[Work in progress]
Choosing wallpaper can be a time-consuming task; there are just so many designs to select from. Most interior decorators will be happy to lend you large books of samples which you can leaf through at your leisure. Yet, how many patterns are there? You might be surprised to know that in fact there are only seventeen. And it's been proven - mathematically.
This is quite a startling result. If word got out that there were so few, it could blow the lid off the wallpaper industry: they have evidently been over-egging it somewhat with all those huge sample books. Their ranges must be full of duplicates! It's not clear how they managed to pull the wool over our eyes for so long, but this series of entries sets out to discover the truth behind wallpaper design. We will include an identification guide for the seventeen varieties below. Any others you find are clearly impostors.
OK, that's not strictly true. Of course there are more than seventeen designs. Let's say, for example, you have a simple motif on your wallpaper, one which repeats itself at regular intervals. Now, depending on your taste in interior decoration, this motif could be a fleur-de-lys, maybe, or the Manchester United club crest, or even a picture of Harry Potter, resplendent in his wizard's robes. Whatever it is, it could be printed in various sizes and different colours. There are infinitely many wallpaper designs, really. When we say 'wallpaper patterns' we mean certain, recognisable groups of them which have the same symmetrical properties. Maybe the pattern would be unchanged, for example, if you hung the wallpaper upside down, or maybe the patterns have some sort of reflectional symmetry - that sort of thing. There are exactly seventeen of these wallpaper pattern groups.
There's nothing special about wallpaper, of course. You will find similar patterns represented on all sorts of things: carpets, curtains, knitted jumpers, frosted glass, even wrought-iron gates. It just happens that mathematicians once labelled these 'wallpaper patterns', and it stuck1.
One feature of wallpaper patterns, and this may sound obvious, is that the pattern extends in two dimensions. Yet, it's worth mentioning, as it distinguishes these patterns from those which extend in only one dimension (known as friezes), and those in three dimensions (crystals). For the record, there are seven different frieze pattern groups and 230 crystallographic groups.
Another feature of patterns important to the mathematician is that they extend forever. There are no border effects where the pattern stops. You shouldn't need an infinitely large wall to spot the symmetries, though - our brain is pretty good at working them out from a small sample.
To understand what defines and differentiates the seventeen wallpaper groups, we first need to understand a little bit about the geometry of patterns. In the rest of this Entry, we'll describe symmetries and associated lattices. It's not too demanding, but if you really don't want to get into this, feel free to skip to the other Entries in this series, where you'll find most of the pretty pictures.
[Embed picture right: symmetries. External link]
Look carefully at a pattern and you'll see that it repeats itself exactly at various points. This is self-evident when you consider that wallpaper comes on rolls, and each must be matchable to the others. There are four ways in which a pattern can repeat itself, and mathematicians describe these symmetries – or, to be precise, Euclidean plane isometries – as follows:
Translation: If you move the wallpaper a particular distance in a certain direction, then the original pattern will repeat itself exactly at that point.
Rotation: If you pivot the wallpaper around a given point by a certain angle, then the pattern will repeat itself. You could demonstrate this by sticking a pin through the paper and rotating it (preferably before you paste it to the wall).
Reflection: If you place a mirror against the wallpaper, angled in a particular direction, then the mirror image will repeat the pattern.
Finally, there's the composite symmetry known as a glide reflection. If you reflect a pattern and then slide it for a certain distance in the direction of the line of reflection, it will repeat itself.
The easiest way to demonstrate these symmetries is to print your outline pattern onto both a piece of paper and a transparent piece of plastic (like those acetate sheets we used to use with overhead projectors). When you lay the transparency on top of the paper pattern, the designs match up, but when you move it slightly, they become confused. It is only when you move it in a certain direction for some distance (translation), or spin it around at some point by a certain angle (rotation), or flip it over at a particular angle (reflection) that you may see the patterns match up once again.
Now, the seventeen types of wallpaper are distinguished by the particular combinations of symmetries they have. Before we get stuck into our spotter's guide, however, we need to say something about lattices.
Lattices and Generating Regions
[Embed picture right: lattices. External link]
A lattice is a kind of simple framework - an example in real life could be a garden trellis or a honeycomb in a beehive. These are very simple patterns made from identical interlocking outline shapes. The individual cells of a lattice can be one of six different shapes: either squares, rectangles, parallelograms, rhombuses, hexagons or parallelo-hexagons.
Now, every wallpaper pattern has an associated lattice. If you draw the lines of the lattice on top of the pattern then you will see that every cell of the lattice contains an identical section of the wallpaper pattern. So, to make a pattern, all we need to do is take one cell of a lattice, fill it in with some design or other, and then repeat it in all the interlocking cells.
Well, that's a pattern, for sure, but the symmetries you end up with in your pattern depend on what you fill the cell with. If, for example, you have in each cell of your lattice a non-symmetrical picture of Harry Potter, then this pattern has no rotational or reflectional symmetries whatsoever. If you rotate it by any angle, then Harry's not the right way up; if you reflect it in any direction, then he's holding a magic wand in his left hand rather than his right. The only pattern we see in this design is that his picture is repeated at regular intervals. This is one of the seventeen pattern groups - the simplest, in fact - one which goes by the natty title of p1 (well, you think of a better name!)
[Embed picture right: fleur-de-lys wallpaper. External link]
To get the more interesting patterns, we need to fill each cell of our lattice with something which has some symmetrical properties which are aligned with the symmetrical properties of that lattice. If we have a rhombic lattice, for example, and we replace Harry Potter with, say, a fleur-de-lys motif with a line of vertical symmetry aligned along one of the diagonals of the rhombic cell, then this produces a wallpaper pattern with lines of vertical reflective symmetry - it's a pattern group known as cm. To create this pattern, all we need to do is draw half a fleur-de-lys and reflect it into the other half of the cell. Then we repeat the cell across the whole lattice, as before, to produce the entire wallpaper pattern.
