Singular Point and Translational Symmetry

In an asymmetric figure, each point is singular and the multiplicity of each point is one. 

Perioilicity Dimensionality n = 0, no periodicity n = 1, [periodicity in one direction n = 2, periodicity in two directions n = 3. jxrriodicity in three directions 

The notation of the symmetry center or inversion center is 1 while the corresponding combined application of twofold rotation and mirror-reflection may also be considered to be just one symmetry transformation. The symmetry element is called a mirror-rotation symmetry axis of the second order, or twofold mirror-rotation symmetry axis and it is labeled 2. Thus, 1 = 2. 

The twofold mirror-rotation axis is the simplest among the mirror-rotation axes. There are also axes of fourfold mirror-rotation, sixfold mirror-rotation, and so on. Generally speaking, a 2 -fold mirror-rotation axis consists of the following operations a rotation by (360/2//) and a reflection through the plane perpendicular to the rotation axis. The symmetry of the snowflake involves this type of mirror-rotation axis. The snowflake obviously has a center of symmetry. The symmetry class m-6 m contains a center of symmetry at the intersection of the six-fold rotation axis and the perpendicular symmetry plane. In general, for all m n m symmetry classes with n even, the point of intersection of the //-fold rotation axis and the perpendicular symmetry plane is also a center of symmetry. When n is odd in an m-n m symmetry class, however, there is no center of symmetry present. 

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