Rotation
The turning of an object or coordinate system by an angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's rotation theorem states that an arbitrary rotation can be parameterized using three parameters. These parameters are commonly taken as the Euler angles. Rotations can be implemented using rotation matrices.
Rotation in the plane can be concisely described in the complex plane using multiplication of complex numbers with unit modulus such that the
resulting angle is given by 
. For example, multiplication by
represents a rotation to the right by 
and by
represents rotation to the left by 
. So starting
with 
and rotating left twice gives 
,
which is the same as rotating right twice, 
,
and 
. For multiplication by multiples
of 
, the possible positions are then concisely
represented by 
, 
, 
, and 
.
The rotation symmetry operation for rotation by 
is denoted "
." For periodic
arrangements of points ("crystals"), the crystallography
restriction gives the only allowable rotations as 1, 2, 3, 4, and 6.
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