The art of tiling originated very early in the history of civilization. Almost every known human society has made use of tilings in some form or another. In particular, tilings using only regular polygons have great visual appeal. Decorated regular tilings with continuous and symmetrical patterns were widely used in decoration field, such as mosaics, pavements, and brick walls. In science, these tilings provide inspiration for synthetic organic chemistry. Building on previous CG&A “Beautiful Math” articles, the authors propose an invariant mapping method to create colorful patterns on Archimedean tilings (1-uniform tilings). The resulting patterns simultaneously have global crystallographic symmetry and local cyclic or dihedral symmetry.

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A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra.

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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.

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.

In this paper, using both hand-drawn and computer-drawn graphics, we establish a method to generate advanced Escher-like spiral tessellations. We first give a way to achieve simple spiral tilings of cyclic symmetry. Then, we introduce several conformal mappings to generate three derived spiral tilings. To obtain Escher-like tessellations on the generated tilings, given pre-designed wallpaper motifs, we specify the tessellations’ implementation details. Finally, we exhibit a rich gallery of the generated Escher-like tessellations. According to the proposed method, one can produce a great variety of exotic Escher-like tessellations that have both good aesthetic value and commercial potential.