Symmetry, groups, and wallpaper

| Marcus | Uncategorised

There is new wallpaper going up in our house.  Since my DIY skills are marginally better than my cat’s, we’ve employed a decorator to do it. (I fixed a dripping tap once – that was the high point of my DIY activity.) 

Looking at the new wallpaper has reminded me that I studied it in my first year at university, as part of my ‘Crystalline Materials’ course. What has wallpaper got to do with crystalline materials? It’s a good example of lattices and symmetry.

Wallpaper, unless it is completely bland, contains repeating patterns. Have a look at some and see if you can identify exactly what the nature of the pattern is. You can describe it by a ‘lattice’ and a ‘motif’. The lattice is a set of equivalent points on the pattern. Imagine doing the following (or do it for real if you are about to strip the wallpaper). Look at your wallpaper, pick a point on it, put a nice big dot with a marker pen on that point. Then go and find all the other points on the wallpaper that are at the exact same point in the pattern as this one, and put a dot there too. Sometimes you have to think carefully about whether two points really are equivalent. The key is that if you picked up the whole sheet of wallpaper and moved it so that one dot landed on another dot (no rotating allowed) the final result on your wall would be unchanged.  What you are left with is a lattice of dots on your wallpaper.

On wallpaper, which is two dimensional, there are only five distinct symmetries the lattice can have. An obvious one is ‘square’ symmetry – your dots will form a square lattice. But there are others – hexagonal, rectangular, centred-rectangular and oblique.

In three dimensions, which is the domain of crystals, there are fourteen distince lattices.

Connected with the lattice, is the motif. This describes the pattern itself. If we take a motif, and apply it at every lattice point, we reconstruct the whole wallpaper.

 Then, there are the symmetries. These are obviously related to the lattice,  but are not the same thing. Think about the ways that you can reflect, rotate and translate the wallpaper so that the pattern remains unchanged. (A translation from one lattice point to another by definition does this). There are seventeen different symmetries a wallpaper can have. Which is yours? In three dimensions, there are 32 different crystal symmetries, which are very important in crystallography. These are called the crystallographic point groups.

After studying crystallography, even for a short while, you’ll never look at wallpaper the same way again.

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