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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.
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
As seen in Figure 4.10, we name the different kinds of coordination based on the geometric shape of the coordinating polyhedra obtained when the centers of coordinating anions are connected. So, we have triangular (3-fold), tetrahedral (4-fold), octahedral (6-fold), cubic (8-fold), and cubeoctahedral (12-fold) coordinations. Triangles, tetrahedra, octahedra, cubes, and cubeoctahedra are the only possible kinds of regular polyhedra—polyhedra with all bonds the same length. Many minerals contain predominantly regular coordination polyhedra, but irregular polyhedra (distorted versions of those shown in this drawing) and other coordinations (e.g., 5-fold, 7-fold) are also possible. When these less common coordinations are present, it is often because bonds are not entirely ionic and, therefore, may be stronger in some directions than others, or because of the presence of large anionic groups such as (CO3)2− instead of individual anions.
The Truss and the Space Frame
Published in Bjørn N. Sandaker, Arne P. Eggen, Mark R. Cruvellier, The Structural Basis of Architecture, 2019
Bjørn N. Sandaker, Arne P. Eggen, Mark R. Cruvellier
A space frame typically consists of top and bottom surface grids that are mutually connected to each other by means of diagonal members, the whole system thereby forming a three-dimensional network of struts or bars. The geometry can also be interpreted as that of a close packing of diverse polyhedra, which are spatial figures composed of at least four faces (called polygons) intersecting along their edges. (Fig. 9.21.) The “edges” in this case are formed by the structural members, and those in turn intersect at the corners (vertices) in structural joints. The so-called regular polyhedra are known as the five Platonic solids.10 Of those, the tetrahedron, which is a pyramid of four equilateral triangles, the octahedron, which may be seen as two pyramids with a square base joined along the base with the finished figure having eight faces of equilateral triangles, and the self-evident cube, all fill space by themselves or by combining with each other. While both the tetrahedron and the octahedron are stable, the cube needs to be braced by diagonals in order to be stable as a spatial figure. If a space frame is composed entirely of closely packed and stable polyhedral “building blocks,” then the space frame as a whole is surely an internally stable structural system.11
Math and Art: An Introduction to Visual Mathematics, 2nd ed.,
Published in Technometrics, 2022
Chapter 6 of Some Three-Dimensional Objects describes regular polyhedra, such as Platonic solids, total five of them — cube, and all tetra-, octa-, dodeca-, and icosa- hedrons. Other polyhedral include various convex objects such as semiregular polyhedra, or Archimedean solids, containing prisms, antiprisms, and 13 types of the so-called chirals, for instance, the so-called rhombicosidodecahedron. For any Platonic or Archimedean polyhedron, if V is the number of its vertices, E – number of edges, and F – number of faces, then the Euler famous characteristic holds: V – E + F 2. Other 3 D objects are considered as well, including sphere, cylinder, cone, and conic sections defined by equations of circle, ellipse, parabola, and hyperbola. 3D Euclidean and hyperbolic geometries, tilings, fractals, and cellular automata are described too, where the higher dimensional analogs of 2D polygons and 3D polyhedral are known as polytopes. In 4D space, there are exactly six regular polytopes, which are presented in graphs and figures for the hyper-tetrahedron, hyper-cube, and some others.