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Definitions

We will study symmetries called isometries, which preserve distances. Many more exotic transformations exist, but for the sake of simplicity we are going to restrict ourselves to isometries that can be described in terms of rubber stamping out many copies of the same image. We'll leave questions of stretching and inflating for another class.

[finite and infinite motifs]

The word motif means a repeated design element. Mathematicians often use this word to refer to the smallest portion of a pattern that can be repeated to recreate the entire pattern. We will follow this convention, often using the word to refer to some "original" design element. We will use the word image to refer to each of the repeated copies of the motif that make up the design. You may wish to think of a motif as a rubber stamp and the images as the inked pictures.
An isometry of an object or space is any contortion or movement of the object or space which doesn't change the distances between the points of that object or space. Isometric objects are congruent; you can turn one into the other by sliding and flipping, without stretching or bending or ripping it.
Two objects or figures are congruent if there is an isometry taking one to the other.

[translation, rotation, reflection, glide reflection]

If you're at a Macintosh with The Geometer's Sketchpad installed, you can experience an interactive version [GSP Help] of this illustration.

A translation is an isometry which is a shift of some specified direction and distance.
A rotation is another isometry, determined by a center and an angle.
A reflection is an isometry specified by a line of reflection, i.e. a mirror.
The product, or composition, of two isometries is the isometry resulting from applying one and then the other in order. For example, you might first rotate an object about a center, then translate it.
A glide reflection is the product of a reflection and a translation along the line of reflection. This produces a "footprint" pattern.
A symmetry of a pattern or picture is any isometry that leaves the appearance of the pattern unchanged. For instance, a five pointed star can be rotated by seventy two degrees without changing its appearance.

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Prev: Introduction

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Author: Heidi Burgiel, adapted from notes used in Math 5337, Spring 1995 by Chaim Goodman-Strauss.
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