Which rotational symmetries appear in the electron diffraction pattern of aluminum-manganese quasicrystals?

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quasicrystal Figure 1: Hexagonal lattice of atomic sites.

The electron diffraction pattern of aluminum-manganese quasicrystals exhibits rotational symmetries that are forbidden in ordinary periodic crystals. Specifically, the diffraction pattern of these quasicrystalline alloys displays:1

Fivefold symmetry: Indicated by rings of 10 bright spots.1
Threefold symmetry: Indicated by rings of six bright spots.1
Twofold symmetry: These axes are also present and self-evident in the diffraction pattern.1

These observed symmetries, along with the angles between the axes, correspond to the geometry of an icosahedron. The presence of these sharp, discrete diffraction spots confirms that, despite lacking translational periodicity, the atomic arrangement in these materials possesses long-range order.1