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What features can a wallpaper pattern have? Texture, color, design,
symmetry, and many others. We will concern ourselves with symmetry
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
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.
- 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
- 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.
- 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
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
- A kaleidoscopic point or corner is a point where
two or more mirrors meet. The corners of a mirror string are
- 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
(1) If you do not already have a copy of Kali,
you may use the Java version
a copy to run on your Macintosh.
(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.
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The Geometry Center Home Page
Author: Heidi Burgiel, adapted from notes
used in Math 5337, Spring 1995 by Chaim
Created: Dec 7 1995 ---
Last modified: Tue Jun 11 10:46:11 1996
Copyright © 1995-1996 by
The Geometry Center
All rights reserved.