Parallelogram tilings and Jacobi-Perron algorithm
type: Item
instance of: scholarly article
title: Parallelogram tilings and Jacobi-Perron algorithm
zbMATH Open document ID: 0805.52011
published in: Tokyo Journal of Mathematics
publication date: 1994-09-28T00:00:00Z
review text: The authors present the following construction of non-periodic tilings of the plane: Consider the face-to-face tiling of 3-space by unit cubes and a plane through the origin, orthogonal to a vector \((1,\alpha, \beta)\) with rationally independent coordinates. The union of those cubes meeting the plane has two boundary surfaces which are the union of (unit) squares (each a translate of some side of the unit cube). The orthogonal projection of one of these surfaces onto the plane yields a tiling by three parallelograms (corresponding to the three types of squares). This tiling is quasi-periodic (in the sense, that every boundary patch occurs in all sufficiently large patches of the tiling), and can be generated by a substitution method related to the Jacobi-Perron algorithm (applied to \((\alpha, \beta)\)). (This corresponds to a previously known construction of one-dimensional tilings related to continued fractions).
Mathematics Subject Classification ID: 52C17
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 645684
zbMATH Keywords: quasi-periodic tilings
zbMATH Keywords: substitution method
zbMATH Keywords: projection method
zbMATH Keywords: Jacobi-Perron algorithm
author: Shunji Ito
author: Makoto Ohtsuki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270128186
OpenAlex ID: W1976512427
DOI: 10.3836/TJM/1270128186
description: scientific article; zbMATH DE number 645684