Platonic solid

Abstract: The synthesis of nanoparticles with a tight control on their size and shape is a requirement for the achievement of many nanotechnology goals, since these are the main parameters determining the material properties at the nanoscale. Among the many different materials currently under investigation, semiconductor and metal nanoparticles are extremely attractive because of the possibility of controlling their electronic and optical properties through tailoring size and shape during synthesis. For instance, the presence of sharp edges or tips has been shown to increase electric-field enhancement, [1] which is important for applications involving metal nanoparticles as sensors. Additionally, nanoparticle morphology will ultimately determine the way in which nanoparticles can be assembled. While spheres, rods, cubes, and flat prisms are the most thoroughly studied shapes, there have been several reports on the synthesis and detailed characterization of other geometries, and it is believed that metal nanoparticles tend to grow with the structure of one of the Platonic solids, that is, with all faces made of the same regular polygon and the same number of polygons meeting at each corner. [2–4] Non-Platonic structures that have been reported for metal nanocrystals include nanorods, either with the geometry of single-crystal or pentatwinned prisms, [5–9] as well as flat platelets, usually with triangular or hexagonal shape, [10–13] in which faces are made of more than one polygon. Recently, other elaborate geometries have also been reported, such as nanocages [14] or nanostars. [15] In this paper we report the synthesis with tight size control and high monodispersity of nanometer-sized, uniform gold nanoparticles with decahedral morphology, in which all ten faces are equilateral triangles, but not the same number of triangles meet at every corner (four or five are possible), and for this reason they do not qualify as Platonic nanocrystals, but can rather be classified as Johnson structures. These decahedral particles have a lower symmetry (D5h) than Platonic solids, but still display a striking beauty and attractive optical properties that are strongly influenced by particle size. Although the geometry (and thus the aspect ratio) is precisely the same for particles with different sizes, clear changes in the optical responses have been observed and modeled through a boundary element method (BEM) for bicones. The synthesis is based on our previous work where N,N-dimethylformamide (DMF) was used as both solvent and reducing agent [16] to generate silver nanoparticles of various shapes, including spheres, [17] nanoprisms [18] and nanowires. [19] The

Abstract: Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the adaptive shrinking cell (ASC) scheme. This optimization problem is solved here (using a variety of multiparticle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space ${\mathbb{R}}^{3}$, except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823\dots{}, 0.836\dots{}, 0.904\dots{}, and 0.947\dots{}, respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other nontiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles and apply it to Platonic solids, Archimedean solids, superballs, and ellipsoids. Provided that what we term the ``asphericity'' (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be the analog of Kepler's sphere conjecture for these solids. The truncated tetrahedron is the only non-centrally symmetric Archimedean solid, the densest known packing of which is a non-lattice packing with density at least as high as $23/24=0.958\text{ }333\dots{}$. We discuss the validity of our conjecture to packings of superballs, prisms, and antiprisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.

Abstract: Herein we describe some properties and the occurrences of a beautiful geometric figure that is ubiquitous in chemistry and materials science, however, it is not as well-known as it should be. We call attention to the need for mathematicians to pay more attention to the richly structured natural world, and for materials scientists to learn a little more about mathematics. Our account is informal and eschews any pretence of mathematical rigor, but does start with some necessary mathematics. Regular figures such as the five regular Platonic polyhedra are an enduring part of human culture and have been known and celebrated for thousands of years. Herein we consider them as the five regular tilings on the surface of a sphere (a two-dimensional surface of positive curvature). A flag of a tiling of a two-dimensional surface consists of a combination of a coincident tile, edge, and vertex. A generally accepted definition of regularity is flag transitivity, which means that all flags are related by symmetries of the tiling (i.e. there is just one kind of flag). In addition to the five Platonic solids, there are three regular tilings of the plane (a surface of zero curvature), and these are the familiar coverings of the plane by triangles, squares, or hexagons tiled edge-to-edge. The corresponding regular tilings of three-dimensional space are also wellknown. Flags are now a polyhedron (tile) with a coincident face, edge, and vertex, and the regular tilings of the three-sphere are the six nonstellated regular polytopes of four dimensional space. We remark that four dimensions is the richest space in this regard; higher dimensions have only three regular polytopes (and of course three dimensions has five). However, in flat threedimensional (Euclidean) space, the space of our day-to-day experience, there is disappointingly only one regular tiling—the familiar space filling by cubes sharing faces (face-to-face). The classic reference to these figures is Coxeter!s Regular Polytopes, in which he remarks on the tilings of threedimensional Euclidean space: “For the development of a general theory, it is an unhappy accident that only one honeycomb [tiling] is regular...”. Unhappy indeed, because, perhaps as a consequence, the rich world of periodic graphs, which are the underlying topology of crystal structures, has been largely neglected by mathematicians. The graph associated with (carried by) the regular tiling by cubes is the set of edges and vertices. It is notably the structure of a form of elemental polonium, and chemists often refer to it as the a-Po net. Recently a system of symbols for nets has been developed and this net has the symbol pcu. Our review is concerned with another such periodic graph, and an associated surface.

Abstract: Proportion in architecture similarity the golden mean graphs tilings with polygons two-dimensional networks and lattices polyhedra - platonic solids transformation of the platonic solids I transformation of the platonic solids II polyhedra - space filling isometries and mirrors symmetry of the plane.