An integer representation for periodic tilings of the plane by regular polygons

José Ezequiel Soto Sanchez¹, Tim Weyrich², Asla Medeiros e Sá³, Luiz Henrique de Figueiredo¹

¹ IMPA, Rio de Janeiro, Brazil
² University College London
³ FGV EMAp, Rio de Janeiro, Brazil

Abstract

We describe a representation for periodic tilings of the plane by regular polygons. Our approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely by a (2 + n) × 4 integer matrix containing lattice coordinates for two translation vectors and n seed vertices. We discuss several properties of this representation and describe how to exploit the representation elegantly and efficiently for reconstruction, rendering, and automatic crystallographic classification by symmetry detection.

Citation Style: Display Alpha ACM SIGGRAPH IEEEtran Publication

An integer representation for periodic tilings of the plane by regular polygons.
José Ezequiel Soto Sanchez, Tim Weyrich, Asla Medeiros e Sá, Luiz Henrique de Figueiredo.
Computer & Graphics, 95, 69–8080, April 2021.José Ezequiel Soto Sanchez, Tim Weyrich, Asla Medeiros e Sá, and Luiz Henrique de Figueiredo. An integer representation for periodic tilings of the plane by regular polygons. Computer & Graphics, 95:69–80, April 2021.Soto Sanchez, J. E., Weyrich, T., Medeiros e Sá, A., and de Figueiredo, L. H. 2021. An integer representation for periodic tilings of the plane by regular polygons. Computer & Graphics 95 (Apr.), 69–80.J. E. Soto Sanchez, T. Weyrich, A. Medeiros e Sá, and L. H. de Figueiredo, “An integer representation for periodic tilings of the plane by regular polygons,” Computer & Graphics, vol. 95, pp. 69–80, Apr. 2021.

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Acknowledgments

J. E. Soto Sánchez was partially supported by a CNPq doctoral scholarship. L. H. de Figueiredo is partially supported by a CNPq research grant. This research was done in the Visgraf Computer Graphics laboratory at IMPA, at FGV EMAp, and at the Department of Computer Science, University College London. Visgraf is supported by the funding agencies FINEP, CNPq, and FAPERJ, and also by gifts from IBM Brasil, Microsoft, NVIDIA, and other companies.

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