Self-Similar Fractal Drawings Inspired by M. C. Escher’s Print Square Limit

Abstract

A fractal tiling ( f -tiling) is a kind of rarely explored tiling by similar polygonal tiles which possesses self-similarity and the boundary of which is a fractal. Based on a tiling by similar isosceles right triangles, Dutch graphic artist M. C. Escher created an ingenious print Square Limit in which fish are uniformly reduced in size as they approach the boundaries of the tiling. In this article, we present four families of f -tilings and propose an easy-to-implement method to achieve similar Escher-like drawings. By systematically investigating the local star-shaped structure of f -tilings, we first enumerate four families of f -tilings admitted by kite-shaped or dart-shaped prototiles. Then, we establish a fast binning algorithm for visualising f -tilings. To facilitate the creation of Escher-like drawings on the reported f -tilings, we next introduce one-to-one mappings between the square, and kite and dart, respectively. This treatment allows a pre-designed square template to be deformed into all prototiles considered in the article. Finally, we specify some technical implementations and present a gallery of the resulting Escher-like drawings. The method established in this article is thus able to generate a great variety of exotic Escher-like drawings.