ABSTRACT
This chapter begins with an exploration of Japanese symmetric designs. The first section defines finite design. The next section discusses the mathematical definition of symmetry in terms of symmetry transformations—transformations of a design that leave it apparently unchanged. The symmetries of a square include four rotations and four reflections. The symmetries of a pinwheel based on a square include only four rotations. Leonardo’s theorem states that the symmetry of a finite design includes either only rotations or the same number of rotations and reflections and that for any positive integer n, there is a finite design with exactly n rotations and a finite design with exactly n rotations and n reflections. The method of determining the signature of a finite design is given along with examples of many finite designs. Instructions are given for the construction of regular polygons as well as instructions for creating designs with point symmetry or bilateral symmetry and designs with rotations only or both rotations and reflection, having any period. The next sections discuss layered designs and how to determine their signature; symbols and logos; and sacred geometry. A section gives tips for finding symmetries of finite designs. The chapter concludes with a list of exercises, readings, and artists who use symmetry.
