Symmetry element and symmetry operation

A symmetry element (not to be confused with a group element) is a point, a line, a plane or an axis with respect to which a symmetry operation is carried out. For three-dimensional objects, there are only 5 types of symmetry elements that need to be considered:


Is  a unique symmetry operation?


Yes, because an object with an  symmetry element may not have a symmetry element of the same . For example, a tetragonal disphenoid has an symmetry element but only a symmetry element.


If an object has more than one symmetry axis of rotation, the one with the highest  is called the principal axis, which is conventionally coincident with the -axis.

Consider an equilateral triangle in a three-dimensional real vector space (see figure I). It has a  axis of symmetry and three planes of reflection. Figure II has a  axis of symmetry, while figure III has an  axis of symmetry and a centre of inversion . All objects have the identity symmetry element.

A symmetry operation is a geometric transformation of an object with respect to a symmetry element in a real vector space such that the object looks the same after the transformation. However, the labelling of components of the object may be different post-transformation. For example, a reflection of the equilateral triangle in the -plane results in the triangle looking the same but the labels  and  being swapped.


Why do we need to label the corners of the equilateral triangle?


Since the object is defined in a real vector space, each point on the object, and hence each label, is represented by a position vector. A symmetry operation preserves the magnitude of the position vector but changes its direction. For example, a reflection  in the -plane results in the triangle looking the same but transforms vector  to vector  (see diagram below). If we carry out two consecutive symmetry operations of  on the original triangle, vector  transforms into itself. This is equivalent to performing the symmetry operation  on the original triangle. Consequently, we are able to form mathematical groups, whose elements are symmetry operators, and use them to analyse the properties of objects like molecules.


In total, there are 6 symmetry elements and 6 associated symmetry operations for the equilateral triangle:


What does the notation  mean and why is the reflection in the -plane not included in the set of symmetry operations for the equilateral triangle?


The angle of rotation is regarded as positive if the rotation is counter-clockwise when we look down the axis of rotation towards the origin (and hence the notation ).  and  can also be denoted as  and respectively, where  is counter-clockwise by default and  is the double application of . The reflection in the -plane is a symmetry operation for the equilateral triangle but has the same effect as  and hence not included in the set.


If we conduct all 6 symmetry operations on the triangle, we find one ordered triple invariant: the origin. The significance of the invariant point and the two definitions above, together with the definition of a group, leads to an important mathematical group – the point group – which we’ll discussed in the next article.


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