Society, 2003

Let b ≥ 2 be an integer base, D = {0, d 1 , · · · , d b−1 } ⊂ Z a digit set and T = T (b, D) the set of radix expansions. It is well known that if T has nonvoid interior, then T can tile R with some translation set J (T is called a tile and D a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of J ; (ii) for a given b, characterize D so that T is a tile.

2007

It is known that any two domino tilings of a polygon are flip-accessible, i.e., linked by a finite sequence of local transformations, called flips. This paper considers flip-accessibility for domino tilings of the whole plane, asking whether two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, we provide three

Experimental Mathematics, 1994