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

Sánchez, José Ezequiel Soto; Weyrich, Tim; Sá, Asla Medeiros e; Figueiredo, Luiz Henrique de

doi:10.1016/j.cag.2021.01.007

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Representing these complex mathematical objects with integers allows new robust algorithms for artistic and computational applications. The representation and its properties are the focus of the recent publication An integer representation for periodic tilings of the plane by regular polygons [8], which appeared in Computers & Graphics, Vol. 95, 2021.…”

Section: Publications and Future Workmentioning

confidence: 99%

“…Yet, a complete classification of these tilings remains elusive. Lenngren's survey [6] described the previous state of the art for this problem and Chavey [7] made the most recent contribution in the subject until our paper [8], derived from the thesis [9].…”

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confidence: 99%

“…These results are described in Chapters 2, 3, and 4 of the thesis. They are concisely presented in [8] and summarized in Section II.…”

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confidence: 99%

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On periodic tilings with regular polygons

Sánchez

Sá²,

Figueiredo

2021

Anais Estendidos Da XXXIV Conference on Graphics, Patterns and Images (SIBRAPI Estendido 2021)

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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.

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Section: Publications and Future Workmentioning

confidence: 99%

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confidence: 99%

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On periodic tilings with regular polygons

Sánchez

Sá²,

Figueiredo

2021

Anais Estendidos Da XXXIV Conference on Graphics, Patterns and Images (SIBRAPI Estendido 2021)

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Symmetrization of quasi-regular patterns with periodic tilting of regular polygons

Yin,

Jin,

Fang

et al. 2024

Comp. Visual Media

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Computer-generated aesthetic patterns are widely used as design materials in various fields. The most common methods use fractals or dynamical systems as basic tools to create various patterns. To enhance aesthetics and controllability, some researchers have introduced symmetric layouts along with these tools. One popular strategy employs dynamical systems compatible with symmetries that construct functions with the desired symmetries. However, these are typically confined to simple planar symmetries. The other generates symmetrical patterns under the constraints of tilings. Although it is slightly more flexible, it is restricted to small ranges of tilings and lacks textural variations. Thus, we proposed a new approach for generating aesthetic patterns by symmetrizing quasi-regular patterns using general k-uniform tilings. We adopted a unified strategy to construct invariant mappings for k-uniform tilings that can eliminate texture seams across the tiling edges. Furthermore, we constructed three types of symmetries associated with the patterns: dihedral, rotational, and reflection symmetries. The proposed method can be easily implemented using GPU shaders and is highly efficient and suitable for complicated tiling with regular polygons. Experiments demonstrated the advantages of our method over state-of-the-art methods in terms of flexibility in controlling the generation of patterns with various parameters as well as the diversity of textures and styles.

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A Note from the Editor in Chief

Jorge¹

2021

Computers & Graphics

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

Cited by 4 publications

References 21 publications

On periodic tilings with regular polygons

On periodic tilings with regular polygons

Symmetrization of quasi-regular patterns with periodic tilting of regular polygons

A Note from the Editor in Chief

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