*symmetry group* which is a kind of "transformation group." Now, the word "group" as used in the English language just means a bunch of things considered together. Mathematicians need a word for something more, and, for better or for worse, they decided on "group."

*group* of things, for a mathematician, means a collection of things with a certain structure. The structure is one of "composition." Given two elements *S* and *T* of a group, you can "compose" them to get another element *ST* of the group. In our case we're composing transformations of the plane that leave a pattern invariant. That just means first perform one transformation *S*, then perform the other transformation *T*. (It's a matter of convention whether you read *ST* from left to right or from right to left. Although the right-to-left convention is more common, lets use the left-to-right convention here.) If each transformation is a symmetry of a pattern, then their composition is a symmetry of the pattern, too.

*I* to denote the identity of a group. A little more precisely, the axiom requires that the identity *I* when composed with any element *T* gives back *T*. Algebraically, we require

In a symmetry group for a pattern, the identity is the identity transformation. That's the transformation of the plane that doesn't move any point. It's trivial. It doesn't do anything!

*T* is another element, usually written *T*^{-1}, whose composition with *T* gives the identity. That is,

So, the inverse undoes whatever *T* does. For example, the inverse of a translation upwards is a translation downwards. The inverse of a rotation 90° clockwise is a rotation 90° counterclockwise. The inverse of a reflection, surprisingly enough, is itself.

*associativity*. Algebraically, whenever *S*, *T*, and *U* are three elements,

It allows us to write the composition of three elements without using parentheses. Composition of transformations is associative.

*commutative*, that is, the equation

usually doesn't hold in our transformation groups. If both *S* and *T* happen to both be translations, then it does hold, but rarely otherwise. For example, if *S* and *T* are reflections with parallel axes, then *ST* and *TS* are both translations, but in opposite directions.

Up to the table of contents

Back to lattices

On to the 17 plane groups

David E. Joyce

Department of Mathematics and Computer Science

Clark University

Worcester, MA 01610

The files are located at http://aleph0.clarku.edu/~djoyce/wallpaper/