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 rn = 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 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.
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 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 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 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