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Onion-Like Inorganic Fullerenes from a Polyhedral Perspective
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2020
Ch. Chang, A. B. C. Patzer, D. Sülzle, H. Bauer
The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids: the cube, dodecahedron, icosahedron, octahedron, and tetrahedron (see Table 15.2). Platonic solids are further characterized by having three unique spheres associated. A circumsphere on which all vertices lie, an insphere which passes through all facial midpoints, and a midsphere which touches upon all edges. Furthermore, all dihedral angles (i.e. the angle between two adjacent faces) are equal, all solid angles as well as all vertex environments are equivalent because every vertex is surrounded by the same number and kind of faces. The dual of a Platonic solid is again a Platonic, i.e. they are dual between themselves, and in fact, the tetrahedron is self-dual. They are also characterized by high symmetry and high sphericity. The most spherical of them is the icosahedron (see Figure 15.3). A summary of their geometric features is given in Table 15.2.
Triangulating Your Region of Formulation
Published in Mark J. Anderson, Patrick J. Whitcomb, Martin A. Bezener, Formulation Simplified, 2018
Mark J. Anderson, Patrick J. Whitcomb, Martin A. Bezener
The pattern of points depicted in Figure 2.1 forms a textbook design called a “simplex-centroid” (Scheffé, 1963; Cornell, 2002). We will introduce a more sophisticated design variation called a “simplex-lattice” later, but let’s not get ahead of ourselves. The term “simplex” relates to the geometry—the simplest figure with one more vertex than the number of dimensions. In this case, only two dimensions are needed to graph the three components on to an equilateral triangle. However, a four-component mixture experiment requires another dimension in simplex geometry—a tetrahedron, which looks like a pyramid, but with three sides rather than four. To show how easy it is to create a simplex centroid, here is how you’d lay it out for four components: Four points for the pure components (A, B, C, D) plotted at the corners of the tetrahedron).Six points at the edges for the 50/50 binary blends (AB, AC, AD, BC, BD, CD).Four three-component blend points at the centroids of the triangular faces of the tetrahedron.The one blend with equal parts of all ingredients at the overall centroid of the tetrahedron.
The Structure of Water
Published in Tadahiro Ohmi, Ultraclean Technology Handbook, 2017
Figure 2a illustrates the dimensions and bonding angle of the water molecule. ∠HOH, the bonding angle between oxygen and the hydrogen atoms, is 104.52°. This figure is rather large compared to the angles in the homologous series H2S (92.16°), H2Se (90.53°), H2Te (90.25°), and so on. The angle 104.52° is close to the angle of a regular tetrahedron, 109.5°.
Computation for biomechanical analysis of aortic aneurysms: the importance of computational grid
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Farah Alkhatib, Adam Wittek, Benjamin F Zwick, George C Bourantas, Karol Miller
For tetrahedral meshes, we used two element quality measures (Parthasarathy et al. 1994): The minimum/maximum allowable interior angles, which are determined between any two edges of the tetrahedron face. The acceptable interior angle ranges between 30° and 120° (Yang 2018).The volumetric skew (AltairEngineering 2022) that determines how close a given element is to the ideal reference shape (in the sense of isoparametric FE analysis) (Ruggiero et al. 2019). where the ideal tetrahedron is an equilateral tetrahedron with the same circumradius as the actual (analysed) tetrahedron. The circumradius is the radius of a sphere passing through all four vertices of the tetrahedron.
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.
The effect of partial substitution of chloride by bromide in the 0-D hybrid material (C4H12N2)[CuCl4]·2H2O: Structural, vibrational, thermal, in silico and biological characterizations
Published in Journal of Coordination Chemistry, 2022
Marwa Abid Derbel, Raja Jlassi, Thierry Roisnel, Riadh Badraoui, Najeh Krayem, Hanan Al-Ghulikah, Walid Rekik, Houcine Naïli
The coordination sphere of the copper ion is formed by four disordered Br- and Cl- ions. Three geometries are then possible: tetrahedron, square plane or seesaw geometry. To determine the geometry of our copper polyhedron we calculated the τ4 parameter (τ4 = 0 for a square plane geometry and τ4 = 1 for a tetrahedral form) using the following formula [51]: where α and β are the two large angles in the polyhedron and θ the angle in a regular tetrahedron, θ = 109.5°.