Polyhedra projections

Figure 5.8 The crystal structure of olivine, (a) The structure projected onto (100) showing serrated chains of octahedra running parallel to the c axis (b) oxygen coordination polyhedra projected about the Ml and M2 positions. Metal—oxygen distances in each coordination site are indicated. Cell parameters and interatomic distances are for fay-alite (from Smyth Bish, 1988).

Fig. 31 Crystal structure of Rb sNb30Ft8. Projection on the plane (001). Numbers in brackets are the atom coordinates on the z-axis in percents of c parameter. Structure of NbOFj chains and NbFf polyhedron. Reproduced from [209], A. I. Agulyansky, V. E. Zavodnik, V. Y. Kuznetzov et al. Neorgan. Mater. 27 (1991) 380, Copyright 1991, with permission of Nauka (Russian Academy of Sciences) publishing.

Figure 9.38. (a) A projection of the unit cell of InNi2 along c is shown. (b) An expanded cell to show the polyhedron for Nii. 

Figure 5.18 The crystal structure of Mg-Fe amphibole (cummingtonite). The figure shows the structure projected onto (001), and a (100) projection of each oxygen coordination polyhedron about a metal position. Metal-oxygen distances in each coordination site are indicated (pm). Ml M2 M3 ° M4. Atomic coordinates and cell parameters from Ghose (1961).

Fig. 18. (a) The space lattice and linkage of metal coordination polyhedrons (b) The projection of the linkage of the metal coordination polyhedrons in the directions of [100],... 

In projecting a closed polyhedron on a fixed plane, the normals to its faces must be drawn either all inwards or all outwards. 

Fig. 8. Parts of the crystal structures of acidic aluminum orthophosphates (polyhedral representation), (a) A1(H2P04)(HP04) H2O polyhedron linkages directed upwards in this projection are indicated by shading. The hydrogen atom positions are shown by wedges. Inter- and intramolecular hydrogen bonds are shown by broken lines, (b) H30[Al3(H2P04)6(HP04)2] 4H2O the solid circles represent the H3O+ sites [reprinted with permission from Kniep (14) copyright 1986 Wiley-VCH].

Figure 2.17 The polyhedron of 12 vertices forming the regular geometric orbit of the T symmetry group (a) as an elliptical projection and (b) in perspective.

This polyhedron is chiral and can be drawn as either enantiomer by appropriate choice of the 60 vertices, either red or blue, in Figure 2.24. All the lower orbit structures, O12, O20 and O30 shown in the second column of projections in Figure 2.24 are achiral and identical to those found by coalescing local sets of 10, 6 and 4 vertices in full Ih point symmetry. 

Figures 5 to 8 show projections of the Point-Molecule representation for Cu(II)-P-diketones quelate compounds, on the plane of two principal components for the Cioslowski-like, Coulomb-like, overlap-like and triple density similarity measure matrices, respectively. As before, the figures represent two-dimensional projections of 10-dimensional polyhedrons and the elements of the set are divided in two classes, depending on their respective extraction constant (K ). We can see that the most active compounds (represented by circles) can be shown split from the less active ones (represented by squares).

The combination of these rescaled dodecahedral points with the vertices of the icosahedron indicated above yields another indexed polyhedron with icosahedral symmetry the triacontahedron discovered by Kepler in 1611. The triacontahe-dron has 32 vertices, 12 icosahedral and 20 dodecahedral ones and 30 rhombic faces, or 60 triangular ones in the latter case it is then denoted ico-dodecahedron. The triacontahedron is the projection in space of a six-dimensional hypercube and has two points [200000] and [11 111 I] as generators with integral indices (Figure 11-7). Further polyhedra with icosahedral symmetry and vertices at icosahedral lattice points are obtainable from one or more generators with integer indices [28]. 

In addition to these requirements, a correct graph must also meet additional criteria as to the sequence in which one may proceed from one permutation to others not obtainable from it in a single pseudorotation. No simple polyhedron in three-dimensional space can meet all requirements and thus more complex graphs, that can be regarded as projections of regular polyhedra in higher-dimensional space on three-dimensional space, must be invoked. One that seems particularly useful is shown in Fig. 1-16. It will be... 

Figure 65. The crystal structures of nefedovite and olgite (a) nefedovite projected onto (100) (b) nefedovite projected onto (001) (Na( )n) polyhedra are daik-shadow-shaded, (Ca( )8) polyhedra are light-shadow-shaded (c) (SrOi2)-icosahedron layer in olgite projected onto (001) (d) the (Sr,NaOio)-(Na07)-polyhedron layer in olgite projected onto (001) (e) the stacking of layers along [001] in olgite (S n) icosahedra are dark-shadow-shaded, (NaOio) polyhedra are dark-shaded, (NaOe) octahedra are dot-shaded.

Figure 67. The crystal structures of stercorite and natrophosphate (a) stercorite projected onto (001) (b) stercorite projected onto (010) (c) natrophosphate projected onto (001). (Na( )6) octahedra are shadow-shaded, (Nac )5) polyhedron is shadow-shaded, (NH4) are shown as small black circles.

In the cuboctahedron case, we were able to introduce the large number of the sites to represent each edge between the vertices of the polyhedron using the simple arithmetic mean to generate coordinates of new sites. In contrast, here such an approach is not possible since we want the new points to be everywhere inside the molecule, not only along the bonds. To arrive at approximately uniformly distributed points in the interior of the van der Waals contour of the molecule we select the coordinates of the points at random and then check that indeed the point is inside the molecular interior. In Figure 22 we illustrate distributions of 1000 and 5000 random points that represent a planar model of the H2O molecule (i.e., the projection of H2O on a plane). 

Figure 24. Projection of an Nb20g cluster in NaNb306 and related compounds together with the surrounding Nb atoms and a coordination polyhedron for one of these atoms. The broad line corresponds to Nb-Nb multiple bond, dotted lines indicate weak Nb-Nb bonding.

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