On the self-similarities of two icosahedral patterns

Abstract

Abstract The vertex set Q of the usual three-dimensional Penrose tiling whose strip projection construction starts from an icosahedron is invariant under an infinity of affine similarities r ↦ τ 3(r − q) + q having as centers certain points q ∈ Q and the scaling factor τ 3, where τ = (1 + [unk])/2. A similar construction done by starting from a dodecahedron leads us to an icosahedral pattern Q′ invariant under the similarities r ↦ (5 + 2 [unk]) r and r ↦ (4 + 2 [unk]) r. These new self-similarities whose determination is based on the decomposition into irreducible components of a representation of the icosahedral group can not be obtained by using the method recently reported by Masáková, Patera, Pelantová (J. Phys. A: Math. Gen. 31 (1998) 1443).