Symmetry and Rigid Motions

Preliminary Activity. The object drawn below does not exhibit any nontrivial symmetry. Use this image to create a pattem or design that does exhibit both reflective and rotational symmetry. This can be done in any nuumer desired. After the pattern has been created, write a short paragraph that describes the methods used to create it. 

To this point, only two types of symmetries have been ntentioned. reflective and rotational. In terms of planar symmetry (symmetries that exist for objects drawn on a plane), there are two more, translational and glide reflective. All tour are listed next. 

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. 

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. 

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The discussion in Chapter 4 is focused on symmetry and rigid motions. The notation e.stabli.shed in Chapter 3 for permutations is used in a mathematical investigation of the symmetries of a variety of different shapes. The symmetries of a pentagon form the basts of the very reliable Verhoeff check digit scheme presented in Chapter 5. Furthermore, the use of rigid motions to create elaborate patterns will serve as an introduction to the discussion of group theory that begins Chapter 5. 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

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. 

In such states, the molecule does not exhibit or exist in the full symmetry appropriate to the stationary states. Why molecules do this is the unanswered question. That they do it is simply not at issue. Given that they do, all we have to do is ask whether delocalization takes place on a time scale that rules out the ideas of separated motions and a rigid molecule. This is an empirical matter. For most chemical states, the separability of nuclear and electronic motions is valid. 

However, to demonstrate how the rigid-body motion conforms to the irreducible representations, in this example we will go over the effect of symmetry operations on the translational and rotational motion of H2O. 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

Although all molecules are in constant thermal motion, when all of their atoms are at their equilibrium positions, a specific geometrical structure can usually be assigned to a given molecule. In this sense these molecules are said to be rigid. The first step in the analysis of the structure of a molecule is the determination of the group of operations that characterizes its symmetry. Each symmetry operation (aside from the trivial one, E) is associated with an element of symmetry. Thus for example, certain molecules are said to be planar. Well known examples are water, boron trifluoride and benzene, whose structures can be drawn on paper in the forms shown in Fig. 1. 

None of the models address the question of how the main chains are packed, and details of crystallinity are neither factored into nor predicted by mathematical models of the structure and properties of Nafion. Chains packed in crystalline arrays are usually considered to be rigid within the context of certain properties for example, with regard to diffusion, crystallites are viewed as impenetrable obstacles. F NMR studies indicate otherwise. Molecular motions that do not significantly alter symmetry can in fact occur in polymer crystals. It would seem, for example, that the response of the Nafion structure to applied stress would depend on the flexibility of the polymer backbone, a certain fraction of which is incorporated in crystalline regions. On the other hand. Starkweather showed that the crystallinity and swelling of Nafion are not correlated. 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

An elegant variant of the Aziz et al treatment was performed by Abarbanel Zwas (Ref 9) who considered the 1-D motion of a rigid piston in a closed-end pipe . The two equivalent systems examined are shown in Fig 3. In the upper sketch, detonation is initiated at a rigid wall, and in the lower sketch at a plane of symmetry. This system differs from that of Aziz et al in that the boundary condition at the rigid wall (or plane of symmetry) is one of zero particle velocity rather than zero pressure... 

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