The T(5) property of packings of squares

  • Ted Bisztriczky University of Calgary, Canada
  • Károly Böröczky Eötvös Loránd University, Hungary
  • Károly J. Böröczky Alfréd Rényi Institute of Mathematics, Hungary
Keywords: Transversals, parallelograms, Minkowski plane


According to a classical theorem of Gruenbaum, if any five of a family of pairwise disjoint translates of a square has a transversal line (the family satisfies T(5)), then the whole family has a transversal line (satisfies T). First we show that this result is optimal in the sense that the “T(5) implies T" property does not necessarily hold anymore if only the slightly shrinked versions of the squares are pairwise disjoint. Next we prove the “T(5) implies T" property for a family of translates of squares if the interiors are pairwise disjoint and there exist two translates meeting at a common vertex.

Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)