On periodic tilings with regular polygons

• José Ezequiel Soto Sánchez IMPA
• Asla Medeiros e Sá FGV / EMAp
• Luiz Henrique de Figueiredo IMPA

DOI: https://doi.org/10.5753/sibgrapi.est.2021.20025

Resumo

The thesis describes a simple integer-based computational representation for periodic tilings of regular polygons using complex numbers, which is now the state of the art for these objects. Several properties of this representation are discussed, including elegant and efficient strategies for acquisition, reconstruction, rendering, and automatic crystallographic classification by symmetry detection. The thesis also describes a novel strategy for the enumeration and generation of triangle-square tilings via equivalence with edge-labeled hexagonal graphs. The equivalence provide triangle-square tilings with an algebraic structure that allows an unfolding interpretation.

Referências

D. K. Washburn and D. W. Crowe, Symmetry Comes of Age: The Role of Pattern in Culture. University of Washington Press, 5 2004.

J. Bonner, Islamic Geometric Patterns - Their Historical Development and Traditional Methods of Construction. Springer-Verlag New York, 2017. [Online]. Available: https://www.springer.com/gp/book/9781441902160

C. S. Kaplan, “Islamic star patterns from polygons in contact,” in Graphics Interface 2005, 2005, pp. 177–185.

B. Grunbaum and G. C. Shephard, Tilings and Patterns. W.H.Freeman & Co Ltd, 1986.

C. S. Kaplan, Introductory Tiling Theory for Computer Graphics. Morgan & Claypool Publishers, 2009.

N. Lenngren, “k-uniform tilings by regular polygons,” Tech. Rep., 2009. [Online]. Available: http://www.diva-portal.org/smash/get/diva2:444746/FULLTEXT01.pdf

D. Chavey, “Tilings by regular polygons III: Dodecagon-dense tilings,” Symmetry-Culture and Science, vol. 25, no. 3, pp. 193–210, 2014.

J. E. Soto Sánchez, T. Weyrich, A. Medeiros e Sá, and L. H. de Figueiredo, “An integer representation for periodic tilings of the plane by regular polygons,” Computers & Graphics, vol. 95, pp. 69–80, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0097849321000078

J. E. Soto Sánchez, “On periodic tilings with regular polygons,” Ph.D. dissertation, Instituto de Matemática Pura y Aplicada (IMPA), 08 2020. [Online]. Available: http://chequesoto.info/thesis.html

J. E. Soto Sánchez, A. Medeiros e Sá, and L. H. de Figueiredo, “Acquiring periodic tilings of regular polygons from images,” Visual Computer, vol. 35, no. 6, pp. 899–907, 2019.

A. Medeiros e Sá, L. H. de Figueiredo, and J. E. Soto Sánchez, “Synthesizing periodic tilings of regular polygons,” in SIBGRAPI 2018, IEEE. Computer Press, 2018, pp. 17–24.

D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination. Chelsea, 1952.

B. L. Galebach, “n-Uniform Tilings,” 3 2002, [Online; accessed 24. Mar. 2020]. [Online]. Available: http://probabilitysports.com/tilings.html

R. Sá and A. Medeiros e Sá, Sobre Malhas Arquimedianas. Editora Olhares, São Paulo, 2017.

J. H. Conway, H. Burgiel, and C. Goodman-Strauss, The Symmetries of Things. CRC Press, 4 2016.

D. K. Washburn and D. W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. University of Washington Press, 1988.

D. Schattschneider, “The Plane Symmetry Groups: Their Recognition and Notation,” American Mathematical Monthly, vol. 85, no. 6, pp. 439– 450, 6 1978.

W. Bruns and B. Ichim, “Normaliz: Algorithms for affine monoids and rational cones,” Journal of Algebra, vol. 324, no. 5, pp. 1098–1113, 9 2010.