Affiliation:
1. Department of Mathematics City University of Hong Kong Kowloon Hong Kong
2. School of Science Guangxi University of Science and Technology Liuzhou China
3. Nicolas Tessellation Quincy‐sous‐Senart France
4. Independent Researcher Grimsby UK
5. Institute of Computer Science University of Silesia Sosnowiec Poland
Abstract
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.
Funder
Specific Research Project of Guangxi for Research Bases and Talents
National Natural Science Foundation of China
Subject
General Engineering,General Mathematics