Mapping Complex Dimensions to 3D Surface Visualization

[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.

 

[jvicens]

Thank you for sharing your project and the challenges you are facing in visualizing a surface in complex dimensions 2. This is a very interesting topic and I would be happy to offer some suggestions on how to approach this problem.

Firstly, it is important to understand that visualizing surfaces in complex dimensions 2 is not a trivial task. In fact, it is not possible to directly visualize a surface in 4D as our brains are limited to comprehending 3 dimensions. However, there are ways to represent and approximate these surfaces in a way that can be visualized on a computer screen.

One approach to visualizing a surface in complex dimensions 2 is to use a technique called stereographic projection. This involves projecting the 4D surface onto a 3D space, which can then be visualized on a 2D screen. This method has been used in various fields such as mathematics, physics, and computer graphics.

Another method that could be used is to represent the 4D surface as a series of 3D cross-sections. This can be achieved by fixing one of the complex dimensions and varying the other one to produce a 3D slice of the surface. By repeating this process for different values of the fixed dimension, you can create a series of 3D cross-sections that can be visualized together to give an approximation of the 4D surface.

Regarding your question about using the equation z^5_1 + z^5_2 = 1 and mapping it to 4 real dimensions, your approach of using z_1 = x+iy and z_2 = z+iw is a valid method. However, it is important to note that you will need to define the ranges of x, y, z, and w in order to produce a meaningful visualization.

I would also like to suggest exploring other visualization techniques such as color mapping and texture mapping, which can help in representing complex surfaces in a more intuitive way.

In conclusion, visualizing surfaces in complex dimensions 2 requires a combination of mathematical understanding and creative visualization techniques. I hope the suggestions I have provided will be helpful in your project. Best of luck with your research!

 

[tacman]

Mapping complex dimensions to 3D surface visualization can be a challenging task, but it is definitely possible. The quintic equation you have chosen is a good starting point, as it is a simple and well-known equation that can produce interesting surfaces.

The algorithm you have mentioned, Marching Cubes, is a commonly used technique for generating 3D surface plots from algebraic equations. However, as you have discovered, it may not work well for equations with complex dimensions. This is because the algorithm relies on intersecting a grid of cubes or hypercubes, which may not accurately capture the behavior of a surface in complex dimensions.

One approach to solving this issue is to use a different algorithm that is specifically designed for complex dimensions. There are several such algorithms available, such as the marching squares algorithm for 2D surfaces and the marching tetrahedra algorithm for 3D surfaces. These algorithms use similar principles to Marching Cubes, but are adapted for complex dimensions.

Alternatively, you can try to transform your complex dimensions into real dimensions, as you have suggested in your last question. This approach can work, but it may not accurately represent the behavior of your surface in complex dimensions. It is important to keep in mind that complex dimensions have unique properties and behaviors that cannot always be accurately represented in real dimensions.

Overall, the best approach for mapping complex dimensions to 3D surface visualization will depend on the specific equation and surface you are working with. It may require some experimentation and testing to find the most suitable method. I would recommend exploring different algorithms and techniques, as well as consulting with experts in the field, to find the best solution for your specific project.