The direct evidence of relation between Penrose tiling and Quasi crystal

We present for the first time a comprehensive, logically consistent, and easily understandable explanation elucidating the connection between two of the most perplexing discoveries over the past 45 years: Penrose tiling in two-dimensional geometry and three-dimensional metallic quasicrystals. We start with two key concepts: the modified Penrose tiling is conceived as a uniform partition framework for atomic arrangements; after establishing a geometric framework that can extend infinitely, the next crucial question is to verify whether this framework truly corresponds to the atomic structure in quasicrystals. This study proposes the so-called 'three-step evidence‘: the first level is global pattern matching, where similar topological structures and symmetry patterns can be observed on a large scale. The second level is vertex matching, observing whether the positions of dodecagonal pentagonal projections (PPSD) in the structure correspond to the special points in the material (especial Pd atom point). The third level is atom-to-atom matching, involving point-by-point correspondence between each Penrose tiling geometric vertex and the actual atomic positions in HAADF-STEM imaging over a very large atomic span. An evidence chain can be gradually established to link the geometric model with the atomic structure.?

Therefore, we propose two independent two-dimensional (semi-three-dimensional) surface models to describe quasicrystals. However, we must clarify the consistency between these two two-dimensional surface models and introduce a universal three-dimensional quasicrystal model applicable to all fivefold symmetric quasicrystals. We observed four potential nucleation centers with fivefold symmetric atomic configurations from PES STEM images. These four clusters are considered parameters of the four most important positions of an icosahedral solid, namely vertices, face centers, edge centers, and body center. We also need to further explain how dodecahedron crystals grow into larger crystals.

Dr. Chung Yuan Kung is Professor Emeritus in the Department of Electrical Engineering at National Chung Hsing University, Taichung, Taiwan. He earned his BS (1969) and MS (1972) in Physics from National Tsing Hua University, an MS in Physics from the University of Alabama in Huntsville (1974), and a PhD in Materials Science from Northwestern University (1979). After postdoctoral research at Georgia Tech and Lawrence Livermore National Laboratory and six years in the U.S. semiconductor industry with Fairchild and National Semiconductor, Dr. Kung joined Taiwan’s ITRI in 1987 and served on NCHU’s faculty from 1991 until his retirement in 2016. His 2024 publications represent a culmination of decades spent unraveling the deep connections between geometry and material.