## Abstract and Applied Analysis

### JPD-Coloring of the Monohedral Tiling for the Plane

#### Abstract

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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015 (2015), Article ID 258436, 8 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.aaa/1429103756

Digital Object Identifier
doi:10.1155/2015/258436

Mathematical Reviews number (MathSciNet)
MR3316989

Zentralblatt MATH identifier
1347.52017

#### Citation

El-Shehawy, S. A.; Basher, M. JPD-Coloring of the Monohedral Tiling for the Plane. Abstr. Appl. Anal. 2015 (2015), Article ID 258436, 8 pages. doi:10.1155/2015/258436. https://projecteuclid.org/euclid.aaa/1429103756

#### References

• B. Grünbaum and G. C. Shephard, “Tiling by regular polygons,” Mathematics Magazine, vol. 50, no. 5, pp. 227–247, 1977.
• B. Grünbaum and G. C. Shephard, “Perfect colorings of transitive tilings and patterns in the plane,” Discrete Mathematics, vol. 20, no. 3, pp. 235–247, 1977.
• S. Eigen, J. Navarro, and V. S. Prasad, “An aperiodic tiling using a dynamical system and Beatty sequences,” in Dynamics, Ergodic Theory, and Geometry, vol. 54 of Mathematical Sciences Research Institute, pp. 223–241, Cambridge University Press, 2007.
• C. Mann, L. Asaro, J. Hyde, M. Jensen, and T. Schroeder, “Uniform edge-$c$-colorings of the Archimedean tilings,” Discrete Mathematics, vol. 338, no. 1, pp. 10–22, 2015.
• R. Santos and R. Felix, “Perfect precise colorings of plane regular tilings,” Zeitschrift für Kristallographie, vol. 226, no. 9, pp. 726–730, 2011.
• B. Grünbaum, P. Mani-Levitska, and G. C. Shephard, “Tiling three-dimensional space with polyhedral tiles of a given isomorphism type,” Journal of the London Mathematical Society, vol. 29, no. 1, pp. 181–191, 1984.
• M. E. Basher, “$\sigma$-Coloring of the monohedral tiling,” International Journal of Mathematical Combinatorics, vol. 2, pp. 46–52, 2009.
• J. E. Freund, I. Miller, and M. Miller, Mathematical Statistics with Applications, Prentice Hall PTR, 7th edition, 2003.
• D. C. Montgomery and G. C. Runger, Applied Statistics and Probability for Engineers, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999.
• L. Bernardin, P. Chin, P. DeMarco et al., Maple Programming Guide, Maplesoft, 2011. \endinput