Threefold axes

Cubic Three axes equal and mutually perpendicular Four threefold axes a = b = c a = /3 = 7 = 90°... 

The outer diameter of the shell is approximately 180 A and the inner diameter about 125 A except around the threefold axes, where the N-ter-minal a helices project about 20 A into the core. The RNA molecule that is present in the core is not visible in the electron density map. 

Figure 16.S Schematic illustration of the way the 60 protein subunits are arranged around the shell of safellite tobacco necrosis virus. Each subunit is shown as an asymmetric A. The view is along one of the threefold axes, as in Figure 16.3a. (a) Three subunifs are positioned on one triangular tile of an Icosahedron, in a similar way to that shown in 16.4a. The red lines represent a different way to divide the surface of the icosahedron into 60 asymmetric units. This representation will be used in the following diagrams because it is easier to see the symmetry relations when there are more than 60 subunits in the shells, (b) All subunits are shown on the surface of the virus, seen in the same orientation as 16.4a. The shell has been subdivided into 60 asymmetric units by the red lines. When the corners are joined to the center of the virus, the particle is divided into 60 triangular wedges, each comprising an asymmetric unit of the virus. In satellite tobacco necrosis virus each such unit contains one polypeptide chain...

In the T = 4 structure there are 240 subunits (4 x 60) in four different environments, A, B, C, and D, in the asymmetric unit. The A subunits interact around the fivefold axes, and the D subunits around the threefold axes (Figure 16.7). The B and C subunits are arranged so that two copies of each interact around the twofold axes in addition to two D subunits. For a T = 4 structure the twofold axes thus form pseudosixfold axes. The A, B, and C subunits interact around pseudothreefold axes clustered around the fivefold axes. There are 60 such pseudothreefold axes. The T = 4 structure therefore has a total of 80 threefold axes 20 with strict icosahedral symmetry and 60 with pseudosymmetry. 

Satellite tobacco necrosis virus is an example of a T = 1 virus structure. The 60 identical subunits interact tightly around the fivefold axes on the surface of the shell and around the threefold axes on the inside. These interactions form a scaffold that links all subunits together to complete the shell. 

Figure 4.17 Crystal and molecular structure of (LiMe)4 showing (a) the unit cell of lithium methyl, (b) the LijCj skeleton of the tetramer viewed approximately along one of the threefold axes, (c) the 7-coordinate environment of each C atom, and (d) the (4 -I- 3 -I- 3)-coordinate environment of each Li atom. After ref. 93, modified to include Li—H contacts.

Cubic point groups have four threefold axes (3 or 3) that mutually intersect at angles of 109.47°. They correspond to the four body diagonals of a cube (directions x+y+z, -x+y-z, -x-y+z and x-y-z, added vectorially). In the directions x, y, and z there are axes 4, 4 or 2, and there can be reflection planes perpendicular to them. In the six directions x+y, x-y, x+z,. .. twofold axes and reflection planes may be present. The sequence of the reference directions in the Hermann-Mauguin symbols is z, x+y+z, x+y. The occurrence of a 3 in the second position of the symbol (direction x+y+z) gives evidence of a cubic point group. See Fig. 3.8. 

Figure 6.7 Residues close to the threefold axes in (a) BfLF (b)Sma-l (c) EcFTN (d)EcBFR. Viewed along the threefold axis from the inside surface. Reprinted from Harrison et al., 1998, by courtesy of Marcel Dekker, Inc.

Fig. 19 The linkages between neighboring triacontahedral clusters in the Tsai-type 1/1 ACs along a the twofold axes, and b the threefold axes. The 2/1 ACs exhibit same linkages although they have different atom identities and symmetries. All decoration atoms of the triacontahedra are omitted for clarity. (Adapted with permission from [83], Copyright 2006 American Chemical Society)...

Figure 2.24 Models of packing of chains in a-form of sPS according to space groups (a) / 3cl52 and (b) P3150. In (a) dotted lines indicate crystallographic glide planes coincident with local glide planes of chains. In (b) triplets of chains are rotated around threefold axes and crystallographic glide planes are lost.

A good example of a higher-than-actual symmetry is provided by hexamethyl-benzenetricarbonylchromium. In the crystal structure the threefold axes of the Cr(CO)3 groups are almost parallel both to each other and to one of the (symmetry determined) crystallographic axes (Fig. 6)65 It follows that the dipole moment... 

In X- and Y-type zeolites which exhibit the faujasite structure, the cations can reside in four basically different types of site which are located on the threefold axes of the cubic faujasite structure (Fig. 18). 

Applying the previous picture this would imply a noteworthy difference in stability between the two triple ions resulting in an ionic recombination process between a triple ion and simple ion. Such a difference in stability would be very plausible judging from the important difference in size and shape of the two ions which would make the triple anion more stable as a structure where the two plcrate ions penetrate along "threefold axes" in two cavities formed by the alkyl-limbs. Applying eq. 

Figure 7.1 Representation of Si044 tetrahedron (left) and A1C>69 octahedron in terms of the view down threefold axes.

Ttie cubic classes stand somewhat apart from the rest. They have as their distinctive feat ure four threefold axes lying along cube diagonals these are secondary axes. The primary axes may be either twofold or fourfold. ... 

Cubic (sometimes called isometric, or lesser al). All crystals having four secondary threefold axes have three mutually perpendicular directions all equivalent to each other. The unit cell is thus a cube, the secondary threefold axes being the cube diagonals. The five classed of the cubic system are 23, m3 (= 2/m3), 43m, 43 (= 432), and (= 4/m 3 2/m). Examples are shown in Fig 37. 

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