Rhombic dodecahedron

The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

Tellurium nitride was first obtained by the reaction of TeBt4 with liquid ammonia more than 100 years ago. The empirical formula TeN was assigned to this yellow, highly insoluble and explosive substance. However, subsequent analytical data indicated the composition is Tc3N4 which, in contrast to 5.6a and 5.6b, would involve tetravalent tellurium. This conclusion is supported by the recent preparation and structural determination of Te6N8(TeCl4)4 from tellurium tetrachloride and tris(trimethylsilyl)amine (Eq. 5.5). The TceNs molecule (5.12), which is a dimer of Tc3N4, forms a rhombic dodecahedron in which the... 

Granat-baum, m. pomegranate tree, -dode-Icaeder, n. rhombic dodecahedron. 

Figure 5.15 Anion-centered polyhedron (rhombic dodecahedron) found in the cubic closest-packed structure (a) oriented with respect to cubic axes, the c axis is vertical (b) oriented with [111] vertical (c) cation positions occupied in the sodium chloride, NaCl, structure and (id) cation positions occupied in the zinc blende (sphalerite) cubic ZnS structure.

To the best of our knowledge, there is one host which conforms to the structure of an Archimedean dual. Harrison was the first to point out that the quaternary structure of ferritin, a major iron storage protein in animals, bacteria, and plants, corresponds to the structure of a rhombic dodecahedron. [45] This protein, which is approximately 12.5 nm in diameter, consists of 24 identical polypeptide subunits (Fig. 9.18), and holds up to 4500 iron atoms in the form of hydrated ferric oxide with... 

Fig. 9.17 The 13 Archimedean duals derived from corresponding Archimedean solids (see Fig. 9.9). Triakis tetrahedron (1), Rhombic dodecahedron (2), Triakis octahedron (3), Tetrakis hexahedron (4), Deltoidal icositetrahedron (5), Pentagonal icositetrahedron (6), Rhombic tri-... 

Fig. 9.18 Ferritin, a spherical host based upon the rhombic dodecahedron. X-ray structure of the polypeptide subunit.

Rhombic dodecahedron, (d) hcc array with enclosed truncated octahedron the 24 vertices (black dots) lie at the centres of the distorted tetrahedra. (Because these 24 sites lie on the unit cell faces they are each shared between two unit cells, making 12 such sites per unit cell on average.) (e) The positions of two tetrahedral sites and a trigonal site between them. 

The theoretical analysis for two-dimensional foams and emulsions has recently been expanded to three dimensions [38], with Kelvin s minimal tet-rakaidecahedron as the unit cell. The system is subjected to a uniaxial extensional strain. As the elastic limit, or yield point, is approached, the cell shape tends towards a rhombic dodecahedron however, at the yield point, the shrinking quadrilateral faces of the polyhedron have finite (albeit small) area. 

Using planar tetrakaidecahedra as the model, on the other hand, causes the square faces to shrink to zero area at the yield point. The unit cell therefore resembles a true rhombic dodecahedron. The elastic response was found to be anisotropic (i.e. dependent on initial cell orientation) for the planar model, up to the elastic limit. This is in contrast to the monodisperse 2D case, which is... 

Figure 16-3 Structure of the protein shell of ferritin (apoferritin). (A) Ribbon drawing of the 163-residue monomer. From Crichton.62 (B) Stereo drawing of a hexamer composed of three dimers. (C) A tetrad of four subunits drawn as a space-filling diagram and viewed down the four-fold axis from the exterior of the molecule. (D) A half molecule composed of 12 subunits inscribed within a truncated rhombic dodecahedron. B-D from Bourne et al.7i...

Very recently the crystallization of non-spherical nanoparticles into ordered arrangements has been observed that had not been seen with spherical particles. Thanks to the high density of DNA on the particle solid faces, hexagonal packing of nanorods, and columnar stacking of nanoprisms, fee crystals of rhombic dodecahedrons were observed in [152] (see Fig. 35). 

GROMACS offers a variety of box types the standard cubical/rectangular box, the truncated octahedron, the hexagonal prism, and the rhombic dodecahedron. As the shape of the box can affect the simulation, a box type that best reflects the geometry of the solute should be used. For example, the truncated octahedron approximates a sphere, so it is the optimal choice for globular proteins and other roughly spherical molecules. 

Figure 9-15. Representation of simple highly symmetrical shapes (a) Cube (b) Tetrahedron (c) Octahedron (d) Rhombic dodecahedron.

Eqs. (4.15) and (4.16) are obtained empirically from experimental data about (p H) dependence for emulsions. Eq. (4.17) is derived theoretically by using rhombic dodecahedron. For other polyhedra (minimal tetrakaidecahedron and pentagonal dodecahedron) these dependences are the same. 

The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The Wigner-Seitz cell of the FCC lattice is the rhombic dodecahedron in Figure A. b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high symmetry points are shown in Table 4.4. 

The aqueous solution (blue, with a reddish tinge, but becoming green at 50° to 70° C.) deposits violet octahedra of the several forms in which chrome alum crystallises, including the rhombic dodecahedron, cube, and trapezohedron, the octahedron is the most stable. The crystals have a density of about 1-84, a specific heat of 0-324, and a coefficient of expansion of 0-0000246. The transformation of the violet to the green variety takes place at 78° C., - the substance melting at 89° C.ii to a green mass. Other physical constants have been determined, and the absorption spectra of its aqueous solution, studied. ... 

FIGURE 2.14. Some of the various shapes (hahits) of crystals that can result from a crystal structure with a cubic unit cell, (a) Models of a cube, octahedron, and rhombic dodecahedron, composed of cubic unit cells, (b) Tetrahedron, (c) cube, (d) octahedron, (e) rhombic dodecahedron, (f) tetrahexahedron, (g) icositetrahedron, and (h) irregular dodecahedron (pyritohedron). The corresponding stereographic projections are also shown, with filled circles for the northern hemisphere and open circles for the southern hemisphere (see Figure 2.13). (Models courtesy Amy Kaufman). 

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