How does decagonal quasicrystal symmetry differ from icosahedral symmetry?

Decagonal and icosahedral quasicrystals differ primarily in their structural symmetry and the dimensions in which they exhibit translational periodicity.

• Icosahedral Symmetry: These quasicrystals possess the symmetry of an icosahedron, which includes axes of fivefold, threefold, and twofold rotational symmetry. This structure lacks translational periodicity in all three dimensions, meaning the atomic arrangement does not repeat at regular intervals in space.312
• Decagonal Symmetry: These quasicrystals exhibit tenfold rotational symmetry within two-dimensional atomic layers. Unlike icosahedral quasicrystals, decagonal structures maintain ordinary translational periodicity perpendicular to these layers.3

While both types of quasicrystals are characterized by long-range order—a property that distinguishes them from amorphous solids—they represent different ways in which atomic arrangements can bypass the constraints of traditional crystallography, which forbids fivefold, eightfold, tenfold, and twelvefold rotational axes in periodic crystals.32