The square reduces to since the basis vectors themselves square to +1. This result holds generally for all bivectors, and as a result the bivector plays a role similar to the imaginary unit. Geometric algebra uses bivectors in its analogue to the quaternion, the "rotor", given by where is a unit bivector that describes the plane of rotation. Because squares to −1, the power series expansion of generates the trigonometric functions. The rotation formula that maps a vector to a rotated vector is then  where is the "reverse" of formula_72 (reversing the order of the vectors in formula_73 is equivalent to changing its sign). Example. A rotation about the axis can be accomplished by converting to its dual bivector, where is the unit volume element, the only trivector (pseudoscalar) in three-dimensional space. The result is  In three-dimensional space, however, it is often simpler to leave the expression for , using the fact that commutes with all objects in 3D and also squares to −1. A rotation of the vector in this plane by an angle is then  Recognizing that and that is the reflection of about the plane perpendicular to gives a geometric interpretation to the rotation operation: the rotation preserves the components that are parallel to and changes only those that are perpendicular. The terms are then computed:  The result of the rotation is then  A simple check on this result is the angle . Such a rotation should map to . Indeed, the rotation reduces to  exactly as expected. This rotation formula is valid not only for vectors but for any multivector. In addition, when Euler angles are used, the complexity of the operation is much reduced. Compounded rotations come from multiplying the rotors, so the total rotor from Euler angles is  but These rotors come back out of the exponentials like so:  where refers to rotation in the original coordinates. Similarly for the rotation, Noting that and commute (rotations in the same plane must commute), and the total rotor becomes  Thus, the compounded rotations of Euler angles become a series of equivalent rotations in the original fixed frame. While rotors in geometric algebra work almost identically to quaternions in three dimensions, the power of this formalism is its generality: this method is appropriate and valid in spaces with any number of dimensions. In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in.