X-ray diffractions from quasi-crystals exhibit tenfold symmetry which in turn is indication of a similar symmetry of the atoms in the quasi-crystal. This tenfold symmetry had never been observed before the invention of quasi-crystals in 1984. Simple crystals do not exhibit ten-fold symmetry. This means that it is impossible to tile a two-dimensional surface with a decagon without overlaps. Therefore, quasi-crystals do not have the translational symmetry that crystals do. It is widely believed that understanding of quasi-crystals in 3D is an extension of their corresponding 2D case. Roger Penrose discovered two rhombuses with different angles and was able to tile without overlaps a 2D surface. So the difference between a 2D crystal and a 2D quasi-crystal is that in the crystal case a 2D surface can be tiled with one surface unit cell while in a 2D quasi-crystal 2 unit cells are required.
A summary of some of the important questions concerning the quasi-crystals are as follows: (a) construction of a quasi-crystal lattice, (b) development of reliable interatomic interaction models for the quasi-crystals, (c) study of diffusion of adatoms on the surface of quasi-crystals.
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