Five-fold rotational symmetry

There are 6 five-fold rotational symmetry elements in an object of Ih point symmetry. Thus, in Figure 2.19b the 120 vertices of the great rhombicosidodecahedron are arranged in sets of 10 about the poles of these axes on the unit sphere. That construction emphasises that uniform contractions of these sets about these axes points will return the 12-vertex Platonic solid, the icosahedron, in which each vertex has Csv site symmetry. There are 10 three-fold rotational axes and, so, in Figure 2.19c the decoration pattern is arranged to divide the 120 vertices into sets of 6 about the 20 poles of these axes on the unit sphere. Again, uniform contraction of these subsets of vertices onto these positions on the unit sphere generates the fifth Platonic solid, the dodecahedron, and the site symmetry each vertex is Csy. 

Note, that since five-fold rotational symmetry cannot be propagated on alattice, there are only 32 Crystallographic point groups, since the icosahedral groups are excluded. 

Figure 12.3. Construction of 2-dimensional quasicrystal pattern with fat (A, B, C, D, E) and skinny (a, b, c, d, e) rhombuses. A seed consisting of the five fat and five skinny rhombuses is shown labeled on the right-hand side. The pattern can be extended to cover the entire two-dimensional plane without defects (overlaps or gaps) this maintains the five-fold rotational symmetry of the seed but has no long-range translational periodicity.

The relaxed structures of the various (rotational) symmetric toroidal forms were obtained by steepest decent molecular-dynamics simulations[15]. For the elongated tori derived from torus C240, the seven-fold rotational symmetry is found to be the most stable. Either five-fold or six-fold rotational symmetry is the most stable for the toroidal forms derived from tori Cjyo and C540, respectively (see Fig. 5). 

Only certain symmetry operations are possible in crystals composed of identical unit cells. In three dimensions these are one-, two-, three-, four- and six-fold rotations and each of these axes combined with inversion through a centre to give I, 2 ( = m, mirror plane), 3, 4, and 6 operations. Five-fold rotations and rotations of order 7 and higher, while possible in a finite molecule, are not compatible with a three-dimensional lattice. 

Since its introduction into clinical use in about 1979 the immunosuppresant cyclosporin has been responsible for a revolution in human organ transplantation.3 The exact mechanism of action in suppressing T-lymphocyte-mediated autoimmune responses is still not completely clear, but cyclosporin, a cyclic lipophilic peptide from a fungus, was found to bind to specific proteins that were named cydophilins.d Human cyclophilin A is a 165-residue protein which associates, in the crystal form, as a decamer with five-fold rotational and dihedral symmetry.6 This protein is also found in almost all... 

Some stable ternary intermetallic phases have been found that are quasiperi-odic in two dimensions and periodic in the third. These are from the systems Al—Ni—Co, Al—Cu—Co, and Al—Mn—Pd. They contain decagonally packed groups of atoms (local tenfold rotational symmetry). It should be noted that there are also known metastable quasicrystals with local eightfold rotational symmetry (octagonal) and 12-fold rotational symmetry (dodecagonal) as well. The dodecahedron is also one of the five Platonic solids (Lalena and Cleary, 2005). 

In the crystal structure [711], a dimes formed by two N-terminal 434 repressor fragments is bound to the 20 base pairs DNA duplex so that the complex has overall 2-fold rotational symmetry. The polypeptide chain is folded into five a-helices HI to H5, with helices H2 and H3 forming the helix/turn/helix motif (Fig. 20.16). Helices H3 and H3 of the 434 repressor dimer insert into two successive major grooves of the operator DNA whereas the N-termini of the flanking helices H2, H4 and H2 H4 contact the sugar-phosphate backbones. 

Because of the restrictions imposed on the values of the rotation angles (see Table 1.4), sincp and cos(p in Cartesian basis are 0, 1 or -1 for one, two and four-fold rotations, and they are 1/2 or Vs/2 for three and six-fold rotations. However, when the same rotational transformations are considered in the appropriate crystallographic coordinate system, all matrix elements become equal to 0, -1 or 1. This simplicity (and undeniably, beauty) of the matrix representation of symmetry operations is the result of restrictions imposed by the three-dimensional periodicity of crystal lattice. The presence of rotational symmetry of any other order (e.g. five-fold rotation) will result in the non-integer values of the elements of corresponding matrices in three dimensions. 

When the various symmetry elements that are present in a two-dimensional (planar) shape are applied in turn, it is seen that one point is left unchanged by the transformations. When these elements are drawn on a figure, they all pass through this single point. For this reason, the combination of operators is called the general point symmetry of the shape. There are no limitations imposed upon the symmetry operators that are allowed, and in particular, five-fold rotation axes are certainly allowed, and are found in many natural objects, such as star-fish or flowers. There are many general point groups. 

Since then, many other alloys that give rise to sharp diffraction patterns and which show five-fold, eight-fold, ten-fold and twelve-fold rotation symmetry have been discovered. The... 

If we consider all the symmetry operations which are associated with a particular molecular geometry, these operations form a point group and all these operations have the property of permuting atoms in identical environments in the molecule. However, if a set of identical Cartesian basis functions is placed on each symmetry-equivalent atom then, in addition to the permutation of symmetry-equivalent basis functions, some of the symmetry operations will send these basis functions into linear combinations of themselves (it is only necessary to think of the action of a three- or five-fold rotation on a set of p basis functions to see this). 

The archetypal discogenic molecule has a rigid, planar aromatic core with three-, four- or six-fold rotational symmetry, and generally six or more flexible side chains, each more than five atoms. However, exceptions to all these rules are now quite common. Systems are known with low symmetry, a non-planar, non-aromatic core, and with side chains as short as three atoms. 

As was mentioned in chapter 1, the discovery of five-fold and ten-fold rotational symmetry in certain metal alloys [5] was a shocking smprise. This is because perfect... 

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