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Mineral Crystals
Published in Dexter Perkins, Kevin R. Henke, Adam C. Simon, Lance D. Yarbrough, Earth Materials, 2019
Dexter Perkins, Kevin R. Henke, Adam C. Simon, Lance D. Yarbrough
In ionic crystals, cations occupy holes between anions. A close look at Figure 4.7 reveals that there are two kinds of holes. As seen in Figure 4.8 (left side), one involves three anions forming an equilateral triangle with a fourth anion sitting directly above the center of the bottom three The other involves three anions forming an equilateral triangle, with three anions in the next layer forming a triangle pointing in the opposite direction (Fig. 4.8, right side). If we draw lines connecting the centers of the anions, we get a tetrahedron in the first case and an octahedron in the second. So, cations occupying the spaces in the centers of the two arrangements are said to be in tetrahedral coordination (also called 4-fold coordination because bonds go to four anions) or in octahedral coordination (also called 6-fold coordination because bonds go to six anions).
Optical and magneto-optical properties of the SC, FCC and BCC phases of the B 24 N 24 crystal
Published in Journal of Modern Optics, 2020
Ali Reza Adabinezhad, Mojtaba Yaghobi, Mohammad Ali Ramzanpour
The BN cages are the most attractive of nanostructured materials since they have perfect symmetrical structures, cage shapes, and surface-charge distributions. In addition, the properties and structures of fullerene-like are sometimes noticeably different from those of their bulk counterparts, hence the more attention paid to fullerenes [15,16]. The optical properties of fullerene pristine solid were strongly related to the properties of the molecule itself [39]. The isolated molecule has a unique octahedral (O) symmetry with48 point group operations for the boron and nitride atoms located at the corners of 12 squares, eight hexagons, and six octagons face of a truncated octahedral [37]. Therefore, understanding the optical properties of this interesting material is extremely important for an expansive review on the properties of fullerenes and nanomaterials.
Geometry in Our Three-Dimensional World
Published in Technometrics, 2023
Chapter 6 is a joy to read and discusses, among other things, Platonic and Archimedean solids. These solids are polyhedral and are the natural spatial equivalents of plane objects such as triangles, squares, and other polygons. The fact that there are only five Platonic solids can be easily established thanks to Euler’s characteristics formula. Using the concept of duality, it is shown that a dodecahedron (with 20 faces and 12 vertices) is the dual of an icosahedron (with 12 faces and 20 vertices). A hexahedron (aka a cube) is shown as the dual of an octahedron. A tetrahedron, however, is self-dual and is without a Platonic partner. Somewhat less popular but equally interesting Kepler-Poinsot solids have also been discussed in considerable detail.