*crystallographic restriction*
*translational periodicity*
*aperiodic lattice*
six-fold symmetry (60^{o}) possible without overlaps
five-fold symmetry (72^{o}) would lead to atoms encroaching on each other

Five-fold symmetry is possible in finite and non-periodic structures such as
*Buckminsterfullerene*, C_{60}

*tilings*
*self-similarity*
*inflation - deflation*
Tilings made from two different types of tile can be regular but aperiodic.

But not all such tilings are aperiodic: The arrangement shown here has ordinary crystalline translational periodicity.

Crystal - quasicrystal
single crystal - *single-grain quasicrystal*

phonons - *phasons*

*electron diffraction*
10-fold (or 5-fold or 8-fold) symmetry
periodic in the other two directions
Dan Shechtman, Nobel prize for chemistry, 2011
*icosahedral quasi-crystals*
aperiodic in all three dimensions

*defects*
exist in quasicrystals as they do in crystals. The diagram above is a simulation
showing the disorder of a two-dimensional quasi-crystal (tiling) increasing as the temperature rises.

Many quasicrystalline materials are ternary alloys, often involving aluminium. The phase diagram shows the small area of thermodynamic stability of the quasi-crystalline phase in the system Al-Cu-Fe (dark red). Adjacent to it are two-phase regions where the quasi-crystal phase is in equilibrium with various crystalline compounds (light red).

Materials Physics lecture: crystallographic phase transitions