Definitions

What features can a wallpaper pattern have? Texture, color, design, symmetry, and many others. We will concern ourselves with symmetry here.

In the last chapter, we learned how to identify the symmetries relating two images of a motif. We will now use these skills to identify symmetries relating the infinitely many images found in a wallpaper pattern.

To illustrate and explore these wallpaper patterns, we will use a computer program called Kali.(1) First, go through the instructions for use of the Macintosh version of Kali to get a feel for the program. Then read through the following definitions, using Kali to explore the terms defined.

• We will say that a wallpaper pattern is a pattern which covers the entire plane and can be produced by repeatedly applying isometric transformations to a finite motif (and to the images of that motif). We generated several plane patterns when we experimented with three mirrors perpendicular to the desk.

  Figure 8

• A wallpaper pattern has reflective symmetry if there is a reflection that transforms one half of the pattern into the other half. If you set a mirror down on a line of mirror symmetry of such a pattern, you will see the same pattern in the mirror that you would if you replaced the mirror with a piece of glass. (The Kali patterns with symmetry group *2222 will have many lines of mirror symmetry.)

  Figure 9

• A pattern is said to have translational symmetry if there is some translation of the pattern that takes each image of the motif to some other image. All wallpaper patterns have translational symmetries in at least two different directions. One family of wallpaper patterns has only translational symmetry.

• A pattern has rotational symmetry if some rotation of the pattern takes each image in the pattern to some other image. If that rotation has angle 360/n degrees, we say that the rotational symmetry is of order n. (This is because we can repeat that rotation n times before returning to our starting position.) If you're using Kali, choose a button with no "*" in its label to construct a pattern with lots of rotational symmetry.

  Figure 10 Figure 11

• A pattern has glide reflective symmetry if there is some glide reflection that transforms the pattern into itself. (It is often difficult to detect glide reflections in a wallpaper pattern. Don't be frustrated if you can't pick them out right away.) Any wallpaper pattern with reflective symmetry will also have some glide reflective symmetry, but some patterns have glide reflective symmetries and no reflective symmetry!

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 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" (2) 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.

• A kaleidoscopic point or corner is a point where two or more mirrors meet. The corners of a mirror string are kaleidoscopic points.
• Gyration points may also be referred to as cone points.
• A handle is denoted by an "o" in the orbifold notation (older versions of Kali use a solid dot). These show up in wallpaper patterns with translational symmetry -- the name is a shortened version of "wonderful wandering", which suggests translations.
• A cross cap is denoted by an "x" in the orbifold notation (Kali may use an o). These appear in the notation for patterns which have glide reflections -- they're "miraculous" ways of moving from a right handed copy of a motif to a left handed image without crossing a "mirror" line.

Notes

(2) 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.

The Geometry Center Home Page

Author: Heidi Burgiel, adapted from notes used in Math 5337, Spring 1995 by Chaim Goodman-Strauss.
Comments to: webmaster@geom.umn.edu
Created: Dec 7 1995 --- Last modified: Tue Jun 11 10:46:11 1996
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