RESUMO
Symmetry can be widely found in natural phenomenon. Regular polygons and polyhedra are the most basic and important symmetrical structures in 2D and 3D Euclidean space. Four-dimensional regular polytopes (4-RPs) are the 4D analogs of regular polyhedra in three dimensions and regular polygons in two dimensions. After introducing the fundamental root systems of 4-RPs, this article presents three interesting methods to visualize 4-RPs using a fundamental region algorithm.
RESUMO
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.
Assuntos
Algoritmos , Arquitetura/métodos , Arte , Cor , Gráficos por Computador , Materiais de ConstruçãoRESUMO
This third installment of the Beautiful Math articles considers the visualization of aesthetic patterns with hyperbolic-triangle-group symmetries. A flexible form of invariant mappings contributes to a simple, efficient way to generate hyperbolic patterns. Combined with conformal mappings, this method can yield an abundance of exotic patterns.
