Approximate icosahedral periodic tilings with pseudo-icosahedral symmetry in reciprocal space

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Abstract
We characterize, by classical results of geometry of numbers, 3D periodic tilings of any Bravais lattice built with the prolate and oblate Ammann rhombohedra (the prototiles of 3D Penrose tilings) such that the Fourier transform of all the vertices of the tiling almost presents the icosahedral symmetry. An analytical expression of the displacements of the intense peaks from the exact icosahedral symmetry is given. We prove that exact icosahedral positions are located on the projection of a lattice of R6, stable under the action of the icosahedral group