Symmetries of Frieze Patterns

Abstract

Ornamental patterns have decorated art and architecture across cultures for millennia, long before their mathematical properties were formally understood. This paper explores the symmetry groups underlying ornamental and frieze patterns. We first examine one-dimensional ornamental patterns, governed by translations and reflections, and show that only two distinct symmetry groups are possible. Extending to two dimensions, we classify frieze patterns – infinitely repeating designs in one direction – into seven distinct groups based on combinations of translations, reflections, rotations, and glide reflections.