*A wallpaper group is a discontinuous
subgroup of the isometries of the Euclidian plane. That
is, for every point P and every circle C containing P in the plane,
There are only a finite number of images P* of P in C where *
is an isometry of the wallpaper group. As implied
in the title, there are seventeen wallpaper groups. The
groups are classified by the symmetries they contain. I
shall use Conway's notation for naming the groups. But first
we need some language to describe the groups.*

*First, a wallpaper group is called
a wallpaper group because of its similarity to wallpaper.
In fact, some people call them wallpaper patterns. Each
wallpaper group is characterized by a unique pattern that repeats
itself periodically. Wallpaper groups are distiguished from
another type of transformation pattern of the plane called "tilings"
by the fact that only a single pattern does the tiling of the
plane in each group. Tilings can have many different patterns
simultaneously.*

*The fundamental unit of a wallpaper
pattern in the smallest area of the plane that contains all of
the information needed such that the entire plane can be filled
using products of the symmetries of the group. The images
generated by the fundamental and the symmetries of the group do
not overlap.*

*Some sources describe a translation
region. A translation region is the smallest area that
can be formed from a fundamental unit with the property that the
remainder of the pattern can be generated by using only the translations
of the group.*

*We are now ready to describe the
groups. Let r in R, the set of rotations of the Euclidian
plane, be such that **r ^{n} = 1, the identity.
Then n can only be 1, 2, 3, 4, or 6. This is called the
crystallographic restriction. Proof of this is beyond what
I want to present here. The wallpaper groups can be classified
by the value of n. The elements of each wallpaper group
are a pair of translations and their integer multiples at a minimum.
Each group can contain rotations, reflections, and glide
reflections, but not every group contains these.*

*If n = 1
then the only rotation is the identity or trivial rotation of
360 degrees. There are four wallpaper groups in this classification. *

*Conway type o**Conway type ****Conway type xx**Conway type *x*

*Conway type
o has the simplest generator set of all the wallpaper groups.
The generator set is the group. It consists of two translations
and their integer multiples. The fundamental region of this
group is also the translation region, hence the unit lengths of
the translation vectors correspond to the lengths of two intersecting
edges of the fundamental region. This group has no non-trivial
subgroups. In mathematical terms this can be expressed as:*

*Let a and b be translations. Then the group G contains
a and b such that ab = ba.
*

*An example
of this type is shown below.*

*The fundamental region is shown
as a rectangle.*

*Conway type
** also has two translations, in fact every wallpaper group has
two reflections and their integer multiples, so I won't mention
them again. In addition, type ** has two reflections with
parallel mirrors on opposite edges of the fundamental region.
The translations form a normal subgroup of this group. In
fact, the translations are a normal subgroup of each of the wallpaper
groups. The product of a translation and a reflection is
a glide reflection, so this group also contains glide reflections.
Since only one translation vector is parallel to the mirrors,
there are only two glide reflections and their inverses.
The reflections are self inverting. That is, if f is a reflection
in the group, then ff = 1. To describe the group in mathematical
terms we have:*

* *
*Let a and b be translations and f a reflection. Then
G consists of a, b, and f such that ab = ba, af = a, bf = b-inverse,
and ff = 1.*

*An example
of this type is shown below.*

*The fundamental region is shown
as a rectangle. In this example, the right and left edges
of the rectangle are the mirrors for the reflections.*

*Conway type
xx contains a glide reflection and its inverse in addition
to the normal translation subgroup. The glide reflection
does not contain an edge, and is in fact collinear with the midpoints
of opposite edges of the fundamental region. The length
of the glide vector is equal to the length of one of the edges
parallel to the mirror of reflection. In mathematical terms
this can be expressed as:*

*
Let a and b be translations and g be a glide reflection.
Then the group G consists of a, b, and g such that ab = ba, ag
= a, bg = b-inverse, and gg = a.*

*An example
of this type is shown below.*

*The fundamental region is shown
as a rectangle. The glide reflection has a horizontal mirror
through the midpoints of the vertical sides.*

*Conway type
*x contains two reflections with parallel mirrors on opposite
edges of the fundamental region and a glide reflection, and its
inverse, with mirror parallel to the reflection mirrors and coincident
with the midpoints of the edges joining the mirrors. The
length of the glide vector is the length of one of the edges parallel
to its mirror. As before, the reflections are self inverting.
This group can be desribed mathematically as:*

