Title : From the two-dimensional geometry of Penrose tiling to the physical problems of three-dimensional quasicrystals
Abstract:
In 1974, Sir Roger. Penrose discovered a five-fold rotationally symmetric tile tiling that can be seamlessly laid out on a two-dimensional plane using two rhombus tiles. Penrose tiling is basically a two-dimensional space distribution geometry problem, where the space is evenly filled using these two rhombuses.
In 1982 Professor Dan Shechtman first observed the electron transmission diffraction pattern in ten directions, which is impossible in the crystal field. This triggered a lot of discussion about whether the so-called pentagonal three-dimensional.
There is a unique connection between quasicrystals and Penrose tilings, both of which have non-periodic five-fold rotational symmetry. The connection between the two can be understood from high-resolution transmission electron microscopy (HRTEM) images. Therefore, for 40 years, many scholars believe that there is a deep connection between the non-periodicity of quasicrystals and Penrose tilings. As a result, there are many studies in this area, and previous studies have mainly focused on the study of the five-fold characteristics of local atomic clusters.
Our research has enabled us to make Penrose tiling images very large, large enough to compare with HRTEM atomic layering images of comparable size. By comparing a large amount of data (Penrose tiling and HRTEM atomic arrangement), we found that the six-sided flattened icons of the dodecahedron (pentagonal cube) in the two-dimensional images are exactly the same in topology when connecting certain specific points (important points). All vertices of a pentagon that is enlarged or reduced by the golden ratio can also coincide at a certain point. From the matching of all pentagonal vertices. The direct correlation of the two problems is logically proved. Through the above two pictures, we made a 3D Penrose tiling dodecahedron pentagonal cube (dodecahedron) and a dodecahedron quasicrystal 3-D model.
But this is just a superimposed 3D, or it can only be said to be a semi-3-D, which at least gives us an opportunity to enter the study of real 3D entities. In the course of the demonstration, I will give another 3D frame pattern, but not the Penrose style.
Nevertheless, the important difference between the two is not only a geometric problem of uniform partitioning, but also a physical problem. The latter requires a reasonable arrangement of chemical bonds to explain the so-called five-fold rotational structure. As it happens, the chemical composition of quasicrystals has the characteristics of polaritons, which are electrical bonds that can be adjusted in any direction, namely Van der Waals bonds. It forms a five-fold rotational structure in the alloy, which we will explain in more depth using van der Waals chemical bonding.
