In a normal crystalline solid the positions of atoms are arranged in a periodic lattice of points, which repeats itself in 3 dimensions the same way that a honeycomb structure repeats itself: each cell has an identical pattern of cells surrounding it. In a quasicrystal, the pattern of atoms is only quasiperiodic. The local arrangements of atoms are fixed, and in a regular pattern, but are not periodic throughout the entire material: each cell has a different configuration of cells surrounding it.
Quasicrystals are remarkable in that some of them display five-fold symmetry. In an ordinary crystal, only 1-, 2-, 3-, 4-, and 6-fold symmetries are possible. This is a geometrical consequence of filling space with congruent solids -- these are the only symmetries that can fill space. Prior to the discovery of quasicrystals, it was thought that five-fold crystal symmetry could never occur, because there are no space-filling periodic tilings, or space groups, which have five-fold symmetry.
There is a strong analogy between the quasicrystal and the Penrose tiling of Roger Penrose. In fact, some quasicrystals can be sliced such that the atoms on the surface follow the exact pattern of the Penrose tiling.Patterns in Quasicrystals