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I was wondering about the best way to visualize complex functions. As they're $$ {\mathbb R}^2 \rightarrow {\mathbb R}^2\ ,$$ I think best way are complex plane image/grid transforms like they used in the Dimensions movie (part 6) or here. Now, is there is a geometric surface (which you could plot with 3dplot) for every complex function which when projected renders the grid transform? (For $$z \rightarrow\ z + k/z $$ this would probably have to cut or overlap itself.) Also what is the further relationship between geometry and complex numbers? And maybe someone knows what software they used for the animations. Is there any sw along these lines able to visualize your own algebras (vs. functions), i.e. not just $(ac - bd), (ad + bc)$ for multiplication.

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    Anyway, you can visualize a function $\mathbb R^2\to\mathbb R$ as a surface in $\mathbb R^3$, but a function $\mathbb R^2\to\mathbb R^2$ would have to be a surface in $\mathbb R^4$. People sometimes visualize the real part, the imaginary part, and/or the absolute value of the function as separate surfaces.2012-09-02
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    I've gotten this far. The question i.a. is how to represent the grid transform in 3D. The modulus plot doesn't contain all information.2012-09-03

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