## Symmetries

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 Crystals are nothing more than the ordered repetition of atoms and molecules. Their internal structure is similar to the arrangement of tiles on a wall or Moorish mosaics. The symmetry that results from this repetition is a fundamental property of crystals and has inspired numerous works of art in painting, sculpture and architecture. This symmetry is the foundation of all the physical properties of crystals. Do you know what symmetry is? Have you ever wondered why pentagonal tiles or paving stones are not made? Do you know what anisotropic means?

The characteristic that defines crystals and differentiates them from glass is order. Crystals are ordered matter. They are built as an ordered stacking of atoms or molecules or of groupings of atoms and molecules. In them there is always a unit, what is called the unit cell, which can be an atom, a molecule or a grouping of atoms or molecules, which is repeated periodically in the space filling a volume.

But any thing ordered in the same way, in other words, repeating the same motif over and again at the same distance and with the same arrangement, is crystalline material. For example, something as familiar as stone-paving on the ground or tiling on the wall. These examples serve as models for crystallographers to explain the properties of crystals and crystalline matter.

For example, periodic order – periodicity – is one of the fundamental properties of crystals. It is not difficult to grasp this concept of periodically ordered matter. We see it in flooring, in wallpapering or in the brick walls of houses. We use it whenever we want to cover a surface. For example, a brick wall:

 In this wall the distance between each block or brick is always the same. This distance is called period and, as we can see, it changes with direction. The property of matter organised in this way is called periodicity. It’s as easy as that! In this classic brick-laying arrangement, the bricks are also placed at an identical distance but the period values change. The structure is different but it is also periodic – that is, it is also crystalline.

These bricks are the equivalent of what we call unit cell crystallography. To create a crystalline structure you only have to imagine changing those bricks into molecular units. We’re going to help you imagine it by representing each repetitive unit of a molecular crystal as a brick cube from a construction set.

The unit cell we have chosen is that of pyrite, a very common iron sulphide, a very pretty mineral that is gold in colour.

If we place blocks or cells next to each other to form a row, we get a periodic distribution in one direction:

If we pack several of these rows in a perpendicular direction, we get a periodic plane in two directions. This is a two-dimensional crystal:

Finally, if we stack blocks or unit cells in the third direction, we get a three-dimensional periodic stacking of identical elements.

This is a crystal. The only difference is that a one-centimetre pyrite crystal contains around 100,000,000,000,000,000,000,000 unit cells instead of the three hundred and forty-three in the illustration.

 Pyrite crystals

Now, imagine that you find yourself inside this crystalline structure. Place yourself in an iron atom and look around. Then move to another iron atom of another cell and you’ll see that your surroundings are the same. This is another property of crystals: homogeneity. You can try this in our giant installation of crystalline structure:

Now try to move in a particular direction. You will have to negotiate the links that join the atoms with difficulty. Stop and change the direction of movement. Now you’ll see that the difficulty of moving has also changed. This is another highly important property of crystals: anisotropy.

Crystallographers have succeeded in uncovering all the different ways of grouping elements periodically. There are 17 ways to do so on a surface and 240 ways in three dimensions. Arab geometricians were especially brilliant in exploring the periodic surfaces. Beautiful examples can be seen in the Alhambra in Granada or in the Alcázar in Seville.

And still today they make and search for new designs in the kingdom of Morocco.

You can try it out yourself in our installation of mosaics.

## Did you know that

• No two snowflakes are alike, as the growth of every ice crystal depends on multiple factors.
• In 1891 Fedorov demonstrated that there are no more than 17 basic structures for the infinite possible decorations of the plane formed by periodic mosaics.
• Symmetry is a concept that lies behind many things: in biology, for example, the H1N1 virus is a symmetrical object and uses symmetry to replicate.
• The tiles of the Alhambra are arranged in obsessive plots and repetitions, governed by a strict series of symmetrical schema, in such a way that if they are rotated they keep the same appearance (quite similar to rotating an equilateral triangle), which perhaps explains something of their mesmerising attraction.

## To find out more

• The Wikipedia page on symmetry is the best source to start to learn about the different types of symmetry and their relation with other scientific disciplines.
• You can also visit the webpage on geometry in nature and learn how to identify geometric forms in nature.
• On the Wikipedia webpage on fractals you will find very interesting information about geometry in the natural world.