Icosahedral crystals from cuts in six-dimensional space

Abstract

In condensed matter physics it is traditionally assumed that crystalline ordering must necessarily be associated with a regular periodic lattice. A crystal structure is formed by placing a basis of atoms at each lattice point. The symmetry of the crystal is described by the space group which is the mathematical group containing all the symmetry operations which leave the crystal invariant. In addition to the discrete translations which reflect the periodicity of the crystal, the space group includes rotation and reflection symmetries forming the so-called point group. One of the consequences of the existence of an infinite periodic lattice is that the point group cannot contain five-fold symmetry operations. When the stability of crystal ordering is investigated by some physical model, the analysis is always restricted to comparing energies of two or more regular crystal structure belonging to the 230 3D space groups. Yet it has never been proven that the only stable way of having long-range positional ordering of atoms is on a periodic lattice. Indeed, in condensed matter physics it is well known that there exist incommensurate crystals, which can be formed by superposition of structures with incommensurable periods. However, for these crystals one can always define amore » basic lattice belonging to one of the usual space groups, and the positions of atoms are modulated around these lattice positions. If the crystal ordering does not take place on a lattice, there is no reason to rule out ab initio the possibility of having a five-fold axis. In fact, from a purely mathematical point of view, it has been known for some time that it is possible to arrange a small number of different ''titles'' in an infinite ordered arrangement which has five-fold symmetry.« less