Wallpaper groups and Crystalline groups

Last time, we talked about the various symmetries of simple molecules, and how that can tell us about the electrostatic properties of certain substances. However, symmetry in chemistry doesn’t stop at the level of methane or ammonia, where there’s an easy central atom that we can analyse with respect to. Today, we will learn about the symmetric properties of wallpaper, and how that can help you, the chemist, analyse a salt.

Go to an old fashioned, traditional, reasonably well-off house, maybe your Nan’s place, and look at the wall of the study, or maybe the master bedroom. If we’re lucky, there are patterns in the wallpaper that can be identified, repeated throughout the wall. Looking at the nice (or maybe horrible, for I have no idea how well yours and your grandmum’s tastes match) embroidery, you begin to wonder – How many different patterns can we design a piece of wallpaper?

Before we go further, let’s restrict the constraints of the question that you have asked yourself. First, let’s say that our wallpaper can fill an entire plane – that is, go infinitely far in two cardinal axes. Although we’ve likely destroyed the wallpaper industry, this means we don’t have to worry about the size of the wallpaper, and even better, this means we can consider translations of the wallpaper freely.

Let’s look at the word ‘pattern’. What we really want to know about is the degree of symmetry that the wallpaper has. Regular, nice patterns always conserve the relative position of tiles in some translation, which, as you recall from the second blog post, is the definition of isometry, a more specific version of symmetry. So what we want is a group of isometries that describes the wallpaper.

Also, we want our patterns to ‘fill the space’, so to say. In other words, we don’t want to have any regions in our wallpaper that don’t have any tiles. How we (for now; we’ll see why removing the following will be important later) deal with this is constraining the problem so that we have that our wallpaper must be conserved under two linearly-independent (which just means they can’t be collinear) vector translations. In this way, we have an infinite array of tiles that fill the entire plane.

We also realise that we want there to be some ‘minimal’ tile. We want our wallpaper to be constructible for bounded subsets of the plane. So we say that there is some bounded region that generates the wallpaper by allowing a partition into congruent parts.

So, the question is rephrased as follows:

How many finite groups of isometries of the Euclidean plane are there that contains two linearly-independent translations?

Turns out the answer is rather simple: 17.

wallpapergroups.pngWe can begin by classifying our operations. All the isometric transformations from blog 2 still work here. You can rotate or reflect (point inversion can be described as a reflection and a rotation), and the fact that we’re working on the Euclidean plane means we can translate.

Our constraints mean that there are only certain rotations allowed; specifically, they are order 2, 3, 4, or 6.

We’ll prove that the maximal order is 6; try to show by yourself that order 5 is not possible.

Let O be the centre of rotation of the rotation operator r, for which o(r)=n. Since there is a translation of minimal distance, let A be the image of O under a translation t of minimal distance. The orbit of A over <r> gives a regular n-gon. t^-1 r^-1 t r gives a translation maps A to r(A), as t^-1 maps to O, r^-1 maps to O, t maps to A, and r to r(A).

For n>6, the distance between r(A) and A, which is the length of the side of the regular n-gon, is less than the distance between O and A. This contradicts our assumption that t is the minimal translation vector, which proves our proposition.

Note that this also proves that any rotation has finite order; the orbits rotations of infinite order are dense, which means there is some rotation with an angle of less than 60º.

As there are only 4 rotational generators, this gives us a bound on the number of distinct groups of symmetry.

Why do we care about wallpaper groups? Because we can generalise wallpaper groups to three dimensions, as crystallographic groups. In crystallographic (or point) groups, there are three linearly-independent translations, and we consider isometries of the Euclidean 3-space.

The lattices that we studied in chemistry illustrated the Bravais lattices, but real crystals have further structure. For example, NaCl, table salt, has a face-centred cubic unit cell. In the case of NaCl, the crystal group can be generated by the three translations, three reflections along the x-y, x-z, y-z planes, and the two rotations. However, gas hydrates, which are solids where molecules are trapped by cages of hydrogen-bonded water molecules, like that of methane at low temperatures, have more exotic crystal structures, called the Weaire-Phelan structure.

Lastly, let’s revisit the point of translations. What would happen if we remove the constraint that we need to be conserved under translations, and say the whole space is simply divided into bounded subregions? Turns out, there are non-trivial groups of this sort. Going back to 2-d analogues, these are called Penrose tilings.

Many of these retain reflectional and rotational symmetries, as the one above, but do not have translational symmetry. Also note how this is of order 5 rotational symmetry. Generalised to three dimensions, the theory of quasicrystals is an ongoing field of research – maybe one that you can delve into!

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