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[Woven 4*2]Figure 7

What features can a wallpaper pattern have? You shouldn't be surprised to learn that the different relationships between images of a motif in a wallpaper pattern correspond to the different transformations we used to create copies of a motif.

Many of the terms defined below are illustrated by a picture. Can you match the pictures to the definitions? Do any of the pictures illustrate more than one of the terms defined?

In the last section we saw that a pattern generated by three mirrors actually has many more than three lines of mirror symmetry -- in some computer generated images you can't even tell which mirrors are the originals.

This is the rule rather than an exception. The fact that a wallpaper pattern is generated by repeatedly applying some set of transformations to a finite motif and its images guarantees that this will happen for all the wallpaper groups. Try to find repeated symmetries in the wallpaper patterns shown above.

We want to find all of the different groups of symmetries that can exist in a wallpaper pattern. We're going to use "orbifold notation" (1) to describe these symmetries. This notation was presented in a summer course at the Center, so we'll use materials from that class. Some words on the page (like "quotient orbifold") will be unfamiliar to you -- you'll learn them soon.

Your mission is to learn the definitions of mirror string and gyration point; take a deep breath, relax, and move along to a page taken from the Geometry and the Imagination Summer Program web pages. (You may wish to refer to a picture of a brick wall while reading the notes.) Return to this page when you're done reading. Later, we'll work through several examples of the procedures described on that page.

We'll need just a few more terms for our classification system. In the next section we'll learn more about the topological features they describe.


(1) There are many different ways of describing the symmetries of a wallpaper pattern. We will encounter several of these during this course. Orbifold notation seems to be the most modern, the easiest to interpret and remember, and the best tool for our goal of enumerating the wallpaper groups.

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Author: Chaim Goodman-Strauss, revised and edited by Heidi Burgiel
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Created: Dec 7 1995 --- Last modified: Jul 31 1996
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