Visualizing a Calabi Yau

Solution 1:

The surface Hanson visualized is a three-dimensional object in a two-dimensional complex space consisting of four real dimensions. Don't think of it as separate real and imaginary parts, just four variables with one constraint to define the surface.

The renormalization $ z_5=(-1/3)^{1/5} $ will produce Hanson's equation from what you have. Then he chooses three variables for the projection, in his case the two real parts and a linear combination of the imaginary parts. The fourth variable, a linearly independent combination of imaginary parts, is ignored. This is an orthogonal projection, like how the sun casts shadows when directly overhead.

As to why the two images on Hanson's website look different, it's because one is rotated relative to the other. You can see that in a fully interactive visualization like the one here.