We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Bradley, G.H.: Algorithms for Hermite and Smith normal matrices and linear Diophantine equations. Math. Comput. 25, 897–907 (1971)
Chavey, D.: Tilings by regular polygons. II. A catalog of tilings. Comput. Math. Appl. 17, 147–165 (1989)
Conway, J.H., Burgiel, H., Goodman-Strauss, C.: The Symmetries of Things. CRC Press (2008)
Galebach, B.: \(n\)-uniform tilings. http://probabilitysports.com/tilings.html
Grünbaum, B., Shephard, G.C.: Tilings by regular polygons. Math. Mag. 50, 227–247 (1977)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman, New York (1989)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2004)
Hilbert, D., Cohn-Vossen, S.: Geometry and the Imagination. Chelsea, New York (1952)
Kaplan, C.S.: Islamic star patterns from polygons in contact. In: Proceedings of the Graphics Interface 2005, pp. 177–185 (2005)
Kaplan, C.S.: Introductory tiling theory for computer graphics. Synth. Lect. Comput. Gr. Anim. 4(1), 1–113 (2009)
Kawarabayashi, K.I., Mohar, B.: Graph and map isomorphism and all polyhedral embeddings in linear time. In: STOC’08, pp. 471–480. ACM (2008)
Lenngren, N.: \(k\)-uniform tilings by regular polygons. Tech. Rep. U.U.D.M. project report 2009:23, Uppsala University (2009)
Liu, S., Ng, T., Sunkavalli, K., Do, M.N., Shechtman, E., Carr, N.: PatchMatch-based automatic lattice detection for near-regular textures. In: Proceedings of ICCV 2015, pp. 181–189 (2015)
Liu, Y., Hel-Or, H., Kaplan, C.S., Gool, L.J.V.: Computational symmetry in computer vision and computer graphics. Found. Trends Comput. Gr. Vis. 5(1–2), 1–195 (2010)
McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014)
Nasri, A., Benslimane, R.: Parametric shape grammar formalism for Moorish geometric design analysis and generation. J. Comput. Cultural Herit. 10, 1–20 (2017)
Soto Sánchez, J.E., Sá, A.M., de Figueiredo, L.H.: Periodic tilings of regular polygons. http://www.impa.br/~cheque/tiling/
Sá, A.M., Echavarria, K.R., Arnold, D.: Dual joints for 3d-structures. Vis. Comput. 30(12), 1321–1331 (2014)
Sá, A.M., de Figueiredo, L.H., Soto Sánchez, J.E.: Synthesizing periodic tilings of regular polygons. In: Proceedings of SIBGRAPI 2018, pp. 17–24. IEEE Computer Press (2018)
Sá, R., Sá, A.M.: Sobre malhas arquimedianas. Editora Olhares, São Paulo (2017)
The On-Line Encyclopedia of Integer Sequences: A299780. https://oeis.org//A299780
Wikipedia: Euclidean tilings by convex regular polygons. https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons
We thank Sá and Sá and Galebach for making their collections of tilings freely available at their websites. The first author is partially supported by a CNPq doctoral scholarship. The third author is partially supported by a CNPq research grant. This research was done in the Visgraf Computer Graphics laboratory at IMPA. Visgraf is supported by the funding agencies FINEP, CNPq, and FAPERJ, and also by gifts from IBM Brasil, Microsoft, NVIDIA, and other companies.
Conflict of interest
The authors declare that they have no conflict of interest.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Soto Sánchez, J.E., Medeiros e Sá, A. & de Figueiredo, L.H. Acquiring periodic tilings of regular polygons from images. Vis Comput 35, 899–907 (2019). https://doi.org/10.1007/s00371-019-01665-y
- Geometrical models