*
Let a and b be translations, f a reflectionand g a glide reflection.
Then the group G consists of a, b, f, and g such that ab = ba,
af = b, bf = a, ff = 1, gg = b-inverse, ag = ga, bg = g-inverse.*

*An example
of this type is shown below.*

*The fundamental region is shown
as a rectangle in this example also. The vertical edges
are the mirrors. The glide reflection has a vertical mirror
through the midpoints of the horizontal edges.*

*Wallpaper groups with n = 2 contain
rotation symmetries of 180 degrees and no other angles.
There are five of these groups. *

*Type 2222**Type *2222**Type 22***Type 22x**Type 2*22*

*Conway type 2222 contains four rotations of angle 180 degrees
and two translations. This is the simplest of the n = 2
class of wallpaper groups. It contains no reflections or
glide reflections. In mathematical terms this group can
be describe as:*

*Let a, anb b be translations and r be a rotation.
Then the group G can be generated by a, b, and r such that ab
= ba, ar = a-inverse, br = b-inverse, and rr = 1. *

*The rotations have centers at the vertices of the fundamental
region and at the midpoints of one pair of opposite sides.
The minimal generating set described has r at one of the midpoints.
If rotations at the vertices are used as generators, then a pair
of adjacent vertices must be used. An example of this pattern
is illustrated below.*

*The fundamental region in this pattern is a rectangle.
The centers of rotation are at the four vertices and the midpoints
of the vertical sides. If you look carefully at the pattern
you can see that only two of the vertices have unique symmetries.
The symmetries of the other pair of vertices are duplicates.
Thus the 2222 instead of 222222.*

*Conway type *2222 differs from type 2222 in that reflections
are added. Each edge of the fundamental region is a mirror.
The four vertices now have unique symmetries which are rotations
of 180 degrees. Since each edge is a reflection mirror,
adjacent sides of the fundamental region must be perpendicular.
(Try to generate a pattern when all sides are mirrors and
adjacent sides aren't perpedicular and the rotation is 180 degrees.)
In mathematical terms this group can be described by:*

* Let a and b be translations, f1, f2,
f3, and f4 be reflections and, r1, r2, r3, and r4 be rotations,
with f1 being the bottom edge and r1 being the lower right-hand
vertex numbered counterclockwise. The the group G contains
the identity, a, b, f1, f2, f3, f4, r1, r2, r3, r4, glide reflections
af1, af3, bf2, bf4, and reflections af2, af4, bf1, and bf3.
The inverses of a and b are f4f2 = a-inverse, and f3f1 = b-inverse.
The reflections and rotations are self inverting.*

*An examplle of this group is illustrated below.*

*In the illustration above, the fundamental region is a
rectangle. All of the edges of the rectangle are mirrors.
Each vertex of the rectangle is the center of a rotation of 180
degrees.*

*Conway type 22* has two reflections on opposite edges of
the fundamental region, two 180 rotations with centers at the
midpoints of the two edges that are not reflection mirrors and
two glide reflections wtih mirrors that are perpendicular to the
two reflection mirrors. Since reflections are present, the
fundamental region is a rectangle or a square. This group
has the following structure:*

* Let a and b be translations, a perpendicular
to b, f1 and f2 be reflections, both mirrors perpendicular to
a, g1 and g2 be glide reflections with glide vectors parallel
to a and length 1/2 a, r1 and r2 be rotations of 180 with
centers at the midpoints of the edges of the fundamental region
that are not mirrors. Then the group G contains a, b, f1,
f2, g1, g2, r1, r2, and glide reflections bf1 and bf2. If
f1 is the right mirror and f2 is the left mirror, then a-inverse
= f1f2. If r1 is the bottom center of rotation and r2 is the top
center of rotation, then b-inverse = r1r2. Notice that there
are two additional glide vectors that are the result of the product
of b with the two reflections. For the glide reflections
g1 and g2, g1-inverse is (a-inverse)g2 and g2-inverse is (a-inverse)g1.*

*An example of this pattern is shown below.*

*The fundatmental region is a rectangle. It is easy
to see the reflections and glide reflections. And it is
easy to see the rotations at the vertices of the fundamental region. *

