Compactness Theorems for Geometric Packings

✍ Scribed by Greg Martin

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 128 KB
• Volume: 97
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2001.3209

✦ Synopsis

Moser asked whether the collection of rectangles of dimensions

.., whose total area equals 1, can be packed into the unit square without overlap, and whether the collection of squares of side lengths 1 2 , 1 3 , 1 4 , ... can be packed without overlap into a rectangle of area p 2 /6 -1. Computational investigations have been made into packing these collections into squares of side length 1+e and rectangles of area p 2 /6 -1+e, respectively, and one can consider the apparently weaker question of whether such packings are possible for every positive number e. In this paper we establish a general theorem on sequences of geometrical packings that implies, in particular, that the ''for every e'' versions of these two problems are actually equivalent to the original tiling problems.