Visualization of Escher-like hyperbolic tessellations

Peichang Ouyang; Kwok Wai Chung; Alain Nicolas; Robert W. Fathauer; David Bailey; Krzysztof Gdawiec

doi:10.1016/j.amc.2025.129710

Visualization of Escher-like hyperbolic tessellations

Peichang Ouyang
, Kwok Wai Chung
, Alain Nicolas
, Robert W. Fathauer
, David Bailey
, Krzysztof Gdawiec^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

1 Citation (Scopus)

23 Downloads (CityUHK Scholars)

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. © 2025 The Author(s).

Original language	English
Article number	129710
Number of pages	18
Journal	Applied Mathematics and Computation
Volume	510
Online published	8 Sept 2025
DOIs	https://doi.org/10.1016/j.amc.2025.129710
Publication status	Published - 1 Feb 2026

Funding

This work has been supported by the Guangxi Science and Technology Base and Talent Project [grant numbers AD25069062 and AA21196008]; National Natural Science Foundation of China [grant numbers 62462004 and 62062042]; Doctor Startup Foundation of Guangxi University of Science and Technology Doctor Startup Foundation [grant number 21Z48]; and Guangxi Philosophy and Social Science Research Project [grant number 23FWY025].

Research Keywords

Conformal mapping
Escher art
Hyperbolic geometry
Symmetry group

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This full text is made available under CC-BY-NC 4.0. https://creativecommons.org/licenses/by-nc/4.0/

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title = "Visualization of Escher-like hyperbolic tessellations",

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 Poincar{\'e} 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. {\textcopyright} 2025 The Author(s).",

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author = "Peichang Ouyang and Chung, \{Kwok Wai\} and Alain Nicolas and Fathauer, \{Robert W.\} and David Bailey and Krzysztof Gdawiec",

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T1 - Visualization of Escher-like hyperbolic tessellations

AU - Ouyang, Peichang

AU - Chung, Kwok Wai

AU - Nicolas, Alain

AU - Fathauer, Robert W.

AU - Bailey, David

AU - Gdawiec, Krzysztof

PY - 2026/2/1

Y1 - 2026/2/1

N2 - 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. © 2025 The Author(s).

AB - 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. © 2025 The Author(s).

KW - Conformal mapping

KW - Escher art

KW - Hyperbolic geometry

KW - Symmetry group

UR - https://www.scopus.com/pages/publications/105015142786

UR - https://www.scopus.com/pages/publications/105015142786#tab=impact

U2 - 10.1016/j.amc.2025.129710

DO - 10.1016/j.amc.2025.129710

M3 - RGC 21 - Publication in refereed journal

SN - 0096-3003

VL - 510

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

M1 - 129710

ER -

Visualization of Escher-like hyperbolic tessellations

Abstract

Funding

Research Keywords

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