*Conway type 22x has no reflections, but does have two glide
reflections that are parallel to the translation vectors
with length 1/2 the corresponding translation vector containing
the midpoints of opposite sides. There is also a rotation
of 180 degrees at each vertex. This group has the following
structure:*

* Let a and b be translations and g1 and
g2 be glide reflections, g1 parallel to a with lenght 1/2 a and
g2 parallel to b with lenght 1/2 b, and r1, r2, r3, and r4 be
rotations of 180 degrees at each vertex of the fundamental region
numbered as before. Then the group G consists of the identity,
a, b, g1, g2, r1, r2, r3, and r4. The product g1g2 = g2g1
= r2 by our numbering. The inverses of the translations
are a-inverse = r1r4, and b-inverse = r2r1. The inverse
of g1 = (a-inverseg2 and the inverse of g2 = (b-inverse)g1.*

*An example of this pattern is shown below.*

*The fundamental region is rectangular in this pattern.
It is easy to see the glide reflections and the rotations.
Translation vectors a and b each have length twice the length
of their corresponding parallel edges.*

*Conway type 2*22 is the last of the groups where n = 2.
This group has three reflections. Two of the reflections
have parallel mirrors that are opposite edges of the fundamental
region. The third reflection mirror is one of the remaining
edges. There are also three rotations of 180 degrees, one
at each vertex where the mirrors intersect and one at the midpoint
of the remaining side that is not a mirror. Of course, where
there are translations and mirrors, there are glide reflections.
The structure of this group is:*

* Let a and b be translations, f1, f2,
and f3 be reflections with f1 parallel to f3 and f2 perpendicular
to f1 and f3, r1, r2, and r3 be rotations with r1 and r3 at the
vertices of the mirrors and r2 at the midpoint of the remaining
side. Then the group G consists of a, b, f1, f2, f3, r1,
r2, and r3, with ab = ba, f1f3 = a-inverse, r2f2r2f2 = b-inverse.
There are also glide reflections f2r2, bf1, bf3, and af2, and
their inverses. *

*An example of this pattern is shown below.*

*It's not as easy to see all of the components of this group
in the pattern above. The glide reflections are fairly easy
to see. In fact you may see glide reflections than I accounted
for in the group description. This occurs occasionally in
wallpaper patterns. The translations a and b are easy enough
to see.*

*I bet you thought n = 3 would be
next. I will come back to n = 3. The reason I skipped
to n = 4 is because this class has one group with a square fundamental
region and two groups with triangular fundamental regions.
The fundamental regions of the remainder of the wallpaper groups
will be triangular (with one exception). The wallpaper groups
with n = 4 are:*

*Conway type 442**Conway type *442**Conway type 4*2*

*Conway type 442 is the one with the
square fundamental region. There are no reflections in this
group. There are two rotation centers with rotations of
90 degrees at opposite vertices and two rotation centers with
rotations of 180 degrees at the other two vertices. The
group structure of this group is:*

* Let a and b be
reflections, r1 and r2 be rotations of 90 degrees at opposite
vertices of the fundamental region and r3 and r4 be rotations
of 180 degrees at the other two vertices. Then the group
G consists of a, b, r1, r1^2, r1^3, r2, r2^2, r2^3, r3, and r4,
where ab = ba, r1^4 = 1, r2^4 = 1, (r1^3)r2 = a-inverse, and (r2^3)r1
= b-inverse.*

*An example of this pattern is shown
below.*

*The 90 degree centers of rotation are ate the middle of
the yellow cross and at the lower right vertex of the fundamental
region. The 180 degree centers of rotation are at the other
two vertices of the fundamental region.*

*Conway type *442 has a triangular fundamental region with
all edges being reflection mirrors. The fundamental region
is an isosceles right triangle with a 180 rotation center at the
90 vertex and 90 degree rotation centers at the other two vertices.
The group structure of the type *442 is:*

* Let a and b be translations, f1,
f2, f3, and f4 be reflections, f1 parallel to a, f3 parallel to
b, f2 perpendicular to f4 and the angle formed by f1 and f2 be
45 degrees, r1 and r3 be centers of 90 degree rotations, and r2
and r4 be centers of 180 degree rotations. Then the group
G consists of the identity, a, b, f1, f2, f3, f4, r1, r1 ^2, r1^3
,r2, r3, r3^2, r3^3, r4, af3, and bf1, where ab = ba, r1
^4 = 1, r3^4 = 1, f1 ^2 = 1, f2^2 = 1, f3^2 = 1, and f4^2 = 1.
The inverses of the translation vectors are a-inverse = r1f4f3,
and b- inverse = f3f1r1.*

