Penrose tiling observed in a quasi-crystal

Abstract

Three-dimensional icosahedral symmetry has been reported¹ for an alloy consisting of aluminium with 14 atomic% manganese. The electron diffraction patterns appear consistent with a three-dimensional Penrose tiling² or quasi-lattice^2,3, although one report has suggested a 20-fold multiple twinning of a close-packed structure⁴. Clearly, however, the determination of the reciprocal lattice is only a first step; it is a further problem to determine the atomic motif(s) required to complete the structure. High-resolution electron microscope images of this alloy, reported here, reveal the localized pentagonal pseudo-symmetry of the structure and show that it projects as a two-dimensional Penrose tiling^2,5. A structural model is derived, based directly on the experimental images and an analysis of the polyhedra occurring in the crystallographic phase Al₆Mn.