The First Table of Atomic Site Coordinates of the Ih Quasicrystal

Abstract

1. Introduction The mathematical derivation and classification of the 230 space groups were completed and perfected independently by Fedorov, Schoenflies, and Barlow before the twentieth century [1]. It was only later, in 1912, that Laue’s discovery of X-ray diffraction initiated the empirical era of determining atomic arrangements in crystals [2]. This historical sequence demonstrates that the determination of these 230 space groups required absolutely no crystal diffraction data. Naturally, I also re-derived the 230 space groups; I carried out this derivation from the perspective of atomic coordinates [5]. I have found that, in order to fill the entire space, an Oh-point group crystals can—while keeping their unit cell unchanged—achieve this by continuously increasing the number of translational directions. How do the Ih quasicrystals fill the entire space, given that their translational directions are fixed to 20, and what kind of "unit cell" do they employ? Undoubtedly, if the translation directions of an Oh crystal are fixed, its unit cells will be unable to fill the entire space. In the process of filling the entire space, Oh crystals exhibit a tendency that could be described as "insatiable"—an endless demand for translational directions. For example, the first step in filling the entire space requires 26 directions; the second step requires 98 directions; and the third step requires 218 directions. Interestingly, the first step corresponds to the number of pieces (cubies) in a 3x3x3 Rubik’s cube; the second step corresponds to the number of pieces in a 5x5x5 Rubik’s cube; and the third step corresponds to the number of pieces in a 7x7x7 Rubik’s cube [3]. Therefore, an Ih quasicrystal fixes its translational directions to 20; it requires no unit cell, needing instead only a highly adaptable "shell." This is the background—namely, the 230 space groups—from which my "n-shell lattice model" is derived. An Ih quasicrystal requires shells that satisfy the condition u+v+w=n, where n is the total number of shells, and u, v, and w describe the size of a given shell. An Ih quasicrystal fills the entire space using a variable shell (u+v+w=n) and 20 invariant translational directions. An Oh crystal fills the entire space by using a fixed unit cell and an ever-increasing number of translational directions. This is the fundamental difference between the lattice model of an Ih quasicrystal and that of an Oh crystal. Similarly, the atomic site coordinates for the icosahedral Ih quasicrystal reported here require no experimental quasicrystal diffraction data; the model is derived entirely from the intrinsic geometric properties of the Ih point group symmetry. Consequently, I present this work as the first complete table of foundational atomic site coordinates for the Ih quasicrystal (QC). Given the tabular format and the vast amount of data, this article is divided into three parts: The First Table of Atomic Site Coordinates of the Ih QC (Article text) is displayed in the first attachment window below; The First List of Atomic Site Coordinates of the Ih QC (n=1 to 11) is shown in the second attachment window; The First List of Atomic Site Coordinates of the Ih QC Cell (n=12 to 20) is shown in the third attachment window. 2. The significance of the symmetry embodied in the first table The symmetry of the symbols and the symmetry of the atomic sites constitute the first characteristic of the first table. The combinatorial enumeration (CE) of atomic site on the shell of the QC cell is the second characteristic of the first table. The third characteristic of the first table is that the labels for the atomic site coordinates within the QC cell are unique, regardless of the size of the quasicrystal. The fourth characteristic of the first table is that the symmetry symbol of the atomic site is related to both the symmetry of the shell and the overall Ih symmetry of the quasicrystal. The fifth characteristic of the first table is the number of shells, n; this parameter is associated with both the combinatorial enumeration of the atomic sites and the size of the QC cell, serves as the sole parameter for the n-shell lattice model, and characterizes the size dependence of the QC cell. The sixth characteristic in the first table specifies the symmetry orbitals and orbital degeneracies of the atomic lattice sites; the product of the two is exactly equal to the order of the Ih point group. Finally, the atomic lattice site coordinates of the QC cell serve as generators for determining all quasicrystal coordinates; applying the symmetry operations of the Ih point group to these coordinates yields the complete set of quasicrystal coordinates. All calculations and operations are performed in a Cartesian coordinate system. Please read the main text in the first attachment window. Please read the list of atomic site coordinate data in the second attachment window. Appendix 1. Full text of the main body: The First Table of Atomic Site Coordinates of the Ih QC Appendix 2. Shell atomic sites, site coordinates, and coordinates generated via point group operations: The First List of Atomic Site Coordinates of the Ih QC (n=1 to 11) Appendix 3. Shell atomic sites and site coordinates: The First List of Atomic Site Coordinates of the Ih QC Cell (n=12 to 20) PPT presentations on the n-shell lattice model of the QC (Ih): An n-shell Lattice Model for 3D Quasicrystals:1-shell QC An n-shell Lattice Model for 3D Quasicrystals:2-shell QC