A perfect icosahedral atomic structure: A two-unit-cell and four-zonohedra description

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Abstract
We establish in this Letter that the atomic structure of a perfect icosahedral quasicrystal can be achieved in a three-dimensional Penrose tiling of two rhombohedral unit cells, where the cluster atomic decoration defines uniquely two types of faces and their matching rule. The τ3 self-similarity ratio defines the hierarchic generation of the tiling, the two types of faces and their matching rule being restored at each step. The structure may be simply understood in terms of four zonohedra: the oblate rhombohedron, a 20-branched stellate zonohedron, the rhombic triacontahedron and the rhombic icosahedron.