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Shape effect of grain in a granular flow
Published in Y. Kishino, Powders and Grains 2001, 2020
D. Petit, F. Pradel, G. Ferrer, Y. Meimon
To stress out the effect of the morphology factor, special polyhedra are selected to replace the spherical grain. It has been decided to select the regular polyhedra which are inscribed into the spherical grain they are replacing. In a tridimensional space, there is only five polyhedra called the Platonic solids and reported on figure 3: tetrahedron, cube, octahedron, dodecahedron, icosahedron.
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
A general polyhedron in three-dimensional space is a solid bounded by a number of polygons (faces), which, two by two have a side in common (edges), while three or more polygonal faces join in common vertices. There is a solid angle Ω associated with each vertex. Broadly, all polyhedra fall into two categories: Convex or non-convex (concave). A convex polyhedron is a polyhedron with the property that for any two points inside of it, the line segment joining them is completely contained within the polyhedron. In general, any polyhedron subtending a solid angle Ω < 2π sr at each of its vertices is called convex while any polyhedron subtending a solid angle Ω > 2π sr at any of its vertices is concave. In other words, a convex polyhedron has each of its vertices protruding outward while a concave polyhedron has any of its vertices indented inwards the surface. Any polyhedron in three-dimensional space can be thoroughly characterized by the total number of its faces f, vertices v, and edges e, which are commonly related by Euler’s theorem: υ−e+f=χ=2(1−g).
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.