Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs
Abstract
:1. Introduction
2. Methods
2.1. Converting of Tessellations into Bi-Colored, Complete Graphs
2.2. Shannon Entropy of the Introduced Graphs and Tessellations
2.3. Transformation of the Voronoi Tessellation into the Bi-Colored Complete Graph
3. Methods
3.1. Statistics of Voronoi Tessellation
3.2. Statistics of Poisson Line Tessellation
3.3. Numerical Simulation
3.3.1. Random Voronoi Tessellation
3.3.2. Random Polygons Produced by Straight Lines (Poisson Line Tessellation)
4. Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Coxeter, H.S.M. Chapter IV, Two-Dimensional Crystallography. In Introduction to Geometry; John Wiley and Sons: New York, NY, USA, 1969; pp. 50–65. [Google Scholar]
- Coxeter, H.S.M. Chapter IV, Tessellations and Honeycombs. In Regular Polytopes; Dover Publications: New York, NY, USA, 1973; pp. 58–73. [Google Scholar]
- Fulton, C. Tessellations. Am. Math. Mon. 1992, 99, 442–445. [Google Scholar] [CrossRef]
- He, Y.H.; van Loon, M. Gauge theories, tessellations & Riemann surfaces. J. High Energy Phys. 2014, 2014, 53. [Google Scholar]
- Wu, S.; Sun, Y. Tessellating tiny tetrahedrons. Science 2018, 362, 1354–1355. [Google Scholar] [CrossRef] [PubMed]
- Meloni, M.; Zhang, Q.; Pak, J.; Bilore, M.N.; Ma, R.; Ballegaard, E.; Lee, D.; Cai, J. Designing origami tessellations composed of quadrilateral meshes and degree-4 vertices for engineering applications. Autom. Constr. 2022, 142, 104482. [Google Scholar] [CrossRef]
- Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J.W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 1984, 53, 1951–1953. [Google Scholar] [CrossRef]
- Collins, L.; Witte, T.; Silverman, R.; Green, D.B.; Gomes, K.K. Imaging quasiperiodic electronic states in a synthetic Penrose tiling. Nat. Commun. 2017, 8, 15961. [Google Scholar] [CrossRef] [PubMed]
- Bursill, L.; Lin, P.J. Penrose tiling observed in a quasi-crystal. Nature 1985, 316, 50–51. [Google Scholar] [CrossRef]
- Bormashenko, E.; Legchenkova, I.; Frenkel, M.; Shvalb, N.; Shoval, S. Voronoi Entropy vs. Continuous Measure of Symmetry of the Penrose Tiling: Part I. Analysis of the Voronoi Diagrams. Symmetry 2021, 13, 1659. [Google Scholar]
- Wilson, R.J. Introduction to Graph Theory, 4th ed.; Addison Wesley Longman Limited: Harlow, UK, 1996; pp. 8–21. [Google Scholar]
- Trudeau, R.J. Introduction to Graph Theory (Corrected, Enlarged Republication ed.); Dover Pub.: New York, NY, USA, 1993; pp. 19–64. [Google Scholar]
- Li, Y.; Lin, Q. Elementary Methods of the Graph Theory; Applied Mathematical Sciences; Springer: Cham, Switzerland, 2020; pp. 3–44. [Google Scholar]
- Katz, M.; Reimann, J. An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics; Student Mathematical Library; American Mathematical Society: Providence, RI, USA, 2018; Volume 87, pp. 1–34. [Google Scholar]
- Graham, R.L.; Spencer, J.H. Ramsey Theory. Sci. Am. 1990, 7, 112–117. [Google Scholar] [CrossRef]
- Graham, R.; Butler, S. Rudiments of Ramsey Theory, 2nd ed.; American Mathematical Society: Providence, RI, USA, 2015; pp. 7–46. [Google Scholar]
- Chartrand, G.; Chatterjee, R.; Zhang, P. Ramsey chains in graphs. Electron. J. Math. 2023, 6, 1–14. [Google Scholar] [CrossRef]
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar] [CrossRef]
- Ben-Naim, A. Entropy, Shannon’s Measure of Information and Boltzmann’s H-Theorem. Entropy 2017, 19, 48. [Google Scholar] [CrossRef]
- Frenkel, M.; Shoval, S.; Bormashenko, E. Shannon Entropy of Ramsey Graphs with up to Six Vertices. Entropy 2023, 25, 1427. [Google Scholar] [CrossRef] [PubMed]
- Voronoi, G. Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire. Recherches sur les paralléloèdres primitifs. Reine Angew. Math. 1908, 134, 198–287. [Google Scholar] [CrossRef]
