# Apparent order but no regularity... Quasicrystals in the eye of the mathematician

The quasicrystalline structure is ordered, but there is no typical regularity. Mathematician Prof. Jacek Miękisz investigates why such an unusual arrangement of atoms can sometimes be optimal.

It is no coincidence that crystals of ice, kitchen salt, sugar, but also diamonds, emeralds and quartz have beautiful, regular shapes. It is connected with their structure: atoms are arranged in a very ordered manner and create patterns that repeat in space. Most of these structures can be inscribed of polyhedrons, which can be used to fill space like bricks by stacking them one on top of another. For example, in table salt, chlorine and sodium atoms are very elegantly arranged in the corners and on the walls of a cube. This means that there is a translational symmetry that makes the system repeat itself in space if it is properly shifted. It seemed that all crystals would have such beautiful and understandable symmetry - until Dan Shechtman discovered the quasicrystals.

**WHERE NO TRADITIONAL SYMMETRY IS FOUND**

Professor Jacek Miękisz from the University of Warsaw, who works on mathematical models of quasicrystals, talks about Shechtman`s research. In the 1980s, this Israeli scientist cooled down the aluminium and manganese alloys and saw a strange structure in the diffraction pattern. It was orderly but non-periodic: the system had a rotational 10-point symmetry. And with such a symmetry, it is mathematically impossible to obtain a structure that periodically fills three-dimensional or two-dimensional space.

**NOT A QUASI-SCIENTIST**

The observation of this previously unknown symmetry in nature outraged physicists of that time. This way of organizing matter was completely out of place in the categories that were used to describe crystals. Shechtman met with ostracism (Nobel laureate Linus Pauling remarked that "there is no such thing as quasicrystals, only quasi-scientists"). Fortunately, the scientist did not give up, his discovery was published and recognized a few years later, and in 2011 the researcher was awarded the Nobel Prize in Chemistry for the discovery of quasicrystals. In addition, it was necessary to change the definition of the crystal to include Shechtman`s discovery.

*Quasicrystal Al71Ni24Fe5 found in a meteorite. Fig. Paul J. Steinhardt et al. - http://www.nature.com/srep/2015/150313/srep09111/full/srep09111.html, Wikipedia*

**DANCING ON A NON-PERIODIC DANCEFLOOR**

The structure, which outraged physicists so much, was already known to mathematicians at the time. "If we were to overlay diffraction images of atoms in Shetzmann`s quasicrystal on vertices in Penrose tiling, the images would overlap" - says Prof. Miękisz.

Penrose tiling is a very interesting arrangement: two types of rhombi (one with angles of 72 and 108 degrees and the other with angles of 36 and 144 degrees) with projections and indents are sufficient to cover a plane, but they can not fill the plane in a periodic manner - unlike periodic tilings, like checkerboard or honeycomb.

*Penrose tiling. Fig. Inductiveload, Wikipedia*

"The discovery of quasicrystals has made an incredible impact on the development of mathematics, which investigates non-periodic structures, in its various branches: topology, algebra, number theory" - says Prof. Miękisz, who himself investigates why quasicrystals exist, what kind of interactions cause the formation of these structures and whether they are resistant to thermal movements.

**SEEKING THE OPTIMUM**

He explains that matter consists of atoms that interact with each other: for example, they detract repel each other when they are too close, and attract each other when they are too far away. When the temperature drops (in the equilibrium state), the atoms are positioned so as to minimize energy consumption. "Matter is lazy: it wants to use as little energy as possible. How to arrange matter, so that the energy of the system is minimal - that is a mathematical task" - he notes. It seemed that the tendency to minimize energy is maintained in classic crystals, those with periodic structures. The question, however, is whether the same happens in quasicrystals.

"The theory is that atoms in quasicrystals are arranged in a certain way because they also minimize energy, but there is no proof of that. And I want to contribute" - the mathematician says. He adds that in order to create a mathematical model of a quasicrystal, one must know the forces that connect atoms and make them arrange themselves in a non-periodic manner. His team wants to create a model that will explain it.

"We are trying to see what interactions between elements would result in a non-periodic system in the ground state, and whether this system is resistant to disruptions. This problem still has no solution in mathematics" - the scientist says.

*Diffraction pattern of the icosaedral quasicrystal Ho-Mg-Zn, fig. Materialscientist, Wikipedia*

The problem of quasicrystals is somewhat connected with one of the famous mathematical problems formulated in 1900 by David Hilbert, namely the second part of the 18th problem. It comes down to the question of whether there is a solid with a shape that we can cover all three-dimensional space with in a non-periodic way.

**COMPREHENDING INFINITY**

The scientist says that in his research he mainly uses a piece of paper, a pencil, an eraser and a trash can. He does need computers for his calculations. "This is research in the field of probability calculations, dynamic systems, the theory of measurement. Computers are inherently limited devices, and we study infinity. Computers can help verify, if we describe infinity with a finite number of rules, but they will not verify an infinite number of rules . While the human brain is also finite, we are already able to work around infinity with it" - the scientist concludes.

The research project is funded with a Harmonia grant from the National Science Center.

PAP - Science in Poland, Ludwika Tomala

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