  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.