This half-cell containing the half fleur-de-lys (shown in green) is known as the generating region of the cell (some people call it the fundamental domain) for that pattern. Each of our seventeen patterns has a generating region of some shape or other, which is reflected, rotated and/or glide-reflected in a unique way to fill the cell of the lattice. The cell is then repeated by translating it to every other cell across the whole lattice to create the recognisable wallpaper pattern.
A Spotter's Guide
The other entries in this series describe the seventeen wallpaper patterns in detail. For reasons of length, the patterns have been grouped into entries according to the rotational symmetries they have – specifically, the highest order of rotational symmetry that exists in each pattern. This is convenient, as it happens to be the first thing you need to look for in order to classify a wallpaper pattern.
[Embed picture right: wallpaper points of rotation. External link]
As an example, take a look at the pattern shown on the right. The distinct points of rotation have been labelled with numbers showing their order, that is, the number of times the pattern would match with itself if you rotated it through 360 degrees about that point.
We can see that this pattern has centres of rotation of orders 2, 3 and 6, so we will classify this as an order-6 pattern (the highest). Two of our seventeen wallpaper pattern groups have order-6 rotations, whereas others have highest orders of 4, 3 or 2. Some patterns have no rotational symmetry at all, and we will call these order-1 patterns.
When you have determined the highest order of rotation in your pattern, you can look it up in one of the following Entries, as appropriate:
Wallpaper Patterns of Order 1
Wallpaper Patterns of Order 2
Wallpaper Patterns of Order 3
Wallpaper Patterns of Order 4
Wallpaper Patterns of Order 6
Each of these Entries will guide you through the steps to identify the exact wallpaper pattern group. For convenience, we have summarised the classification procedure below.
|No Rotational Symmetry (Order 1)|
|If no reflections, then:|
|If glide reflections, then:||pg|
|If reflections, then:|
|If rectangular reflections, then:||pm|
|If rhombic reflections, then:||cm|
Rotational Symmetry of Order 2
|If no reflections, then:|
|If glide reflections, then:||pgg|
|If reflections in one direction only, then:||pmg|
|If reflections in two directions, then:|
|If rectangular reflections, then:||pmm|
|If rhombic reflections, then:||cmm|
Rotational Symmetry of Order 3
|If no reflections, then:||p3|
|If all the 3-centres are on axes of reflection, then:||p3m1|
Rotational Symmetry of Order 4
|If no reflections, then:||p4|
|If the reflections are in four directions, then:||p4m|
Rotational Symmetry of Order 6
|If reflections, then:||p6m|
Those names we have used for the pattern groups – p1, cm, pgg, etc – are examples of what mathematicians call the crystallographic notation. It's not necessary to know what the symbols mean, but here's a brief explanation for the insatiably curious.
Fifteen of the groups are orientated by choosing a lattice cell with the highest orders of rotation at the vertices, and these are denoted 'p' (for primitive cell). The other two start with 'c' (for centred cell).
This is followed by a number indicating the highest order of rotation. Confusingly, mathematicians often drop this number, making p2mm into pmm, for example.
The third symbol indicates 'm' (for mirror) if there's a perpendicular axis of reflection, 'g' if there's a glide-reflection axis or '1' if there's no symmetry axis (as in p31m).
The final symbol describes the type of symmetry axis at a certain angle: 180° for orders 1 and 2, 45 ° for order 4, or 60° for order 3 or 6. Again, this symbol can be either 'm', 'g', or '1'. If there are no reflections or glide reflections, then mathematicians drop the third and fourth symbols (e.g. for pattern p6).
We can thank the International Union of Crystallography for the above, but it's certainly not the only nomenclature available. JH Conway used the branch of mathematics known as topology to come up with his orbifold notation for the seventeen groups. We won't go into detail here, but the patterns are listed as o, 2222, **, xx, *2222, 22*, 22x, x*, 2*22, 442, *442, 4*2, 333, *333, 3*3, 632 and *632.
Armed with all this knowledge, you probably want to get cracking and categorise all the patterns around your home. You may even be able to collect all seventeen!
We're not saying that this is the sort of thing that all mathematicians get up to in their spare time, but one place in which they have regularly been seen scouring for pattern groups is the Alhambra Palace in Grenada, Spain. Lavishly decorated throughout with intricate plasterwork and tilings, this site has been a Mecca2 for pattern-spotters since the middle of the 20th Century. Many have claimed to identify all seventeen pattern groups, although disputing each other's findings has been an equally absorbing pastime. The current thinking is that all seventeen types do indeed exist at the Alhambra, although there is still some niggling doubt over pattern p3m1.
Finally, if you don't have the resources to visit Moorish palaces in your search for these patterns, then why not create your own3? There are plenty of pattern generators available online, including an excellent free one on Richard Morris's site.
If you enjoyed this, you may be tempted to investigate further aspects of the mathematics. To discuss these in any detail would be well beyond the scope of this Entry, but there's one topic that amateur mathematicians might find interesting.
Earlier, we referred to these wallpaper patterns as groups, and indeed they follow all the rules of mathematical group theory. A wallpaper group is comprised of symmetries, each of which is a transformation which maps the pattern to itself, whether by rotation, reflection or glide reflection. The seventeen patterns form distinct groups, and it is through group theory that we are able to prove that there are exactly seventeen of them.