Symmetric polyomino tilings, tribones, ideals, and Groebner bases

TL;DR

This paper uses Groebner bases to analyze symmetric polyomino tilings, revealing specific conditions under which certain triangular regions can be signed tiled by tribones, enhancing understanding of tiling symmetries and algebraic methods.

Contribution

It introduces an algebraic approach using Groebner bases to determine tiling possibilities of symmetric polyominoes, extending prior combinatorial results with algebraic techniques.

Findings

Triangular regions T_N admit signed tribone tilings under specific N conditions.

Signed tilings occur if and only if N=27r-1 or N=27r for some integer r.

The method links algebraic ideals with geometric tiling properties.

Abstract

We apply the theory of Groebner bases to the study of signed, symmetric polyomino tilings of planar domains. Complementing the results of Conway and Lagarias we show that the triangular regions T_N=T_{3k-1} and T_N=T_{3k} in a hexagonal lattice admit a signed tiling by three-in-line polyominoes (tribones) symmetric with respect to the 120 degrees rotation of the triangle if and only if either N=27r-1 or N=27r for some integer r.