This chapter develops the mathematical framework necessary for a rigorous treatment of wallpaper group symmetry and its perceptual consequences. We begin with the fundamentals of group theory as applied to planar isometries, then systematically construct the 17 wallpaper groups.

A wallpaper group is, formally, a discrete subgroup of the isometry group of the Euclidean plane that contains two linearly independent translations. The four types of planar isometries — translation, rotation, reflection, and glide reflection — combine under specific constraints to produce exactly 17 distinct groups, a fact proven independently by Fedorov, Schoenflies, and Barlow in the late 19th century.

We introduce a complexity metric based on the number and type of generating symmetries in each group. This metric, which we call "symmetry density," provides a continuous measure where the mathematical literature has traditionally offered only discrete classification. Symmetry density will serve as our primary independent variable in the experimental work that follows.

The chapter concludes by establishing formal connections between the algebraic properties of each group and the geometric properties of patterns that instantiate them. These connections form the bridge between the mathematical classification and the perceptual experiments described in Chapter 3.