Visualization of Escher-like hyperbolic tessellations

• Fast algorithms to construct hyperbolic tilings in the Poincaré disk model have been established. • Algorithms to render hyperbolic tilings, which result in Escher-like patterns, have been proposed. • A novel hyperbolic kaleidoscopic effect has been proposed. • Examples of intriguing Escher-like patterns have been presented. 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.