Complex numbers in 6 dimensions
  Two distinct systems of commutative complex numbers in 6 dimensions of the polar and planar types of the form u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4+h_5x_5 are described in this work, where the variables x_0, x_1, x_2, x_3, x_4, x_5 are real numbers. The polar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude \rho, and the polar angles \theta_+, \theta_-, the planar angle \psi_1, and the azimuthal angles \phi_1, \phi_2. The planar 6-complex numbers introduced in this paper can be specified by the modulus d, the amplitude \rho, the planar angles \psi_1, \psi_2, and the azimuthal angles \phi_1, \phi_2, \phi_3. Exponential and trigonometric forms are given for the 6-complex numbers. The 6-complex functions defined by series of powers are analytic, and the partial derivatives of the components of the 6-complex functions are closely related. The integrals of polar 6-complex functions are independent of path in regions where the functions are regular. The fact that the exponential form of ther 6-complex numbers depends on cyclic variables leads to the concept of pole and residue for integrals on closed paths. The polynomials of polar 6-complex variables can be written as products of linear or quadratic factors, the polynomials of planar 6-complex variables can always be written as products of linear factors, although the factorization is not unique. 
Silviu Olariu (National Institute of Physics and Nuclear Engineering,   Tandem Laboratory, Magurele, Bucharest, Romania)