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Visualization of Escher-like hyperbolic tessellations

Author

Listed:
  • Ouyang, Peichang
  • Chung, Kwok Wai
  • Nicolas, Alain
  • Fathauer, Robert W.
  • Bailey, David
  • Gdawiec, Krzysztof

Abstract

By combining mathematical principles, modern computer graphics techniques, and the efforts of mathematically inclined artists, we present a visualization method for generating aesthetic patterns in hyperbolic space. To this end, we first establish fast algorithms to construct hyperbolic tilings in the Poincaré disk model. Then, using templates designed by graphic artists, we specify computer techniques to render hyperbolic tilings, which results in Escher-like patterns. Moreover, we present a simple method to realize novel hyperbolic kaleidoscopic effect. To obtain more diverse patterns, we introduce several conformal mappings to create visually appealing tessellations on the other spaces. The proposed methods can be easily implemented using shaders to obtain high-quality tessellations, which have good potential for application in the field of artistic decoration.

Suggested Citation

  • Ouyang, Peichang & Chung, Kwok Wai & Nicolas, Alain & Fathauer, Robert W. & Bailey, David & Gdawiec, Krzysztof, 2026. "Visualization of Escher-like hyperbolic tessellations," Applied Mathematics and Computation, Elsevier, vol. 510(C).
    • Handle: RePEc:eee:apmaco:v:510:y:2026:i:c:s0096300325004369
    DOI: 10.1016/j.amc.2025.129710
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    References listed on IDEAS

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    1. Ouyang, Peichang & Fathauer, Robert W. & Chung, Kwok-wai & Wang, Xinchang, 2019. "Automatic generation of hyperbolic drawings," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 653-663.
    2. Barbara Hausmann & Britta Slopianka & Hans-Peter Seidel, 1997. "Exploring Plane Hyperbolic Geometry," Springer Books, in: Hans-Christian Hege & Konrad Polthier (ed.), Visualization and Mathematics, pages 21-36, Springer.
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