TABLE OF CONTENTS Symmetry brings to mind decorative patterns, tilings, and repetitive mirror-images. As a subject of study, it was once thought to be unrelated to serious mathematics and scientific thinking. Not until the 19th century did mathematicians discover a close relationship between analytical geometry and symmetry. Then they invented group theory to study the symmetry of mathematical objects.
Aside from the obvious application of symmetry to the decorative arts and engineered constructions, the understanding of symmetry is now central to mathematical physics and the study of elementary particles and cosmology. In the equations of quantum mechanics and general relativity, the presence or absence of certain algebraic symmetries profoundly affects how we interpret them. The highly intuitive geometric nature of the elementary principles of symmetry and groups makes this subject easy to teach and learn.
This chapter takes an analytical approach to symmetry. It defines a group and the more specialized symmetry group, including cyclic, dihedral, and finite symmetry groups. The symmetry of ornamental groups (frieze groups and wallpaper groups) reveals a surprising quantitative aspect to these aesthetic forms. The crystallographic restriction shows how symmetry depends on the nature of space itself. This is also true of plane lattices and tilings. The chapter concludes with a look at the rotational symmetry of polyhedra.
When we say that a figure is symmetric, we mean that certain motions or rearrangements of its parts leave it unchanged as a whole. A figure is symmetric if it...
