WALLPAPER GROUPS - IJS ziherl/ Intuitively a wallpaper pattern is a design used for making...
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In this paper we present the wallpaper groups or plane crystallographic groups. The namewallpaper groups refers to the symmetry group of periodic pattern in two dimensions. We startby presenting the symmetries of the Euclidean plane. Than all 17 types of wallpaper symmetriesare described.
1 Introduction 1
2 Symmetry in the plane 22.1 Composition of isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Lattices and point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 The Wallpaper groups 63.1 Parallelogram lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Rectangular and centerd rectangular (rhombic) lattices . . . . . . . . . . . . . . . . . 73.3 Square lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Hexagonal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
A 17 Wallpaper groups 16
Intuitively a wallpaper pattern is a design used for making wallpaper. One can make such a patternby taking a motif and repeating it horizontally and vertically. The example on Figure 1 (left) wasmade by taking a motif , and translating it horizontally and vertically. There are other ways ofrepeating a basic motif to get a pattern. For instance, one could repeat the the same motif in ananother way to obtain different wallpaper pattern (Figure 1 right). As we will see there are exactly17 number of ways a motif can be arranged to form a patter.
Figure 1: Two different ways of wallpaper pattern construction by repeating the same motif.
Wallpaper patterns occur frequently in architecture and decorative art. A nice example is theAlhambra. This Islamic palace lays in the valleys of Granada, Spain and was built after theMuslims conquered Spain in the 8th century. The palaces interior design exhibits the Islamicnecessity to use geometric shapes. This mathematical labyrinth, as it has been referred to, exhibitswalls that are decorated using all possible wallpaper patterns. It is not known whether the architectshad a mathematical knowledge of all the different wallpaper patterns or simply exhausted all thepossibilities.
2 Symmetry in the plane
An isometry of R2 is a distance-preserving transformation of the plane . It is a map f : R2 R2such that for any points P and Q in the plane
d(P,Q) = d(f(P ), f(Q)), (1)
where d(P,Q) is the distance between P and Q. As we will later show there are only four types ofisometries of the plane: translations, rotations, reflections and glide reflections.If vector v R2, then a function T : R2 R2 defined by Tv(r) = r + v for all r R2 isa translation by a vector v. Rotation R0, by an angle about the origin is in Cartesiancoordinates given by
(cos sinsin cos
Rotation by about a point P R2 can be written as RP, = T R0, T1, where T is translationby P . We say that a planer object has rotational symmetry about a point P if it is carried intoitself by at least one non-trivial rotation around P . Let l be a line in R2 going through the originparallel to a unit vector w, then the reflection Ml of vector r across l is given by
Ml(r) = 2(rw)w r. (3)
An object is symmetric with respect to a line l if it is carried into itself by reflection across l. LetMl be a reflection about a line l passing through the origin. If vector t R2 is not perpendicularto l, than a composition of reflection and translation G = Tt Ml is a glide reflection parallel tol and displaced in perpendicular direction by 12t.
The composition of two isometries is an isometry . The isometries of the plane form a groupunder composition of function, the so called Euclidean group E2. Euclidean group is built from thetranslational group T the orthogonal group O2(R) . Every element of O2(R) can be representedby a 2 x 2 matrix, where any such matrix satisfies the condition ATA = I. Let A be the matrixwith respect to the standard basis for an element in O2(R). If
(a bc d
then the condition ATA = I gives(a2 + c2 ab+ cdab+ cd d2 + d2
(1 00 1
The above relation can be satisfied if there is an angle with a = cos , c = sin and (b, d) =( sin , cos ) or (b, d) = (sin , cos ). The first choice gives a matrix A with determinant 1 thatrepresents a rotation about the origin by an angle . Such matrix is an element of special orthogonalgroup SO2(R) a subgroup of O2(R) whose elements preserve orientation. On the other hand thesecond choice gives a matrix of determinant -1 that represents a reflection across a line through theorigin. Elements of SO2(R) are rotations and elements of O2(R) \ SO2(R) are reflections. Sinceevery element of E2 is the composition of a translation with an element of O2(R), it follows thatevery isometry of the plane is either a translation, a rotation, a reflection or a glide reflection .Every isometry of the plane can be written in a specific notation as an ordered pair (A,u), whereu R2 and A O2.
