Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings

This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the "admissible" neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance to this relation. We prove that any validly tiled path defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.Comment: Submitted to Theoretical Computer Scienc

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

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

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

50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

FORMULAE AND ASYMPTOTICS FOR COEFFICIENTS OF ALGEBRAIC FUNCTIONS

International audienc

A counterexample to the periodic tiling conjecture

The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.Comment: 50 pages, 13 figures. Minor changes and additions of new reference

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum