### 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.