Symmetry and Art

TL;DR

This paper employs complex analysis and domain coloring to generate symmetric images and animations that visually demonstrate properties of Euclidean and non-Euclidean geometries, blending mathematical rigor with artistic expression.

Contribution

It introduces a novel application of complex analysis and domain coloring to create symmetric, geometrically meaningful artistic images and animations.

Findings

Generated designs exhibit various symmetries such as rotational, translational, and mirror symmetry.

Animations reveal geometric properties dynamically.

Designs effectively illustrate Euclidean and hyperbolic geometries.

Abstract

We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can have rotational, translational, or mirror symmetry according to our chosen mapping functions. An appealing feature of these designs is how they reveal important properties of Euclidean and non-Euclidean geometries. We can also generate animations of our designs. Our goal is to create designs and animations having significant artistic content.