Mathematics Masterclasses for Young People
Symmetry and Frieze Patterns
by
Jan Abas and Heather McLeay
Theory
Introduction
Decorative art and design has been practiced by human societies from the earliest of times and is a major activity today. Much of such design exploits symmetry, where a motif is repeated in some systematic way to create the whole design. When the motif is repeated on a line we obtain a frieze pattern. At the back of these notes you will find several examples of frieze patterns.
The idea of symmetry is a very important one, not only in art and design but also in many other sciences, particularly mathematics.
What do we mean by symmetry? Think of a square. If we rotate the square about its centre by 90, 180, 270 or 360 degrees then although the vertices are permuted, the square looks the same. Similarly, if we reflect it in any of the lines which join the mid points of the sides or form diagonals then again the square looks unchanged.
Now examine the pattern on top of page 4 which is assumed to extend indefinitely in both directions. Here the distance between a point and its next repeat is L. Clearly, if we drag the pattern horizontally by a distance L, then the pattern looks exactly the same as before. This kind of movement is called a translation and we say that the pattern doesn't change if we apply a horizontal translation L.
Next examine the footprint frieze pattern on page 4 which is again assumed to extend indefinitely. Here the distance from one heel to the next one is L. Then like the last case, the pattern looks exactly as before if we apply a horizontal translation 2L. In this case, however, we can do something else which leaves the pattern unchanged. If we apply a horizontal translation L and then reflect in the line passing through the middle of the pattern, then again the appearance does not change. A combination of a translation and a reflection is called a glide reflection. We can describe a glide reflection by giving the length and direction of the translation and the line which acts as the mirror line.
We can now define symmetry. An object is symmetric if there are some translations, rotations, reflections or glide reflections which, when applied to the object, leave the appearance of the object unchanged. Such translations, rotations, reflections and glide reflections are called the symmetries of the object. For example, rotations by multiples of 90 degrees, reflections through lines joining the midpoints of the sides and reflections in the diagonal lines are the symmetries of the square.
It can be proved that there are only 7 different types of friezes, each type having its own set of symmetries. In this session you will learn how to recognise the seven types of friezes and also how to make them on a computer.
The Notation
The symbols used in Figures 1, 2 and 3 have the following meanings:
T denotes a rectangular tile in which a pattern is to be placed; T will be called a template tile.
We will also assume that the pattern in T has no symmetry, i.e. that it has no vertical or horizontal lines of reflection and no 180 degree rotation (half turns). Denoting T as ABCD, the length of AB will be taken to be L, and m and n will denote the mid points of AD and BC respectively. In Figure 1 we have marked m and n differently to show the positions taken by these points as various transformations are applied to T. |
 Figure 1 |
| T | denotes the basic template tile with motif. |  |
| TH | denotes a tile obtained by reflecting T in a horizontal line. |  |
| TV | denotes a tile obtained by reflecting T in a vertical line |  |
| TR | denotes a tile obtained by rotating T through 180 degrees. |  Figure 2 |
Note: Horizontal reflection and vertical reflection combine to give TR.
“+” denotes the gluing of two tiles. Thus T + TH denotes the tile obtained by gluing T and TH.
Algorithms
The procedure for producing a frieze will be to obtain a unit tile U by gluing to T some transformed versions of T as shown in Figure 3. Having obtained U we simply tile in one direction with U. All we need is to know is how to make the correct unit tile U for each type of frieze. This is given below.
The name of each type shown in Figure 3 has been given in the internationally agreed notation for describing the symmetry of the seven types of frieze pattern (there are several other notations in use). It consists of the letter p followed by three symbols; for example pmm2. The first symbol after the p is used to indicate vertical mirror lines, the second to indicate horizontal lines, and the third to indicate 180 degree rotations (half turns). Thus if vertical lines of reflection exist, the first symbol is m, otherwise it is 1. If horizontal mirror lines exist the second symbol is m; for a glide reflection it is a and for no horizontal mirror line the symbol is 1. The last symbol is 2 if there are centres of 180 degree rotations; otherwise it is 1.
| Type 1: p111 : | U = T. |
 |
| Type 2: p1a1 : | U = T + TH |  |
| Type 3: p112 : | U = T + TR |  |
| Type 4: pm11 : | U = T + TV |  |
| Type 5: p1m1 : | U = T + TH |  |
| Type 6: pmm2 : | U = T + TV + TH + TR |  |
| Type 7: pma2 : | U = T + TV + TH + TR |  |
Figure 3
Examples of Friezes
Shown below are some simple examples of the seven frieze patterns. In each case the box marked with a dotted line indicates the template tile T which has been used to produce all the other components of the pattern. The symmetries of the frieze pattern obtained are described in each case and it is recommended that the reader examine the structure of the corresponding tile U shown in Figure 3 to verify what is said.
p111
The only symmetry is translation of period L.
p1a1
Symmetries are translations of period 2L and glide reflections. The glide vector is of length L and is located along mn.
p112
Symmetries are translations of period 2L and half turns about m and n.
pm11
Symmetries are translations of period 2L and reflections in sides AD and BC.
p1m1
Symmetries are translations of period L and reflections in AB.
pmm2
Symmetries are translations of period 2L, reflections in AB, AD and BC, half turns about the points A and B and glide reflections. The glide vector is of length 2L and is located along AB.
pma2
Symmetries are translations of period 4L, reflections in vertical lines at distances L and 3L from the vertical edges of U, half turns about m and n and glide reflections. The glide vector is of length 2L and is located along AB.
Further Reading
E.H. Lockwood and R.H. Macmillan (1978) ”Geometric Symmetry”, CUP.
References
(1) P.J. Davis and R. Hersh (1981) “The Mathematical Experience”, Pelican (page 169).
(2) Welsh Office funded project “Strategies and Skills in Mathematics”.