The Wallpaper Groups

 There are 17 such patterns, and because their symmetry groups were first studied by 19th Century crystallographers they are also called plane crystallographic groups. Apart from their mathematical classification, they were popularised by the Dutch artist M. C. Escher (1898-1972). They can also be found in Islamic and Chinese Art, for example the tiliongs in the Alhambra palace in Grenada and Chinese lattice designs.

 For some examples, click here.

 Let G be a subgroup of Iso(2) containing independent translations X and Y of minimum length. Let P be any point in the plane. Then 
   P<X, Y> = {PXⁿ.Y^m: n, m in Z}
is an infinite array of points at the corners of parallelograms, called a lattice (of points or parallelograms). Thus any plane crystallographic group must be based on an underlying pattern of such parallelograms, called the basic parallelograms. In particular, any isometry fixes the point lattice.

The Crystallographic Restriction

Theorem 15.1 If R is a rotational symmetry of a point lattice, then it is by an angle q = p/3, p/2, 2p/3 or p.

Proof Let R have order n and let P be a centre of rotation. The translations map P into infinitely many other centres of rotation of order n. Let Q be one whose distance from P is minimal. Let P' = QR and let Q' = P'R. Thus |PQ| = |QP'| = |P'Q'|. If P = Q', then PQP' is an equilateral triangle so the angle is p/3 so n = 6. If not, by the minimality of |PQ|, |PQ'|>= |PQ|.

 If n = 5, then the angles at Q and P' are 2p/5 so |PQ'| < |PQ|, a contradiction.

 If n > 6 then the angle at Q < p/3, so |PP'| < |PQ|, a contradiction.

 Hence the only possibilities are n = 2, 3, 4 or 6, giving angles of rotation of p, 2p/3, p/2 or p/3. We shall see later that all of these can occur. 

1. G = <X, Y> is isomorphic to Z x Z. It is the symmetry group of a pattern with no other symmetries in one of the basic parallelograms in the lattice. Crystallographers call this isometry group p1.
2. Suppose the vectors X and Y are not at an angle of p, p/3, p/2 or 2p/3, and let G be the isometry group of the lattice of parallelograms itself. Apart from products of X and Y, G contains p-rotations about the midpoints of the line segment joining any pair of lattice points. Let R be any such p-rotation. Then the group <X, Y, R> contains all such rotations, so G = <X, Y, R>

 Crystallographers call this group p2 . The Crystallographic Restriction ensures there are no further isometries.

Recognition of Wallpaper patterns

And here are simple examples of each of the 17 types.

The structures are described in the following table:

 
1. Translation vectors form non-rectangular parallelogram.

1.	p1	no rotations, no reflections and no glides two linearly independent translations
2.	p2	half-turns at corners, edge centres, and face centres two independent translations.

2. Translation vectors form rectangle.

3.	pm	no rotations, no glides, two reflections through parallel sides translations parallel to and orthogonal to these sides.
4.	pg	no rotations, no reflections, glides through two parallel sides, two orthogonal translations.
5.	cm	no rotations, reflections through diagonal, two parallel glides.
6.	pmg	half-turns at corners, edge centres and face centres; two parallel reflection lines; central glide line.
7.	pgg	half-turns at corners, edge centres and face centres; no reflections; two orthogonal glide lines.
8.	pmm	half-turns at corners, edge centres and face centres; reflections in sides and centre lines; no glides.
9.	cmm	half-turns at corners, edge centres and face centres; two orthogonal reflections; four parallel glides.

3. Translation vectors form square.

10.	p4	4-fold rotations at corners and face centres,  half-turns at side centres; no reflections or glides.
11.	p4m	4-fold rotations at corners and face centres, half-turns at side centres; eight reflections; four glides.  4-fold rotation centres occur on reflection lines.
12.	p4g	4-fold rotations at corners and face centres, half-turns at side centres; four reflections; six glides. 4-fold rotation centres do not occur on reflection lines.

4. Translation vectors of equal length at 60 degrees.

13.	p3	3-fold rotations at corners and triangle centres; no reflections or glides.
14.	p3m1	3-fold rotations at corners and triangle centres; five reflections; nine glides.  3-fold rotation centres occur on reflection lines.
15.	p31m	3-fold rotations at corners and triangle centres; five reflections; four glides. 4-fold rotation centres do not occur on reflection lines.
16.	p6	6-fold rotations at corners, 4-fold rotations at side centres and face centres; 3-fold rotations at triangle centres; no reflections or glides.
17.	p6m	6-fold rotations at corners, 4-fold rotations at side  centres and face centres;3-fold rotations at triangle centres; ten reflections and ten glides.

Here is a link to a different analysis of wallpaper patterns.

Here is a picture of Chinese screens realizing each of the wallpaper patterns.

Here are some more links to more pages illustrating wallpaper patterns and other tilings of the plane.

[The  diagrams above are modified versions of diagrams from  Doris Schattschneider: The Plane Symmetry Groups, The Amer. Math. Monthly, Vol 85, No. 6, 1978, 439-450, and from John H. Cadwell, Topics in Recreational mathematics,  Cambridge University press, 1966, and from Paul Scott: The Grandeur of Granada, Australian Mathematics Teacher, Vol 54, 2000.]
 

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 Last update May 2, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au