Periodic Tilings of Regular Polygons

Tiling the plane with regular polygons is a fascinating subject, both mathematically and graphically [1,2,3], with a long history [4]. It is now exactly 400 years since the mathematics of tilings were first discussed in Kepler's book Harmonices Mundi of 1619.

A tiling of the plane by polygons is a subdivision of the plane into bounded polygonal regions that either are disjoint, share a vertex, or share an edge. Restricting the faces of the tiling to be regular polygons brings lots of rigidity while still allowing much interesting variety. Standard results on the subject can be found in Wikipedia.

The application implements the method described in [a] and [b] to acquire, represent, and compute periodic tilings of the plane by regular polygons. The proposed approach is reminiscent of the regular systems of points discussed in the classic book Geometry and the Imagination [5]. The user can choose one of the tilings from two acquired sets, named acording to the sources [6] and [7], setup the parameters, and save the image. The proccess of acquisition is described in [b]. The original images are available here and in [7]. They are subject to copyright but are freely available for research.

Publications

1. A. Medeiros e Sá, L. H. de Figueiredo, J. E. Soto S., Synthesizing periodic tilings of regular polygons, Proceedings of SIBGRAPI 2018, 17–24. [doi]
2. J. E. Soto S., L. H. de Figueiredo, A. Medeiros e Sá, Acquiring periodic tilings of regular polygons from images, The Visual Computer 35 #6-8 (2019) 899–907 (CGI 2019 special issue). [doi]

References

1. B. Grunbaum and G. C. Shephard, Tilings and patterns, W. H. Freeman, 1989.
2. J. H. Conway, H. Burgiel, and C. Goodman-Strauss, The symmetries of things, AK Peters, 2008.
3. C. Kaplan, Introductory tiling theory for computer graphics, Morgan and Claypool, 2009.
4. B. Grunbaum and G. C. Shephard, Tilings by regular polygons, Mathematics Magazine 50 #5 (1997) 227–247
5. D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, 1952.
6. R. Sá and A. Medeiros e Sá, Sobre malhas arquimedianas, Editora Olhares, 2017.
7. B. Galebach, n-uniform tilings.

Based on work at http://w3.impa.br/~cheque/tiling/.