×

JPD-coloring of the monohedral tiling for the plane. (English) Zbl 1347.52017

Summary: We introduce a definition of coloring by using joint probability distribution “JPD-coloring” for the plane which is equipped by tiling \(\mathfrak{I}\). We investigate the JPD-coloring of the \(r\)-monohedral tiling for the plane by mutually congruent regular convex polygons which are equilateral triangles at \(r=3\) or squares at \(r=4\) or regular hexagons at \(r=6\). Moreover we present some computations for determining the corresponding probability values which are used to color in the three studied cases by MAPLE-Package.

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
60D05 Geometric probability and stochastic geometry

Software:

Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grünbaum, B.; Shephard, G. C., Tiling by regular polygons, Mathematics Magazine, 50, 5, 227-247 (1977) · Zbl 0385.51006 · doi:10.2307/2689529
[2] Grünbaum, B.; Shephard, G. C., Perfect colorings of transitive tilings and patterns in the plane, Discrete Mathematics, 20, 3, 235-247 (1977) · Zbl 0377.05014 · doi:10.1016/0012-365x(77)90063-2
[3] Eigen, S.; Navarro, J.; Prasad, V. S., An aperiodic tiling using a dynamical system and Beatty sequences, Dynamics, Ergodic Theory, and Geometry. Dynamics, Ergodic Theory, and Geometry, Mathematical Sciences Research Institute, 54, 223-241 (2007), Cambridge University Press · Zbl 1215.52008 · doi:10.1017/cbo9780511755187.009
[4] Mann, C.; Asaro, L.; Hyde, J.; Jensen, M.; Schroeder, T., Uniform edge-\(c\)-colorings of the Archimedean tilings, Discrete Mathematics, 338, 1, 10-22 (2015) · Zbl 1308.52017 · doi:10.1016/j.disc.2014.08.015
[5] Santos, R.; Felix, R., Perfect precise colorings of plane regular tilings, Zeitschrift für Kristallographie, 226, 9, 726-730 (2011) · doi:10.1524/zkri.2011.1391
[6] Grünbaum, B.; Mani-Levitska, P.; Shephard, G. C., Tiling three-dimensional space with polyhedral tiles of a given isomorphism type, Journal of the London Mathematical Society, 29, 1, 181-191 (1984) · Zbl 0533.52006 · doi:10.1112/jlms/s2-29.1.181
[7] Basher, M. E., \(σ\)-Coloring of the monohedral tiling, International Journal of Mathematical Combinatorics, 2, 46-52 (2009) · Zbl 1198.05038
[8] Freund, J. E.; Miller, I.; Miller, M., Mathematical Statistics with Applications (2003), Prentice Hall PTR
[9] Montgomery, D. C.; Runger, G. C., Applied Statistics and Probability for Engineers (1999), New York, NY, USA: John Wiley & Sons, New York, NY, USA
[10] Bernardin, L.; Chin, P.; DeMarco, P.; Geddes, K. O.; Hare, D. E. G.; Head, K. M.; Labahn, G.; May, J. P.; McCarron, J.; Monagan, M. B.; Ohashi, D.; Vorkoetter, S. M., Maple Programming Guide (2011), Maplesoft
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.