Symmetries of isolated objects

A point group is defined as a set of symmetry operations that leave unmoved at least one point within the object to which these symmetry operations are being applied. No translation operations (that is, simple movements along a straight line) can be included in this description  

FIGURE 4.4. Two molecules related by a mirror plane. A mirror plane converts one molecule into another with the opposite handedness. For example, a left hand is converted to a right hand, (a) View of the two molecules, (b) Formulae of the two molecules, (c) Representations of the two molecules as hands with a mirror between them. 

FIGURE 4.6. A rotatory-inversion axis involves a rotation and then an inversion across a center of symmetry. Since, by the definition of a point group, one point remains unmoved, this must be the point through which the rotatory-inversion axis passes and it must lie on the inversion center (center of symmetry). The effect of a fourfold rotation-inversion axis is shown in two steps. By this symmetry operation a right hand is converted to a left hand, and an atom at x,y,z is moved to y,—x,—z. (a) The fourfold rotation, and (b) the inversion through a center of symmetry. 

FIGURE 4.6. (c) A rotatory-inversion axis. The view from above, where filled circles lie at +z and open circles at —z. Two steps are involved (1) a fourfold rotation, and (2) inversion about the origin. 

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Discrete Rotational Symmetry This is a subset of continuous rotations and reflections in three-dimensional space. Since rotation has no translational components their symmetry groups are known as point groups. Point groups are used to specify the symmetry of isolated objects such as molecules. 

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