A) The Surface Geometry of Imaginary Logarithms
As with the complex natural log function, complex imaginary logs also exist as a four dimensional function except oriented perpendicular to the horizontal plane, i.e. in the positive imaginary or ‘depth’ direction.
B) Surface Functions
Going beyond the usual ‘equation of a sphere,’ a rectangular complex region is input to the function and results in a mapping of that region into the surface of a sphere for any radius. This forms a four coordinate function, 
for the surface of a sphere.
Inverses values for the sphere function can be found by:
1)Taking the natural logs of both sides.
2)Setting the real and imaginary parts equal to each other.
3)Solving for 
An extension of the ideas allows for both a ‘Cube’ Surface Function:
And, in addition to many others, a Hexagon Surface Function looking much like the one on the planet Saturn:
C) Object Wave Duality Surfaces
The sphere above uses 
on the horizontal or 
-axis. I.e.
If, instead, 
is used as the coordinate for :
A remarkable result occurs, an associated helicoid wave. This occurs for many surface objects, both closed and open:
The interval determines the overall length of the wave.
D) Circular Surface Functions
In addition to the circular torus surface functions pictured on the e-book cover
Many more circular surfaces are possible with a parametric approach:
For example, a circular helicoid surface function:
Torus surfaces are typically of a circular internal shape and a circular orbit. They may be made to have virtually any inner shape and orbit. Triangular, square, looped, pentagonal etc. The same is true of the helicoid above and others.
