Building symmetries

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low-IMG_0381 This interactive installation offers three tables with methacrylate pieces for building two-dimensional patterns that enable experimentation with key concepts of crystallography.

Let’s do jigsaw puzzles!

Crystals are formed by millions of tiny pieces of matter, whether atoms, molecules or macromolecules.

These tiny pieces of matter are like the pieces of a large jigsaw, which is the crystal.

But this jigsaw is different. In normal jigsaws each piece is different from the rest and only fits in one particular position. In crystals all the pieces, that is, all the molecules, are equal, which means that doing the crystalline jigsaw is relatively easy. It is like a mosaic whose pieces – which in English have the pretty name tesserae – are all identical.



Above is a crystalline jigsaw puzzle, a mosaic. The tesserae, the molecules of a crystal, are all identical but tend to have complex shapes (shown in colour orange). We crystallographers like to convert these complex forms into simpler geometric shapes (in green) to make the calculations simpler and the demonstrations easier and more elegant. For example, it can be demonstrated that there are only five different types of tesserae on a plane and fourteen in space.


Above the five two-dimensional tesserae are illustrated, which in crystallography are called “unit cells”, the repeated elements with which the five two-dimensional lattices are built – in other words, the only five possible ways of filling a plane. On the table you will find the tesserae to construct the five types of crystalline mosaics.

You will have noticed that the unit cells are triangular, square, rectangular and hexagonal pieces. Why aren’t there pentagons? Why is there no lattice showing the packing together of pentagons? Try covering the plane with pentagons and you will find out why…


Crystals, glass an quasicrystals

low-Fichitas-2In this panel we can check out the difference between a crystal, a quasicrystal and amorphous glass.

Make the crystal with two types of tesserae. They can be the same or different. But they must be put into place periodically – the distance between the pieces of one same type in one same direction must always be the same.


If we stack the same tesserae symmetrically but not periodically, we build a quasicrystal like the one below.


Finally, if we do away with both symmetry and periodicity the result is an amorphous pattern, like this one below.


Motifs and unit cells

Although there are only five two-dimensional lattices, using different types of tesserae or different orientations of the same tesserae, it is possible to build a large variety of crystallographic patterns.

These patterns are crystallographic as long as they respect the rules of symmetry and periodicity that you will discover as you play with the tesserae on the table.

You have probably wondered: if there can be so many patterns, what about those only five lattices, those only five types of unit cells? Well, you can see for yourself that they carry on working. Try to find the repeated unit (the unit cell) in the patterns you make. In the example patterns on this panel the cell unit is shown in grey for you.

If you want more, check out the patterns on the right. As well as the unit cell (in grey), a second unit cell is marked (in pink) that corresponds to the unit cell if we take into account the colours of the pieces. Identifying this periodicity is no easy feat, even for many crystallographers. Congratulations if you get it!


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