Visualization of Escher-like Spiral Patterns in Hyperbolic Space

Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.

Automated summary

This paper presents a simple method for creating Escher-like hyperbolic spiral patterns. It includes a fast algorithm for constructing Euclidean spiral tilings and two simple approaches for constructing spiral tilings in hyperbolic models.