AN INVESTIGATION OF SYMMETRY OPERATIONS WITH CLIFFORD ALGEBRA

The geometric algebra produces the new elds of view in the modern mathematical physics, denition of bodies and rearranging for equations of mathematics and physics. The new mathematical approaches play an important role in the progress of physics. Wessel, Argand and Gauss used the complex numbers in the solutions of two-dimensional problems. The exponential form of complex numbers is useful in the theory of rotational motions. The quaternion algebra, which was dened by Sir W. R. Hamilton, was generalized for the three dimensional complex numbers [1]. The quaternion algebra is the Clifford algebra of the twodimensional anti-Eucliedean space. Quaternions in the three-dimensional spaces have more useful appearances for the subalgebras of Clifford algebra. We know that Grassmann was affected from Hamilton. This can be easily understood from Grassmann’s studies. In the n-dimensional spaces, Grassmann carried on the studies for the multi-dimensional bodies and dened the central product, which includes the both interior and exterior products. The Grassmann’s central product is the Clifford product of vectors. This result was found by Grassmann independently from Clifford. Later, Clifford tried to combine the Grassmann’s algebra and quaternions in a mathematical system. Then this study, which was entitled Application of Grassmann’s Extensive Algebra, was published [2]. Today, Clifford algebra has an important role in the investigations of the symmetry properties of systems, crystallography, molecular and solid state physics. The method of point groups in the multi-dimensional spaces is derived by transforming in to the parameters of the reections and possible rotational operations. In the three-dimensional spaces, Altmann showed that the Euler’s angles are not useful for the rotational operations but the Euler-Rodriques’ parameters are more advantageous [3]. To know the rotational pole and