Rigid motion

Lord Kelvin lla> recognized that the term asymmetry does not reflect the essential features, and he introduced the concept of chiralty. He defined a geometrical object as chiral, if it is not superimposable onto its mirror image by rigid motions (rotation and translation). Chirality requires the absence of symmetry elements of the second kind (a- and Sn-operations) lu>>. In the gaseous or liquid state an optically active compound has always chiral molecules, but the reverse is not necessarily true. 

The general numerical approach employs rigid body motions and least-squares fitting. Given two sets of points xt,i = 1,2,..., N and yui= 1,2,..., N (here xt and yt are vectors specifying atomic coordinates), find the best rigid motion of the points y h) such that the sum of the squares of the deviations E xt — Yt 2 is a minimum. 

The relationship between an object and its mirror image is at the heart of the incongruity of counterparts. In particular, the statement that enantiomorphs can be brought into congruence only by rigid motions combined with an odd number of reflections in the object s space requires further elaboration. 

XXV), M2TH5 (T = Rh, h- M = Mg, Ca, Sr, Eu HI) and Mg6T2Hii (T = Co XVII, and Ir XLV). In some structures, the hydride complexes undergo rigid motions, such as in K2PtH4 for which LT-NMR data snggest rotational jumps of the spl-PtELi units in the plane of the square. ... 

Modes (a) and (b) involve the rigid motion of the water molecules the increase in the reduced mass of the vibrations upon deuteration and for substitution... 

To obtain information on the role of dynamics of molecular motions in the reactive systems, the approach of phonon spectroscopy is used. Phonons are low-frequency cooperative lattice vibrations of a solid and, therefore, probe the lattice interactions and dynamics directly. Phonons can be observed as optical transitions in the Raman spectra and in the electronic spectra (in the latter as a phonon side band). Some information regarding averaged librational and translational phonon motions can also be obtained from the rigid-motion analysis of the thermal parameters of x-ray diffraction studies. 

Handedness is intrinsic (to the asymmetric object). Therefore, enantiomorphy is also intrinsic because any rigid motion by definition changes only external relations but leaves intrinsic shape unaltered. On this view, enantiomorphy and asymmetry are necessarily coextensive. Handedness, however, is a further property because being asymmetric does not determine whether an object is right or left handed. 

How are we to choose between these positions Nerlich recommends position (3) on the basis of its clarity and explanatory power. Treating handedness and/or enantio-morphy as intrinsic is not the provision of an analysis, but the refusal to provide one. Positions (1) and (2) just leave these properties obscure. Position (3), in contrast, offers an explanation of why handedness must be extrinsic because in certain nonstandard spaces rigid motions of an asymmetrical object can be equivalent to reflection. We can readily visualize this in the case of two-dimensional enantiomorphs, as we already saw, and although the corresponding three-dimensional case is beyond our visual imagination, we can at least appeal to analogy. 

A detailed analysis shows that the thermal vibrations of the Cs atoms occur mainly relative to the 0 atoms (in addition to a rigid motion of the cluster) The vibration amplitude is largest for the Csl atoms and the direction of the largest displacement for these atoms is perpendicular to the strong (but not directional) Cs-0 bond. The thermal motion for the Cs2 and Cs3 atoms bound to two and three oxygen atoms,... 

Not only do objects, images, and patterns exhibit these four symmetries, but these symmetries can also be used to create objects that are synunetric. In other words, a base image can be taken and specific methods can be used to move it around the plane on which it is drawn to create an image that has symmetry. The movement methods come from the four symmetries mentioned above and are called rigid motions. There are four rigid motions for objects in the plane. 

Reflection. A reflective rigid motion is obtained by taking a base object and reflecting it over an axis of reflection. This can be done once or any number of times to generate a picture that is symmetric. See Figure 4.9. 

Rotation. A rotational rigid motion is obtained when a base object is rotated around a point a certain number of times. The point around which the rotation occurs can... 

These four rigid motions can be combined in any number of ways to create a picture that is symmetric. This combination can occur a finite number of times or an infinite number of times to fill up the entire plane. 

When an infinite number of rigid motions is applied to a base object to fill the plane, a pattern is created. 

Patterns can be created by using any number and combination of rigid motions. First the base object is picked, then copies of that object are created. The copying is done by... 

When a design displays only one translational axis, regardless of what other rigid motions are involved, the pattern is referred to as a border. Figure 4.14 shows three examples of borders. 

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