Natural harmony by an irrational metric in hierarchic icosahedra

How does diffraction occur in quasicrystals where the order is irrational and in geometric series? The structure is known to be aperiodic with multiple interplanar spacings. These features, individually and collectively, falsify Bragg diffraction in these solids. An extra variable is needed to account for the irrational and geometric diffraction. This is examined numerically by a modified structure factor method, and analyzed with Fibonacci series in threefold dimensionality. The natural, semi-integral part of any irrational index is separated from its irrational residue which, in turn, forms a special metric function that is universal in the diffraction. The metric harmonizes the irrational index in coherent diffraction from the hierarchic icosahedra. It does this by scaling the harmonic and digital diffraction to the irrational and geometric indexation. [Journal of Modern Physics, (2020) 11, 581-592. doi: 10.4236/jmp.2020.114038 ]