2.1 Composition of isometries
Now we will look at some compositions of isometries, that will later help us construct the wallpapergroups.Rotation translationComposition of rotation RP, about point P by an angle and a translation Tt is another rotation
RQ, by about a point Q that is displaced vertically by|t|2 cot
2 along bisector of t (Figure 2).
Figure 2: Composition of rotation and translation. Initial object (1) is rotated about P by an angle to asecond object (2), which is than translated by t to third object (3). The first and third object are directlyrelated with rotation about Q by an angle .
Reflection translationIf Mm is a reflection across line m and Tt translation perpendicular to reflection line, than thecomposition of reflection and translation is a reflection Mm across line m
at the bisector of t. Ifhowever the translation is not perpendicular, than the composition is a glide reflection Gt parallel
to reflection line and displaced by t2 in perpendicular direction (Figure 3).
Figure 3: Composition of reflection and translation. Initial object (1) is reflected across line m to a secondobject (2), which is than translated by t to third object (3). If the vector of translation is perpendicular tothe mirror line m, the first and third object are directly related with reflection across m, otherwise they arerelated with glide reflection g.
Glide reflection translationComposition of a glide reflection G and a perpendicular translation Tt is a glide reflection G
atthe bisector of t. Composition of a glide reflection Gt and a non-perpendicular translation is a
reflection M perpendicular to glide reflection and displaced by t2 in perpendicular direction.
Reflection rotationIf Mm is a reflection across line m and P center of rotation by that lies on m, than the compo-sition of rotation and reflection is a reflection M m across a line m
, that is passing through P andis inclined by an angle 2 to line m (Figure 4). If center of rotation does not lie on reflection line,than the composition is a glide reflection.
Figure 4: Composition of rotation and reflection. Initial object (1) is reflected across line m to a secondobject (2), which is than rotated about P by to third object (3). We can see that the first and third objectare related by reflection across line m, which is inclined at an angle 2 relative to line m.
Reflection reflectionSuppose we have two parallel reflection lines m and m separated by t. Reflecting an object acrossreflection line m followed by reflection across the line m is the same as translating the originalobject by 2t (Figure 5 left). In the more general case the two reflection lines are inclined to oneanother by an angle and we have an intersection point P (Figure 5 right). The composition oftwo reflections is in this case a rotation RP, by 2 about point P .
Figure 5: Composition of two parallel (left) and non-parallel (right) reflections. Initial object (1) is reflectedacross line m to a second object (2), which is than reflected across line m to third object (3).
2.2 Lattices and point groups
Wallpaper pattern is a two dimensional repeating pattern that fills the whole plane. The sym-metry group of a wallpaper pattern is said to be a wallpaper group. A subgroup W E2 is awallpaper group if its translation subgroup H is generated by two independent translations and itspoints group J is finite .A two-dimensional lattice L in R2, is the set of all the points that the origin gets mapped tounder the action of H, the translation subgroup of a wallpaper pattern. It is a subgroup of theform nt1 + mt2, where n,m Z for some basis t1, t2 of R2 . By examining all possible basicparallelograms determined by the vectors t1 and t2 we will later see that there are only five differenttypes of lattices; parallelogram (oblique), rectangular, centred rectangular, square and hexagonal.
Theorem (Crystallographic restriction): The order of a rotation in a wallpaper group canonly be 2, 3, 4, or 6.
Proof: Let Q be a lattice point and center of rotation of period n. Transformation operator ofthe lattice transforms Q into infinitely many other centers of rotation of the same period. Let P
Figure 6: Crystalographic restriction.
be one of these other centers at the least possible distance form Q (Figure 6). A third center ofrotation P is obtained by rotation of P through = 2/n about Q; and a fourth Q by rotationof Q by about P .Points generated by rotation must coincide with lattice points, therefore Q an