## Symmetry

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Together with periodicity, symmetry is the principal defining trait of crystals, like this snowflake – that is, this ice crystal. It consists of the invariability of crystals (of their morphology or structure) regarding certain simple operations, such as rotation: if we turn this ice crystal 60 degrees without you knowing, you would not be able to discern the difference. This would be the same if we turned it 120º, 180º, 240º, 300º or, evidently, 360º (a complete spin thus returning the crystal to its original position). We say that the crystal has“rotational symmetry of order 6”or“hexagonal symmetry”.

As well as rotational symmetry, there is reflection symmetry (reflection as in mirrors), inversion centres, translational symmetry and combinations of these such as helical axes or slip planes.

Crystallographers love symmetry, sometimes to the point of obsession. We are symmetry freaks. You might think that what attracts us is the beauty of crystals and crystalline structures, whose harmony is without doubt linked to its symmetry. Indeed, we like this just as the designers of the Alhambra did, and as Moroccan artisans still do.

But above all, we like symmetry because it simplifies the study of crystalline structures enormously. For example, periodicity itself derives from an operation of symmetry: translation. If we translate the cell unit of a crystal a periodic distance, the structure does not change: it is invariant. This allows us to represent an enormous crystalline structure by the few atoms of its cell unit, forgetting about the almost infinite number of other atoms in the structure.

Translational symmetry enables us to know the immense crystalline structure by studying only the atomic configuration in the unit cell.

The other symmetries of rotation, reflection and inversion reduce the representation even more because part of the atoms that are in the unit cell are symmetrical copies of other atoms in the same cell.

Symmetry makes crystallographers’ lives easier, making it possible to classify and understand crystalline structures. How many types of crystal exist? That is, in how many different forms can matter organise itself? Although they may seem boundless, actually there are very few options for filling a space in an ordered way periodically repeating one same piece.

For example, only five types of plane lattices exist:

In other words, if we want to fill a surface, we can do so with rectangles, triangles, squares or hexagons, but not with pentagons. And it also the case that in crystalline structures, rotational symmetry can only be of order 1, 2, 3, 4 or 6.  Thanks to the study of the combinations of symmetry, we know that there are only 17 different shapes that can tile a surface, shapes that were already known by Arab geometricians.

 Arab artisans of the 18th century knew the 17 plane groups of symmetry. And it has also been shown that there are only 230 different forms of periodically packing a volume with identical units. Not one more, not one less.

## Quasicrystals

Molecules can have rotational symmetry 5 – in other words, if they spin 72º they are invariant. But it is impossible to fill a surface with them, or make a network; that is, it is impossible to build a crystal with them. That is why you won’t find pentagonal floor tiles for sale; or if you do, they are combined with the rhombi needed to fill the unavoidable gaps between the pentagons. Molecules can have pentagonal symmetry; but periodic structures cannot.

However, the existence of crystals with order 5 symmetry has recently been discovered. The explanation lies in the fact that the surfaces and volumes can be completely filled following regular, but not necessarily periodically perfect, patterns. For example, this can take place with dilational symmetry, following patterns like the sequence of the Golden Ratio. In other words, materials that are organised quasi-periodically. They are the so-called quasi-crystals.