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.

Generation of Escher‐like spiral drawings in a modified hyperbolic space

“…The angle sum of △ ÔQQ 1 is less than 𝜋; the four boundaries of □ ÔQQ ′ O ′ are all circular arcs, see Figure 1. Recall that both Γ in (19) and Υ = S 2 S 1 are Möbius transformations, so they preserve generalized circles [40] (circles, and line segments regarded as circles with the center at infinity). As a result, the boundaries of each tile of 𝒯 (m, 𝛼, 𝛽, 𝜁, 𝜂) are circular arcs or line segments.…”

“…Based on the principles of symmetry, Escher created plenty of popular and fascinating artworks and had a durable and profound influence in the normally disparate fields of art and mathematics [7][8][9]. Interestingly, due to the aesthetic attraction as well as commercial potential, there appeared a lot of studies dedicated to the creation of Escher-like artworks, such as Escherization [10][11][12], 3D Escher-like tilings [13,14], metamorphosis [15,16], Escher transmutation [17,18], fractal [19], and hyperbolic drawings [20,21]. The most outstanding feature of the resulting images is that motifs are easily recognizable, which presents a true and natural artistic flavor.…”

Visualization of Escher‐like Kaleidoscopic Spherical Patterns of Regular Polyhedron Symmetry

“…Moreover, Fish and Reptiles have symmetry group $$[3, 3]$$, Heaven and Hell has symmetry group $$[3^+, 4]$$ and Eight Grotesques has symmetry group [2]. These artworks later inspired people to generate Escher‐like spherical arts with the help of a computer [Kap02, OCNG21, OCB*22, YS01]. Compared with a flat surface, creating patterns on a spherical surface is not easy since it is a curved and finite space.…”

Simulation Studies on the Design and Analysis of Interdigital and Fractal-Based Micro-Supercapacitors

“…13 The fractal geometries display a unique property of repetition of similar patterns while designing smaller as well as larger scale which is described as self-similarity, expanding or unfolding symmetry. 14 Multiple research teams have been focusing on fractal designs for a wide range of applications, including the characterization of complex irregular surfaces, medical diagnosis, image compression, antenna design, neural stimulation, high-density CMOS capacitors, optoelectronic devices, biosensors, stretchable electronics, radio frequency (RF) MEMS capacitors and peak energy storage devices. [15][16][17][18] The fractal electrode design has high energy density, volumetric capacitance as well as areal capacitance compared to the interdigitated electrode design.…”

“…Theoretical studies prove that the fractal design can provide more active surface area enhancing the electrochemical performance. 14 19 Hilbert, Peano, Moore fractal electrodes and IDE were fabricated and compared for performance. Moore structure was reported to have the highest performance.…”