Tiling the unit square with squares and rectangles
type: Item
instance of: scholarly article
title: Tiling the unit square with squares and rectangles
zbMATH Open document ID: 0574.05011
DOI: 10.1016/0097-3165(85)90054-8
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: Call a rectangle small if it will fit inside the unit square; call a rectangle binary if its dimensions are powers of 2. Theorem 1. If \({\mathcal S}\) is a set of small binary squares having total area greater than 1, some finite subset of \({\mathcal S}\) will tile the unit square. Theorem 2. If \({\mathcal R}\) is a set of small binary rectangles having total area greater than 5/3, some finite subset of \({\mathcal R}\) will tile the unit square. The above bounds are sharp. Various corollaries are drawn. In particular we show that if \({\mathcal R}\) is a collection of rectangles, then \({\mathcal R}\) can cover the plane iff \(\Sigma \{\min (a,1)\cdot \min (b,1)| R\in {\mathcal R}\}=\infty,\) where for a given rectangle, a is its width and b is its length.
Mathematics Subject Classification ID: 05B45
Mathematics Subject Classification ID: 52C17
zbMATH DE Number: 3916288
zbMATH Keywords: small binary squares
zbMATH Keywords: unit square
zbMATH Keywords: small binary rectangles
author: James C. jun. Owings
MaRDI profile type: Publication
full work available at URL: 0097-3165(85)90054-8
OpenAlex ID: W2022929595
cites work: On packing of squares and cubes
description: scientific article; zbMATH DE number 3916288