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Crystallographic restriction

Fig.: Crystallographic restriction: six-fold symmetry conforms.
Fig.: Crystallographic restriction: five-fold symmetry doesn't conform.

crystallographic restriction

translational periodicity

aperiodic lattice

six-fold symmetry (60o) possible without overlaps
five-fold symmetry (72o) would lead to atoms encroaching on each other

Five-fold symmetry in nature

Fig.: Buckminsterfullerene molecule.
Image © Michael Ströck via Wikimedia.

Five-fold symmetry is possible in finite and non-periodic structures such as Buckminsterfullerene, C60

Aperiodic tilings

Fig.: A quasicrystalline Penrose tiling.
Image from Inductiveload via Wikimedia.

tilings

self-similarity

inflation - deflation

Tilings made from two different types of tile can be regular but aperiodic.

Fig.: A periodic tiling of Penrose tiles.
Image from Geometry guy via Wikimedia.

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

Quasicrystals

Fig.: Single-grain quasicrystal.
Image from Ames Laboratory via Wikimedia.

Crystal - quasicrystal single crystal - single-grain quasicrystal

Fig.: .
Image from A Szallas, A Jagannathan, S Wessel; Phys Rev B 79 (2009) 172406.

phonons - phasons

Fig.: Dodecahedron.
Image © DTR via Wikimedia.
Fig.: Icosahedron.
Image © DTR via Wikimedia.

Fig.: Electron diffraction pattern of a ZnMgHo quasicrystal.
Image © Materialscientist via Wikimedia under this CC licence.

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

Fig.: Defects in tilings.
Image © KJ Strandburg, PR Dressel, Argonne National Laboratory; Phys Rev B 41 (1990) 2469.

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.

Fig.: Dodecahedron.
Image © V Raghavan, J Phase Equil Diff 31 (2010) 449.

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).