*An example of this pattern is shown below.*

*Conway type 4*2 also has a triangular fundamental region.
Again, the fundamental region is a right isosceles triangle.
The hypotenuse is a reflection mirror and there are two glide
reflections, both containing the midpoint of the hypotenuse in
their respective reflection mirrors and perpendicular to one another.
There is a 90 degree center of rotation at the 90 degree vertex
and a 180 degree center of rotation at each of the other vertices.
The group structure of this type is:*

* Let a and b be translations, f1 and
f2 be reflections, g1 and g2 glide reflections with g1 parallel
to a and g2 parallel to b, r1 and r4 rotations of 90 degrees,and
r2 and r3 rotations of 180 degrees (f2 is the reflection that
results when f1 is rotated 90 degrees by r1, r4 is the 90 degree
rotation that results when r1 is reflected about f1). Then
the group G consists of a, b, f, g1, g2, r1, r1 ^2, r1 ^3, r2,
r3, r4, r4 ^2, r4 ^3, af, bf, ag1, bg1, ag2, bg2, and fr1, such
that ab = ba, af1 = f1a, bf1 = f1b, f1 ^2 = 1, r1 ^ 4 = 1, r2
^2 = 1, r3 ^2 =1, r4 ^4 = 1, a-inverse = f1(r1 ^2)r2f1, b-inverse
= (r1 ^2)r2, g1-inverse = f1r1, g2-inverse = f1(r1 ^3).
The rotations and reflections form a subgroup of this group. *

*An example of this pattern is shown below:*

*This pattern has an isosceles right triangle for a fundamental
region. Both the horizontal and vertical glide reflections
are readily apparent here. The second rotation of order
four can be seen as well.*

*There are three wallpaper groups
with n = 3. The groups in this class are :*

*Conway type 333**Conway type 3*3**Conway type *333*

*Conway type 333 contains no reflections
and three rotations of order three. This is a fairly simple
group. The group struture of this type is:*

* Let a and b be
translations, r1, r2 and r3 be rotations of order three.
Then the group G consists of the indentity, a, b, r1, r1^2, r2,
r2^2, r3, and r3^2, where ab = ba, r1^3 = 1, r2^3 = 1, and r3^3
=1. The inverses of the translations are: a-inverse = r2r3,
b-inverse = r3r1. Each rotation forms a subgroup of order
three. The entire group can be generated by r1 and the translations.*

*An inllustration of this group appears
below.*

*This pattern has a rhombus with opposite vertices of 120
degrees and the other vertices 60 degrees for a fundamental region.
There is a rotation of order three at each vertex. Wait
a minute, a rhombus has four vertices and we said there were only
three rotations. Look at the fundamental region in the pattern.
It is impossible to distinguish between the patterns at the two
vertices of 60 degrees. That is why we only have three
rotations.*

*Conway type 3*3 has three reflections as well as three
rotations of order three. The fundamental region for this
group can be an isosceles triangle with a 120 vertex and a reflection
mirror on the side opposite the 120 degree vertex. There
is a rotation of order three at each vertex. The group structure
of this type is:*

* Let a and b be translations, f1, f2,
and f3 be reflections, r1, r2, and r3 be rotations of order three.
Then the group G consists of a, b, f1, f2, f3, r1, r1^2, r2, r2^2,
r3, and r3^2, where ab = ba, f1 ^2 = 1, f2^2 = 1, f3^2 = 1, r1^3=1,
r2^3 = 1, and r3^3 = 1. The inverses of the translations
are: a-inverse = r3r2, b-inverse = r3(r2^2). There are two glide
reflections, af1 and bf3. There are more, but these are
easiest to see. *

*An illustration of this pattern is shown below.*

*The fundatmental region of this pattern is an isosceles
triangle with a vertex of 120 degress that was described above.
You can see the three distinct rotations, r1 is a the 120 degree
vertex, r2 is at the 30 degree vertices and r3 is a reflection
of r1 about f1, the horizontal mirror. The other two mirrors
are rotations of f1 by 120 and 240 degrees.*

*Conway type *333 contains three reflections and three rotations
of order three. Sounds like 3*3, doesn't it? The difference
is that the translations are not parallel to any of the reflections.
The group structure of this type is:*

