A binary planar configuration A associates to each point in Z2 an element in {0,1}. Provided a finite window probe P, we locally inspect A by moving P in all its possible positions and counting the 1s elements that fit inside it. In case all the computed values have the same value k, then we say that A is k-homogeneous w.r.t. P. A recent conjecture states that a binary planar configuration is k-homogeneous with respect to an exact polyomino P, i.e., a polyomino that tiles the plane by translation, if and only if it can be decomposed into k configurations that are 1-homogeneous with respect to P. In this paper we define a class of exact polyominoes called perfect pseudo-squares (PPS) and we investigate the periodicity behaviors of the homogeneous configurations that are related to them. Then, we show that some elements in PPS allow 2-homogeneous or 3-homogeneous non-decomposable planar configurations, so providing evidence that the conjecture does not hold for the whole class of exact polyominoes.
On the Decomposability of Homogeneous Binary Planar Configurations with Respect to a Given Exact Polyomino / Ascolese, Michela; Frosini, Andrea. - ELETTRONICO. - 13493:(2022), pp. 139-152. (Intervento presentato al convegno Discrete Geometry and Mathematical Morphology) [10.1007/978-3-031-19897-7_12].
On the Decomposability of Homogeneous Binary Planar Configurations with Respect to a Given Exact Polyomino
Ascolese, Michela;Frosini, Andrea
2022
Abstract
A binary planar configuration A associates to each point in Z2 an element in {0,1}. Provided a finite window probe P, we locally inspect A by moving P in all its possible positions and counting the 1s elements that fit inside it. In case all the computed values have the same value k, then we say that A is k-homogeneous w.r.t. P. A recent conjecture states that a binary planar configuration is k-homogeneous with respect to an exact polyomino P, i.e., a polyomino that tiles the plane by translation, if and only if it can be decomposed into k configurations that are 1-homogeneous with respect to P. In this paper we define a class of exact polyominoes called perfect pseudo-squares (PPS) and we investigate the periodicity behaviors of the homogeneous configurations that are related to them. Then, we show that some elements in PPS allow 2-homogeneous or 3-homogeneous non-decomposable planar configurations, so providing evidence that the conjecture does not hold for the whole class of exact polyominoes.File | Dimensione | Formato | |
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