*Wallpaper groups consist of isometries
of the Euclidian plane. I'll just refer to them as isometries
or for simplicity, symmetries. A symmetry of an object is
a transformation that takes the object onto a copy of the object.
Symmetries preserve dimensions, but not always orientations.
There are two general classes of symmetries of the plane, order
preserving, where each copy has positive congruence, and order
reversing where each copy has negative congruence. *

*There are three order preserving
symmetries, the identity, translations and rotations. *

*The identity is the trivial symmetry
where an object is moved onto itself.*

*A translation is defined by a direction
and a distance. In a translation, every point on the object
being translated is moved the same distance and direction.
The direction and distance moved is called the translation vector
by mathematicians. There are an infinite number of possible
translations in the Euclidian plane. The group T of translations
of the Euclidian plane is a normal subgroup of all the isometries
of the plane. In wallpaper groups you will see that some
limitations are placed on the possible translations. Translations
move every point in the plane. See the illustration below.*

*A rotation is defined by a fixed
point and an angle of rotation. In a rotation, every point
on an object is rotated about the same fixed point using the same
angle. A rotation fixes one point in the plane, the center
of rotation. There are also an infinite number of possible
rotations in the Euclidian plane. The group of rotations
is also a normal subgroup of the isometries of the plane.
For simplicity, I will denote this subgroup as R. In wallpaper
groups there are only a finite number of possible angles of rotation.
See the illustration below.*

*There are two order reversing isometries,
reflections and glide reflections.*

*A reflection is defined by a line
called a mirror. In a reflection every point on an object
is moved perpedicular to the mirror so that any segment connecting
corresponding points on the object is bisected by the mirror line.
A reflection fixes a line in the Eulidian plane.
The set of reflections is not a subgroup of the isometries of
the plane. As with translations and rotations, there are
restrictions on reflections in wallpaper groups. See the
illustration below for an example of a reflection.*

*A glide reflection is a hybrid of a translation followed
by a reflection about a mirror parallel to the translation vector.
As with a translation, a glide reflection normally fixes no points
in the plane, except when the translation is the zero vector.
See the illustration below for an example of a glide reflection.*

*These are all the possible symmetries of the Euclidian
plane. This is so because it can be shown that any transformation
that fixes three or more points that are not on the same line
fixes the whole plane and is thus the identity.*

*Amazingly, the isometries of the Euclidian plane are a
group. The product of any two isometries is another isometry.
Every isometry obviously has an inverse. Isometries are
associative. And there is an identity.*

*Wallpaper groups contain all the possible isometry types. *

*To continue, click on one of the options below.*

*The seventeen wallpaper groups*

*For a good overview of wallpaper groups go to http://www.clarku.edu/~djoyce/wallpaper*

This page was created on April 21, 2001 by Michael E. McCallum. For information or comments please contact mmccallu@bellsouth.net.