How can the shape of a quasicrystal grain show its internal symmetry?

The shape of a quasicrystal grain often reflects its internal atomic symmetry, serving as a macroscopic indicator of the underlying structure. By using a scanning electron microscope, researchers can observe the geometric forms of these grains, which frequently exhibit symmetries that are otherwise forbidden in conventional crystals.1

For example, grains of aluminum-copper-iron have been observed in the shape of a pentagonal dodecahedron. This shape features 12 regular pentagonal faces, with axes of fivefold rotational symmetry passing through them. In this configuration, rotating the grain by 72° leaves its appearance unchanged, meaning the grain repeats its orientation five times during a full 360° rotation.1

This external geometry is not merely superficial; it corresponds to the internal arrangement of atoms. High-resolution imaging confirms that the fivefold symmetry suggested by the grain's shape is mirrored in the actual placement of atoms, which are arranged in parallel rows rotated by 72° relative to one another. This relationship demonstrates how the macroscopic morphology of a quasicrystal grain provides direct evidence of its unique, non-periodic internal order.1