4D surface

1. Nov 18, 2006

Neoliten

I am trying to do a computer visualization of a surface in complex dimensions 2. I choose simple quintic equation:
$$z^5_1 + z^5_2 = 1$$

I also implemented algorithm for producing plots of 3D surfaces that are defined with algebraic equations. It is called Marching Cubes and it simply checks how surface intersects a grid of cubes, so it produces a triangles to approximate the surface. The same technique could be used for 4D, where instead of cubes, it uses hypercubes and extracts tetrahedras to approximate surface. I tested algorithm for hyperspheres and it works perfectly, however this equation (mentioned above) gives strange results.

Generally, in algorithm, it should iterate through all 4 coordinates (x,y,z,w) to produce plot, like 0.0 < x < 1.0, 0.0 < y < 1.0, 0.0 < z < 1.0, 0.0 < w < 1.0. The problem is, that I have two complex dimensions and I try to map them to 4 real dimension to produce a surface.

How can I do this and what methods could be used? Or could it be simply accomplished by:
$$z_1 = x+iy$$
$$z_2 = z+iw$$?

Thank you.

Last edited: Nov 18, 2006