Polyominoes which tile rectangles

Pericas, Enric;Ordóñez, Estrell

Tiling with Polyominoes, Polycubes, and Rectangles

In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes

A new mathematical model for tiling finite regions of the plane with polyominoes

We present a new mathematical model for tiling finite subsets of $\mathbb{Z}^2$ using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear programming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB

An aperiodic monotile

A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to "triskelion", added the name "turtle", added references, new H7/H8 rules (Fig 2.11), talk about reflection

Snakes in the Plane

Recent developments in tiling theory, primarily in the study of anisohedral shapes, have been the product of exhaustive computer searches through various classes of polygons. I present a brief background of tiling theory and past work, with particular emphasis on isohedral numbers, aperiodicity, Heesch numbers, criteria to characterize isohedral tilings, and various details that have arisen in past computer searches. I then develop and implement a new ``boundary-based'' technique, characterizing shapes as a sequence of characters representing unit length steps taken from a finite language of directions, to replace the ``area-based'' approaches of past work, which treated the Euclidean plane as a regular lattice of cells manipulated like a bitmap. The new technique allows me to reproduce and verify past results on polyforms (edge-to-edge assemblies of unit squares, regular hexagons, or equilateral triangles) and then generalize to a new class of shapes dubbed polysnakes, which past approaches could not describe. My implementation enumerates polyforms using Redelmeier's recursive generation algorithm, and enumerates polysnakes using a novel approach. The shapes produced by the enumeration are subjected to tests to either determine their isohedral number or prove they are non-tiling. My results include the description of this novel approach to testing tiling properties, a correction to previous descriptions of the criteria for characterizing isohedral tilings, the verification of some previous results on polyforms, and the discovery of two new 4-anisohedral polysnakes