- Okabe, A.; Boots, B.; Sugihara, K. Spatial Tessellations Concepts and Applications of Voronoi Diagrams; John Wiley & Sons: Chichester, UK, 2000. [Google Scholar]
- Barthélemy, M. Spatial networks. Phys. Rep. 2011, 499, 1–101. [Google Scholar] [CrossRef]
- Bormashenko, E.; Frenkel, M.; Vilk, A.; Legchenkova, I.; Fedorets, A.A.; Aktaev, N.E.; Dombrovsky, L.A.; Nosonovsky, M. Characterization of Self-Assembled 2D Patterns with Voronoi Entropy. Entropy 2018, 20, 956. [Google Scholar] [CrossRef] [PubMed]
- Habib, F.; Megahed, N.A.; Badawy, N.; Shahda, M.M. D4G framework: A novel Voronoi diagram classification for decoding natural geometrics to enhance the built environment. Archit. Sci. Rev. 2024, 1–28. [Google Scholar] [CrossRef]
- Angelucci, G.; Mollaioli, F. Voronoi-like Grid Systems for Tall Buildings. Front. Built Environ. 2018, 4, 78. [Google Scholar] [CrossRef]
- Zhu, S.; Borodin, E.; Jivkov, A.P. Discrete modelling of continuous dynamic recrystallisation by modified Metropolis algorithm. Comput. Mater. Sci. 2024, 234, 112804. [Google Scholar] [CrossRef]
- Bolshakov, P.; Kharin, N.; Agathonov, A.; Halinin, E.; Sachenkov, O. Extension of the Voronoi Diagram Algorithm to Orthotropic Space for Material Structural Design. Biomimetics 2024, 9, 185. [Google Scholar] [CrossRef]
- Jungck, J.R.; Pelsmajer, M.J.; Chappel, C.; Taylor, D. The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics. Mathematics 2021, 9, 2726. [Google Scholar] [CrossRef]
- Hayen, A.; Quine, M. The proportion of triangles in a Poisson-Voronoi tessellation of the plane. Adv. Appl. Prob. (SGSA) 2002, 32, 67–74. [Google Scholar] [CrossRef]
- Calka, P. The explicit expression of the distribution of the number of sides of the typical Poisson Voronoi cell. Adv. Appl. Probab. 2003, 35, 863–870. [Google Scholar] [CrossRef]
- Hinde, A.L.; Miles, R.E. Monte Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Stat. Comput. Simul. 1980, 10, 205–223. [Google Scholar] [CrossRef]
- Suárez-Plasencia, L.; Herrera-Macías, J.A.; Legón-Pérez, C.M.; Socorro-Lanes, R.; Rojas, O.; Sosa-Gómez, G. Analysis of the Number of Sides of Voronoi Polygons in PassPoint. In Computer Science and Health Engineering in Health Services. COMPSE 2020, 4th EAI Virtual International Conference, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering; Marmolejo-Saucedo, J.A., Vasant, P., Litvinchev, I., Rodriguez-Aguilar, R., Martinez-Rios, F., Eds.; Springer: Cham, Switzerland, 2021; Volume 359, pp. 184–200. [Google Scholar] [CrossRef]
- Crain, I.K. The Monte Carlo generation of random polygons. Comput. Geosci. 1978, 4, 131–141. [Google Scholar] [CrossRef]
- Kumar, S.; Kurtz, S.K. Properties of a two-dimensional Poisson-Voronoi tessellation: A Monte-Carlo study. Mater. Charact. 1993, 31, 55–68. [Google Scholar] [CrossRef]
- Tanemura, M. Statistical distributions of Poisson Voronoi cells in two and three dimensions. FORMA-TOKYO 2003, 18, 221–247. [Google Scholar]
- Brakke, K.A. 200,000,000 Random Voronoi Polygons; Dept. Math. Sciences, Susquehanna University: Selinsgrove, PA, USA, 2015; pp. 1–11. [Google Scholar]
- Chiu, S.N. Aboav-Weaire’s and Lewis’ laws—A review. Mater. Charact. 1995, 34, 149–165. [Google Scholar] [CrossRef]
- Saraiva, J.; Pina, P.; Bandeira, L.; Antunes, J. Polygonal networks on the surface of Mars; applicability of Lewis, Desch and Aboav–Weaire laws. Phil. Mag. Lett. 2009, 89, 185–193. [Google Scholar] [CrossRef]
- Limaye, A.V.; Narhe, R.D.; Dhote, A.M.; Ogale, S.B. Evidence for convective effects in breath figure formation on volatile fluid surfaces. Phys. Rev. Lett. 1996, 76, 3762–3765. [Google Scholar] [CrossRef]
- Tanner, J.C. Polygons Formed by Random Lines in a Plane: Some Further Results. J. Appl. Probab. 1983, 20, 778–787. [Google Scholar] [CrossRef]
