Keller's Conjecture Revisited

Abstract

In 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a tile of more complex shape is true.