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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
Polyhedra have been known since antiquity. In about 520 B.C., Pythagoras already knew the existence of three of the five regular polyhedra viz. the cube, the tetrahedron, and the dodecahedron. Plato (about 350 B.C.) reported all five of them adding the octahedron and the icosahedron. He mystically related them as ‘cosmic building stones’ to the five so-called ‘elements’: Fire, air, water, earth, and ‘heavenly bodies’. Therefore, these five regular bodies are today designated as Platonic solids. Archimedes (about 250 B.C.) studied the 13 uniform or semi-regular polyhedra, which today bear his name as Archimedean solids. Then, after a very long time period, Kepler (1619) arrived at an integrated description of the 5 Platonic and the 13 Archimedean solids. He also had the idea that it was possible to construct regular star-shaped so-called stellated polyhedra. This idea was completed two centuries later by Poinsot (1810) presenting the complete list of all fully regular stellated polyhedra. Subsequently, piece by piece, stellated and other general uniform polyhedra were being published. Today, the mathematical analysis of various kinds and families of polyhedra, as well as extensions to higher space dimensions greater than three (named polytopes) and an active area of mathematics are well established (cf. books by Coxeter (1973), Cromwell (1997), Ziegler (1995), Alexandrov (2005), and Grünbaum (2003)).
Overview of the Manifold VNPs Used in Nanotechnology
Published in Nicole F Steinmetz, Marianne Manchester, Viral Nanoparticles, 2019
Nicole F Steinmetz, Marianne Manchester
Description of icosahedral symmetry. The word icosahedron derives from the Greek language and means “twenty seat.” An icosahedron is a polyhedron with 20 triangular faces. It has fivefold, threefold, and twofold rotational symmetry axes, in short: 5:3:2 symmetry (Fig. 2.2).
Magnetic Structures of 2D and 3D Nanoparticles
Published in Jean-Claude Levy, Magnetic Structures of 2D and 3D Nanoparticles, 2018
Here we want to focus first on cluster magical properties. These structural properties do not depend strongly on the nature of the interaction when leading to dense structures since they mainly depend on geometry. Starting with very small clusters, the most stable structures are the more symmetric ones. That is a good introduction to magical clusters and so magical numbers. So for four atoms the regular tetrahedron is stable [119]. For seven atoms the most stable configuration is a regular planar pentagon surrounded by two atoms on the top and the bottom [119]. This unusual fivefold symmetry is found also in the icosahedron of 13 atoms [35] where the icosahedral symmetry group Yh contains 120 symmetry operations, more than the octahedral group Oh, the largest crystalline symmetry group which contains 48 symmetry operations. Then from that, cluster sizes 7 or 13 up to the largest sizes which favor crystalline symmetry, icosahedral symmetry is expected to occur as a rich local symmetry. According to this remark, a large cluster structure with this icosahedral symmetry was built from a variational minimization of energy [105]. An interesting feature of this cluster shown is in Fig. 1.4: this icosahedral symmetry is present everywhere in the cluster at the expense of defects such as holes or voids of different sizes [105].
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.
HIV-1 immature virion network and icosahedral capsids self-assembly with patchy spheres
Published in Molecular Physics, 2023
Brian Ignacio Machorro-Martínez, Anthony B. Gutiérrez, Jacqueline Quintana, Julio C. Armas-Pérez, Paola Mendoza-Espinosa, Gustavo A. Chapela
Figure 5 shows the capsids obtained with models M3 without the 2-Fold interaction. Hexagons are painted in a cyan colour, pentagons in black and molecules not belonging to any polygon are shown in red for the large sphere and green for the small one. T = 1, T = 3 and T = 4 are the capsids formed in this work. All of them show icosahedral symmetry. Following the Kaspar and Klug [19] classification method, the capsomers number of a capsid is given by ; with hexamers and 12 pentamers. Capsomer numbers of the three assembled capsids are: For pentamers; for hexamers and 12 pentamers, and for hexamers + 12 pentamers. Angles required to assemble the capsids correctly are: , and , for T = 1, T = 3 and T = 4, respectively. Same Figure 5 shows a T = 7 half capsid with the hexamers and pentamers are positioned correctly, which means that, for an angle of , a complete capsid could be formed. However, it was not possible in this work to close such a capsid with the correct number of capsomers. This might be since for triangulation numbers with k>0 as , or the capsids are enantiomorphus, which means they exist in their left (levogyrous) and right (dextrogyrous) configurations. Namely, a capsid may be assembled in two ways, starting in a pentamer, turning left and advancing k capsomers until the other pentamer is reached or turning right and following a similar path. Not imposing a restriction in the simulation, to take one of these alternatives, two enantiomers may be combined avoiding the correct assembly of capsid.