Synthesizing Periodic Tilings of Regular Polygons

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

“…Having proven its worth within our previous works [10,11] on rendering and acquisition of tilings, our representation is both convenient and general enough to show great potential for many more operations and analyses of tilings. Accordingly, the remainder of this paper offers for the first time a comprehensive account of the representation, presenting its properties, common topological and geometric operations, and sample applications.…”

“…This paper describes a general, uniform, integerbased representation for periodic tilings of the plane by regular polygons, and discusses its properties in full detail. It significantly expands our previous work on the subject [10,11]. New material includes: full details on the representation and its motivation, an in-depth discussion of its properties, and new algorithms for tiling reconstruction and symmetry detection that are simple and robust, since they do not rely on nearest-neighbors searches and geometric computations with numerical tolerances.…”

“…Preliminary forms of our representation were described briefly in the context of rendering and acquisition [10,11]. Our previous reconstruction algorithm [10] relied on nearest-neighbors searches and geometric computations with numerical tolerances, which worked well for that task but may not work reliably for more general applications. Our previous method for deciding the equivalence of two representations [11] relied on the Hermite normal form of integers matrices.…”

On periodic tilings with regular polygons

“…95, 2021. The first results of the research were presented on Synthesizing periodic tilings of regular polygons [11] at the 31st Conference on Graphics, Patterns and Images (SIBGRAPI 2018). We also presented Acquiring periodic tilings of regular polygons from images [10] at Computer Graphics International 2019, which was published in The Visual Computer Journal, Vol.…”

“…The first contribution is the formulation of a simple integer-based representation for periodic tilings of regular polygons using complex numbers. This representation allowed us to acquire geometric state-ofthe-art models from two large collections of images [10], to synthesize new images of the tilings at any scale with arbitrary precision [11], and to classify tilings by crystallographic groups using symmetry detection. These results are described in Chapters 2, 3, and 4 of the thesis.…”