Some Symmetry Operations in Crystallography

Coordinate transformations are commonly used in crystallography to characterize the crystal symmetry. The four symmetry operations that can occur in three-dimensional geometry are rotation, reflection, inversion, and translation. This Demonstration shows rotation and rotation with inversion (roto-inversion). The original coordinate system (labeled , , ) is colored blue and the new coordinate system (labeled , , ) is colored orange. The black arrow is the axis about which the rotation occurs; the direction of the arrow is controlled with the (theta) and (phi) sliders; is the angle between the black arrow and and is the angle in the - plane between the arrow and the axis, where and (both angles are in radians). Rotation operations are described as "-fold" where refers to the number of steps to complete a full rotation. For example, a fourfold rotation means 4 steps of for a full rotation about the axis.