Universal tilings and universal (0,1)-matrices
type: Item
instance of: scholarly article
title: Universal tilings and universal (0,1)-matrices
zbMATH Open document ID: 0597.05021
DOI: 10.1016/0012-365X(86)90189-5
author: C. R. J. Clapham
published in: Discrete Mathematics
publication date: 1986-01-01T00:00:00Z
review text: A periodic edge-to-edge tiling of the plane by black and white squares is called k-universal if it contains all possible (k\(\times k)\)-blocks of black and white tiles. The paper discusses the problem of finding the k- universal tilings with the smallest fundamental block. This problem is shown to be related to the problem of finding the k-universal (0,1)- matrices of minimal size; here, k-universal means that any (0,1)-matrix of size \(k\times k\) occurs as a submatrix. It is proved that there are k- universal matrices of size \(k2^{k/2}\times k2^{k/2}\) or \((3k+1)2^{(k-3)/2}\times (3k-1)2^{(k-3)/2}\) if k is even or odd, respectively. Then, the 3-universal (10\(\times 8)\)-matrix is used to construct a 3-universal tiling of the plane.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 3961633
zbMATH Keywords: coloured tilings
zbMATH Keywords: (0,1)-matrices
reviewed by: Egon Schulte
MaRDI profile type: Publication
cites work: When a Fabric Hangs together
cites work: Satins and Twills: An Introduction to the Geometry of Fabrics
cites work: Q5585195
description: scientific article; zbMATH DE number 3961633