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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. [ABSTRACT FROM AUTHOR]

Schwarz-Christoffel mapping is an elegant theory that can preserve the structure of a pattern perfectly. One of the most important elements of conformal mapping theory, Schwarz-Christoffel mapping has important applications in many engineering fields, such as electromagnetism, aerodynamics, and thermal field theory. Building on their 2014 CG&A article titled "Beautiful Math, Part 3: Hyperbolic Aesthetic Patterns Based on Conformal Mappings," the authors apply the professional numerical methods of Schwarz-Christoffel mapping to achieve real polygon boundaries. [ABSTRACT FROM PUBLISHER]

A simple, fast method generates various visually appealing spiral patterns. The method is based on the concept that spiral patterns comprise a symmetry group of tilings. It employs invariant mappings and a dynamical system to create seamless colored patterns. [ABSTRACT FROM PUBLISHER]