Asymptotically optimal box packing theorems
type: Item
instance of: scholarly article
title: Asymptotically optimal box packing theorems
zbMATH Open document ID: 1179.05028
author: J. Martínez
published in: The Electronic Journal of Combinatorics
publication date: 2009-04-07T00:00:00Z
full work available at URL: https://eudml.org/doc/117253
full work available at URL: v15i1r78.html
review text: Summary: Given a protoset of \(d\)-dimensional polyominoes, we ask which boxes can be packed by the protoset. In some cases, it may be too difficult to give a complete answer to this question, so we ask the easier question about determining all sufficiently large boxes that can be packed. (We say that a box is ``sufficiently large'' if all edge lengths are \({} \geq C\) for some large \(C\).) We give numerous examples (mostly 2-dimensional) where we can answer this easier question. The various techniques involved are: checkerboard-type colorings/numberings (tile homology), the boundary word method of Conway and Lagarias (tile homotopy), ad hoc geometric arguments, and a very nice theorem of Barnes. Barnes' Theorem asserts that all necessary conditions for a box to be packable can be given in a certain form, and these conditions are also sufficient for large boxes. Barnes' Theorem has not received the appreciation it deserves. We give a new, purely combinatorial proof of this important result. (Barnes' original proof uses techniques of algebraic geometry.) In the special case that all the prototiles are boxes themselves, we show how to determine all sufficiently large boxes that they pack. We prove a theorem based on Barnes' result that reduces this to a straightforward calculation.
Mathematics Subject Classification ID: 05B50
Mathematics Subject Classification ID: 05B40
zbMATH DE Number: 5540983
zbMATH Keywords: polyominoes
zbMATH Keywords: protoset
zbMATH Keywords: box packing
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5540983