Diffraction in quasicrystals is in irrational and geometric series with icosahedral point group symmetry. None of these features are allowed in Bragg diffraction, so a special theory is required. By means of a hierarchic model, the present work displays exact agreement between an
analytic metric, with a
numeric description of diffraction in quasicrystals—one that is founded on quasi-structure-factors that are completely indexed in 3-dimensions. At the quasi-Bragg condition, the steady state wave function of incident radiation is used to show how resonant response, in metrical space and time, enables coherent interaction between the periodic wave packet and hierarchic quasicrystal. The quasi-Bloch wave is invariant about all translations
, where
is the quasi-lattice parameter. This is numerically derived, analyzed, measured, verified and complete. The hierarchic model is mapped in reverse density contrast, and matches the pattern and dimensions of phase-contrast, optimum-defocus images. Four tiers in the hierarchy of icosahedra are confirmed, along with randomization of higher order patterns when the specimen foil is oriented only degrees off the horizontal. This explains why images have been falsely described as having “no translational symmetry”.