Abstract
A binary planar configuration A associates to each point in \(\mathbb {Z}^2\) 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 (\(\mathcal {PPS}\)) and we investigate the periodicity behaviors of the homogeneous configurations that are related to them. Then, we show that some elements in \(\mathcal {PPS}\) 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.
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Notes
- 1.
In this study, we include in the class of periodic tilings also those called half-periodic in [2].
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Ascolese, M., Frosini, A. (2022). On the Decomposability of Homogeneous Binary Planar Configurations with Respect to a Given Exact Polyomino. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_12
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DOI: https://doi.org/10.1007/978-3-031-19897-7_12
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