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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.
Replacing Turing Tape with a Fractal Tape
Published in Anirban Bandyopadhyay, Nanobrain, 2020
A platonic love letter to Big Data: In principle, we could randomly choose geometric shapes and find them in the natural events and use those very shapes to assign complex geometric shapes to the integers in the PPM. However, 1D, 2D, and 3D geometries are selected mathematically to serve as an analogy to the Platonic geometries (five Platonic solids are tetrahedron (or pyramid), cube, octahedron, dodecahedron, and icosahedron). For 2D, we take triangle, quadrilateral, pentagon, hexagon, for 1D a few letters like L, V/U, C (S = 2C), O. Sensors search only for these fifteen geometries, each with a distinct value of S in a massive rapidly changing database of events, not by looking into the content, but, in a matrix of dataset, which is converted into a rapidly changing topology. In TCA, any information is converted into a 1D, 2D or 3D…10D geometric shape. Conversion to a shape ignores the actual content, converts the big data stream into a spatial flow of fluid as a function of time or phase. In the rapidly changing 3D distribution of data, imagine it could be a 3D cubic glass chamber filled with clouds one finds the most inactive and the active points that appear, disappear or change periodically. Active points are those which bursts like a lightening; inactive points are those who remain silent in the 3D glass chamber.
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
Natural convection in platonic solids
Published in Numerical Heat Transfer, Part A: Applications, 2023
Eliton Fontana, Ricardo Persi de Souza, Claudia Angela Capeletto
Platonic solids have not only mathematical significance but also esthetic appeal, which has made them popular in the design of various structures and objects. In architecture, while not common, the icosahedron and dodecahedron have been used to create visually striking shapes. For example, Vasilieva [19] highlights instances where these solids have been utilized in architectural design. The dodecahedron, in particular, has also been utilized in the design of loudspeakers with the aim of producing an omnidirectional radiation pattern [20]. Additionally, the scattering characteristics of plasmonic nano-particles shaped as each of the regular polyhedra have been evaluated by Tzarouchis et al. [21], and the results showed that features such as the sharpness of corners and edges have a significant impact on the scattering spectra. In terms of natural convection, these solids can be considered for applications such as the design of electronic devices, solar collectors, chemical reactors, buildings, and any other application where passive cooling is desirable.
Reimagining the history of GIS
Published in Annals of GIS, 2018
Rather than lay a quadtree on a flattened Earth, DGGs are hierarchical structures on the curved surface of the Earth itself, thus avoiding all of the distortions inherent in map projections. Unfortunately only five ways exist to create a 3D solid using pieces of equal size and shape. These are the five Platonic solids, and have been known since classical times: the 4 triangles of a tetrahedron, the 6 squares of a cube, the 8 triangles of an octahedron, the 12 pentagons of a dodecahedron and the 20 triangles of an icosahedron. Instead, DGGs begin with one of the Platonic solids or a simple modification, and then use a hierarchical scheme to subdivide each of its faces. Since triangles are the favoured display element of 3D graphics systems, it is desirable that the basic elements at any level in the hierarchy be triangles. Triangles are also convenient since it is desirable that basic elements at each level nest within the corresponding element at the next higher level, and it is easy to create four nesting triangles from a larger triangle by connecting the midpoints of its edges. On the other hand the pentagons of the dodecahedron do not lead to hierarchies with simple and desirable properties.