Generation of advanced Escher-like spiral tessellations

Ouyang, Peichang; Chung, Kwok Wai; Bailey, David; Nicolas, A.; Gdawiec, Krzysztof

doi:10.1007/s00371-021-02232-0

Cited by 5 publications

(3 citation statements)

References 24 publications

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“…Substituting ( 14), (23), and ( 24) into (25) and noting that there is only one unknown 𝛿 1 , we solve the equation for tan 𝛿 1 and obtain (20) and (21).…”

Section: Transformations For Constructing Mhstsmentioning

confidence: 99%

“…These drawings later motivated many studies focused on the creation of computer-generated spiral patterns [23,24]. Recently, Ouyang et al proposed methods for generating interesting advanced Escher-like spiral [25] and interlocking spiral [26] drawings. Kaplan developed a convenient web application to produce rich Escher-like spiral drawings [27] based on Dixon's antiMercator projection [28].…”

Section: Introductionmentioning

confidence: 99%

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Generation of Escher‐like spiral drawings in a modified hyperbolic space

Chung

Ouyang

Nicolas³

et al. 2023

Math Methods in App Sciences

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Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral drawings in hyperbolic space. In this paper, we consider a modified hyperbolic geometry by removing the condition that a geodesic is orthogonal to the unit circle in the Poincaré model. We show that spiral symmetry and the similarity property exist in this modified geometry so that the creation of uncommon hyperbolic spiral drawings is possible. To this end, we first establish the theoretical foundation for the proposed method by deriving a contraction mapping and a rotation for constructing modified hyperbolic spiral tilings (MHSTs) and introduce symmetry groups to analyze the structure of MHSTs. Then, to embed a pre‐designed wallpaper template into the tiles, we derive a one‐to‐one mapping between a tile of MHST and a rectangle. Finally, we specify some technical implementation details and give a gallery of the resulting MHST drawings. Using existing wallpaper templates, the proposed method is able to generate a great variety of exotic Escher‐like drawings.

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“…Substituting ( 14), (23), and ( 24) into (25) and noting that there is only one unknown 𝛿 1 , we solve the equation for tan 𝛿 1 and obtain (20) and (21).…”

Section: Transformations For Constructing Mhstsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Chung

Ouyang

Nicolas³

et al. 2023

Math Methods in App Sciences

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View full text Add to dashboard Cite

show abstract

“…Moreover, Fish and Reptiles have symmetry group

[3, 3]

, Heaven and Hell has symmetry group

[3^+, 4]

and Eight Grotesques has symmetry group [2]. These artworks later inspired people to generate Escher‐like spherical arts with the help of a computer [Kap02, OCNG21, OCB*22, YS01]. Compared with a flat surface, creating patterns on a spherical surface is not easy since it is a curved and finite space.…”

Section: Introductionmentioning

confidence: 99%

Visualization of Escher‐like Kaleidoscopic Spherical Patterns of Regular Polyhedron Symmetry

Gdawiec,

Chung,

Nicolas

et al. 2024

Computer Graphics Forum

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In this paper, we present a method for creating Escher‐like spherical patterns with regular polyhedron symmetries. Using the generators of the symmetry groups associated with regular polyhedra, we first provide fast algorithms to construct spherical tilings. Then, to obtain Escher‐like patterns, we specify texturing techniques to decorate the resulting tilings. Moreover, we present a strategy to create a novel dynamic effect of Escher‐like kaleidoscopes in which the motifs have a complete body. The method has the advantages of simple implementation, fast calculation, good graphics, and artistic effects, which can be used to create rich elegant spherical patterns.

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Interlocking Spiral Drawings Inspired by M. C. Escher’s Print Whirlpools

Ouyang

Gdawiec

Nicolas³

et al. 2022

ACM Trans. Graph.

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Whirlpools , by the Dutch graphic artist M.C. Escher, is a woodcut print in which fish interlock as a double spiral tessellation. Inspired by this print, in this paper we extend the idea and present a general method to create Escher-like interlocking spiral drawings of N whirlpools. To this end, we first introduce an algorithm for constructing regular spiral tiling T . Then, we design a suitable spiral tiling T and use N copies of T to compose an interlocking spiral tiling K of N whirlpools. To create Escher-like drawings similar to the print, we next specify realization details of using wallpaper templates to decorate K . To enhance the aesthetic appeal, we propose several measures to minimize motif overlaps of the spiral drawings. Technologically, we develop algorithms for generating Escher-like drawings that can be implemented using shaders. The method established is thus able to generate a great variety of exotic Escher-like interlocking spiral drawings.

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Generation of advanced Escher-like spiral tessellations

Cited by 5 publications

References 24 publications

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Visualization of Escher‐like Kaleidoscopic Spherical Patterns of Regular Polyhedron Symmetry

Interlocking Spiral Drawings Inspired by M. C. Escher’s Print Whirlpools

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