There are four transformation options: rotation, rotation with inversion (roto-inversion), reflection, and inversion. The vector defines the direction about which the rotation occurs or the direction normal to the plane of reflection (depending on the transformation type selected). Rotation operations are described as "-fold", where refers to the number of steps to complete a full rotation. For example: a 4-fold rotation means 4 steps of for a full rotation about the axis. The a-matrix is a matrix and the alpha-matrix is ; the elements of the a-matrix are used to calculate the alpha-matrix. Both matrix types are used for coordinate system transformations. For example, a matrix can be transformed to a new coordinate system by the a-matrix with the following formula . The alpha-matrix can be used in a similar manner for a matrix that can be transformed using both the a-matrix and alpha matrix by .