* Let a and b be translations, f1, f2
and f3 be reflections, r1, r2, and r3 be rotations of order 3.
Then the group G consists of the identity, a, b, f1, f2, f3, r1,
r1^2, r2, r2^2, r3, and r3^2, where ab = ba, f1^2 = 1, f2^2, =1,
f3^2 = 1, r1^3 =1, r2^3 = 1, r3^3 = 1. The inverses of the
translations are: a-inverse = r2r3^2, b-inverse = r2r1^2.
Each of the rotations, along with the reflections, forms a subgroup
of order six. The rotations by themselves form subgroups
of order three.*

*An illustration of this pattern is shown below.*

*The fundamental region in this pattern is an equilateral
triangle with each side a reflection and each vertex a rotation
of order three. Looking at the pattern shown above, it is
easy to see all of these.*

*There are only two wallpaper groups
with n = 6. These groups are characterized by rotations
of order six. The two types are:*

*Conway type 632**Conway type *632*

*Conway type 632 has one rotation
of order six, two rotations of order three, and one rotation of
order two. There are no reflections. The group structure
of this type is:*

* Let a and b b
translations, r1 be a rotation of order six, r2 and r2 be a rotations
of order three, and r4 be a rotation of order two. Then
the group G consists of the identity, a, b, r1, r1^2, r1^3, r1^4,
r1^5, r2, r2^2, r3, r3^2, and r4 where ab = ba, r1^6 = 1, r2^3
= 1, and r3^2 = 1. The inverses of the translations are:
a-inverse = r2r3^2, b-inverse = r4r1^3. Each of the rotations
forms a subgroup of the given order.*

*An illustration of this pattern is
shown below.*

*In this example, the fundamental region is an equilateral
triangle. The rotation of order six is at the lower vertex,
the rotations of order three are at the other two vertices and
the rotation of order two is at the midpoint of the side connecting
the rotations of order three. *

*Conway type *632 has six reflections in addition to the
rotations in type 632. This is a fairly complicated group.
The structure of this group is:*

* Let a and b be translations, f1, f2,
f3, f4, f5, and f6 be reflections, r1 be a rotation of order six,
r2 and r3 be rotations of order three, and r4 be a rotation of
order two. Then the group G consists of the identity, a,
b, f1, f2, f3, f4, f5, f6, r1, r1^2, r1^3, r1^4, r1^5, r2, r2^2,
r3, r3^2, and r4, where ab = ba, fi^2 = 1, i = 1,6, r1^6 = 1,
r2^3 = 1, and r3^3 = 1. The inverses of the translations
are: a-inverse = (r1^2)r2r4, b-inverse = r2r1^3. There are
glide reflections af6 and bf4. There are more but not as
easy to see. (Any product of a translation and a reflection
is a glide reflection unless the translation is perpendicular
to the mirror.) This group is rich in subgroups. There
is one of order 12 formed by the rotation of order six and the
reflections. There are three of order six; on formed by
the rotation of order six alone and the other two formed by the
rotations of order three and three reflections. there are
two of order three formed by the rotations of order three.
There is one of order four formed by the rotation of order two
and two reflections. And finally, there is one of
order two formed by the rotation of order two. *

*An illustration of this pattern is shown below.*

*The fundamental region in this pattern is a 30-60-90 triangle
with every side a reflection mirror. The rotation of order
six is at the 30 degree vertex, a rotation of order three is at
the 60 degree vertex, and the rotation of order two is at the
90 degree vertex. The other rotation of order three is a
reflection of the first about the vertical mirror. The mirrors
and centers are all illustrated above.*

*This covers all seventeen of the wallpaper groups.
Are the wallpaper groups a group? No, they are all distinct
subgroups of the group of isometries of the Euclidian plane.
I have tried to make this presentation as accessible as possible.
In my zest to do so, I may have left some things out that you
think should be included. If so, contact me at mmccallu@bellsouth.net.
Feel free to contact me if you want to suggest something.
Don't bother if you just want to complain.*

*For more on the connections of algebra to geometry, continue
on to*

*Frieze Groups and Other Things.*

*Return to Just What is a Group Anyway*

*Return to Isometries of the Euclidian
Plane*

This site was created on April 21, 2001 by Michael E. McCallum