How are a pentagonal dodecahedron and an icosahedron related?

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quasicrystal Figure 1: Hexagonal lattice of atomic sites.

In the study of quasicrystals, the pentagonal dodecahedron and the icosahedron are related as geometric duals.1

This duality means that the two shapes share a reciprocal relationship in their construction: the dodecahedron places a vertex at the center of each face of an icosahedron, and conversely, the icosahedron places a vertex at the center of each face of a dodecahedron.1

Because of this geometric connection, they both exhibit the same underlying icosahedral symmetry. In the context of quasicrystalline grains—such as those observed in aluminum-copper-iron—this symmetry manifests as fivefold rotational symmetry, a feature that is crystallographically forbidden in standard periodic crystals.12