An algorithm for deciding if a polyomino tiles the plane

An algorithm for deciding if a polyomino tiles the plane by translations

For polyominoes coded by their boundary word, we describe a quadratic O(n) algorithm in the boundary length n which improves the naive O(n) algorithm. Techniques used emanate from algorithmics, discrete geometry and combinatorics on words.

On the problem of deciding if a polyomino tiles the plane by translation

A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n log2 n)-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provença...

A parallelogram tile fills the plane by translation in at most two distinct ways

We consider the tilings by translation of a single polyomino or tile on the square grid Z. It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallogram tilings. In this paper, w...

Polyominoes simulating arbitrary-neighborhood zippers and tilings

This paper provides a bridge between the classical tiling theory and cellular automata on one side, and the complex neighborhood self-assembling situations that exist in practice, on the other side. A neighborhood N is a finite set of pairs (i, j) ∈ Z, indicating that the neighbors of a position (x, y) are the positions (x + i, y + j) for (i, j) ∈ N . This includes classical neighborhoods of si...