- Calka, P. Precise Formulae for the Distributions of the Principal Geometric Characteristics of the Typical Cells of a Two-Dimensional Poisson-Voronoi Tessellation and a Poisson Line Process. Adv. Appl. Probab. 2003, 35, 551–562. [Google Scholar] [CrossRef]
- Botnar, A.; Novokov, O.; Korepanov, O. Crystallization Control of Anionic Thiacalixarenes on Silicon Surface Coated with Cationic Poly(ethyleneimine). ACS Appl. Mater. Interfaces, 2024; submitted. [Google Scholar]
- Bormashenko, E.; Fedorets, A.A.; Frenkel, M.; Dombrovsky, L.A.; Nosonovsky, M. Clustering and self-organization in small-scale natural and artificial systems. Phil. Trans. R. Soc. A 2020, 378, 20190443. [Google Scholar] [CrossRef]
- Nosonovsky, M.; Roy, P. Scaling in Colloidal and Biological Networks. Entropy 2020, 22, 622. [Google Scholar] [CrossRef]
- Wang, W.; Wang, H.; Fei, S.; Wang, H.; Dong, H.; Ke, Y. Generation of random fiber distributions in fiber reinforced composites based on Delaunay triangulation. Mater. Des. 2021, 206, 109812. [Google Scholar] [CrossRef]
- Pan, Y.; Iorg, L.; Pelegri, A. Numerical generation of a random chopped fiber composite RVE and its elastic properties. Compos. Sci. Technol. 2008, 68, 2792–2798. [Google Scholar] [CrossRef]
- Anikeenko, A.V.; Gavrilova, M.L.; Medvedev, N.N. The coloring of the voronoi network: Investigation of structural heterogeneity in the packings of spheres. Jpn. J. Indust. Appl. Math. 2005, 22, 151–165. [Google Scholar] [CrossRef]
- Li, M.; Xi, T. Topological and atomic scale characterization of grain boundary networks in polycrystalline and nanocrystalline materials. Prog. Mater. Sci. 2011, 56, 864–899. [Google Scholar] [CrossRef]
- Pan, S.P.; Feng, S.D.; Qiao, J.W.; Wang, W.M.; Qin, J.U. Crystallization pathways of liquid-bcc transition for a model iron by fast quenching. Sci. Rep. 2015, 5, 16956. [Google Scholar] [CrossRef]
Authors | Crain, 1978 [34] | Hinde & Miles, 1980 [32] | Kumar & Kurtz, 1993 [35] | Calka, 2003 [31] | Tanemura, 2003 [36] | Brakke, 2005 [37] |
---|---|---|---|---|---|---|
Number of polygons, N | 57,000 | 2,000,000 | 650,000 | Monte Carlo numerical integration was used | 10,000,000 | 208,969,210 |
P3 | 0.011 | 0.01131 | 0.011 | 0.01124 | 0.01125 | 0.01125 |
P4 | 0.1078 | 0.1071 | 0.1071 | 0.106838 | 0.10685 | 0.10683 |
P5 | 0.2594 | 0.2591 | 0.26 | 0.25946 | 0.25946 | 0.25945 |
P6 | 0.2952 | 0.2944 | 0.294 | 0.29473 | 0.29473 | 0.29471 |
P7 | 0.1984 | 0.1991 | 0.199 | 0.19877 | 0.19877 | 0.1988 |
P8 | 0.0896 | 0.0902 | 0.09 | 0.0897 | 0.0897 | 0.09012 |
P9 | 0.0296 | 0.0295 | 0.03 | 0.0295 | 0.0295 | 0.02964 |
P10 | 0.00751 | 0.00743 | 0.007 | 0 | 0 | 0.00745 |
P11 | 0.00142 | 0.00149 | 0.0015 | 0 | 0 | 0.00148 |
P12 | 0.000175 | 0.00025 | 0.00023 | 0 | 0 | 0.00024 |
P13 | 0.000053 | 0.00003 | 0.00004 | 0 | 0 | 0.00003 |
SV | 1.68927 | 1.69066 | 1.68902 | 1.64087 | 1.64092 | 1.69031 |
ζ | 0.273072 | 0.272438 | 0.272718 | 0.27251 | 0.272515 | 0.272729 |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Gilevich, A.; Shoval, S.; Nosonovsky, M.; Frenkel, M.; Bormashenko, E. Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs. Mathematics 2024, 12, 2426. https://doi.org/10.3390/math12152426
Gilevich A, Shoval S, Nosonovsky M, Frenkel M, Bormashenko E. Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs. Mathematics. 2024; 12(15):2426. https://doi.org/10.3390/math12152426
Chicago/Turabian StyleGilevich, Artem, Shraga Shoval, Michael Nosonovsky, Mark Frenkel, and Edward Bormashenko. 2024. "Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs" Mathematics 12, no. 15: 2426. https://doi.org/10.3390/math12152426
APA StyleGilevich, A., Shoval, S., Nosonovsky, M., Frenkel, M., & Bormashenko, E. (2024). Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs. Mathematics, 12(15), 2426. https://doi.org/10.3390/math12152426