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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?
- We will say that a wallpaper pattern is a pattern which
covers the entire plane and can be produced by repeatedly applying
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 like the samples above 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 as 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.)
- 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.
- 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, click on a button with no "*" below it 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 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
- 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 wonder ring or 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
- A miracle or 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.
(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
Created: Dec 7 1995 ---
Last modified: Jul 31 1996
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