Presentation Title

Fearful Symmetry: Symmetry Groups of Two-Dimensional Manifolds

Format of Presentation

Poster to be presented Friday March 31, 2017

Abstract

A two-dimensional, repetitive pattern in the Euclidean plane (e.g., a checkerboard design, or one of M. C. Escher’s famous “regular divisions of the plane”) is characterized by certain distance-preserving planar transformations (e.g., rotations, translations, reflections, and glide reflections), that leave the pattern unchanged. The set of all such transformations, along with the rules for their composition, is known as the pattern’s “symmetry group”. Remarkably, every pattern can be described mathematically according to one of 17 distinct symmetry groups, known as the “wallpaper groups”. We investigate the analogues to the wallpaper groups in various two-dimensional manifolds other than the Euclidean plane, and explore possible applications to art and design.

Department

Mathematics and Statistics

Faculty Advisor

Peter Smoczynski

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Fearful Symmetry: Symmetry Groups of Two-Dimensional Manifolds

A two-dimensional, repetitive pattern in the Euclidean plane (e.g., a checkerboard design, or one of M. C. Escher’s famous “regular divisions of the plane”) is characterized by certain distance-preserving planar transformations (e.g., rotations, translations, reflections, and glide reflections), that leave the pattern unchanged. The set of all such transformations, along with the rules for their composition, is known as the pattern’s “symmetry group”. Remarkably, every pattern can be described mathematically according to one of 17 distinct symmetry groups, known as the “wallpaper groups”. We investigate the analogues to the wallpaper groups in various two-dimensional manifolds other than the Euclidean plane, and explore possible applications to art and design.