Possibilities and impossibilities in square-tiling

A. M. Berkoff, J. M. Henle, A. E. McDonough, A. P. Wesolowski

Research output: Contribution to journalArticlepeer-review


A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of sidelength n for every n in the set. From Ref. 8 we know that itself tiles the plane. From that and Ref. 9 we know that the set of even numbers tiles the plane while the set of odd numbers doesn't. In this paper we explore the nature of this property. We show, for example, that neither tiling nor non-tiling is preserved by superset. We show that a set with one or three odd numbers may tile the planebut a set with two odd numbers can't. We find examples of both tiling and non-tiling sets that can be partitioned into tiling sets, non-tiling sets or a combination. We show that any set growing faster than the Fibonacci numbers cannot tile the plane.

Original languageEnglish (US)
Pages (from-to)545-558
Number of pages14
JournalInternational Journal of Computational Geometry and Applications
Issue number5
StatePublished - Oct 2011
Externally publishedYes


  • Tiling

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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