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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