complex exponential and logrithm

Higher-dimensional geometry (previously "Polyshapes").

complex exponential and logrithm

Postby Secret » Mon Apr 09, 2012 1:37 pm

http://www.math.brown.edu/~banchoff/gc/ ... toIInv.mpg

http://www.math.brown.edu/~banchoff/gc/ ... p-RtoI.mpg
Is it just me or did I see cockscrews in the form of a clifford torus?

also must R4 is required for any complex functions to be plotted?
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Re: complex exponential and logrithm

Postby quickfur » Wed Apr 25, 2012 9:06 pm

Secret wrote:http://www.math.brown.edu/~banchoff/gc/script/exp-RInvtoIInv.mpg

http://www.math.brown.edu/~banchoff/gc/ ... p-RtoI.mpg
Is it just me or did I see cockscrews in the form of a clifford torus?

also must R4 is required for any complex functions to be plotted?

I know this is reeeally late... but better late than never, right? This forum's been kinda quiet lately. I should rile it up a bit. :P

First, the videos: These corkscrews are not clifford tori, they are actual corkscrew-shaped 2D sheets in 4D. They're a combination of sine/cosine waves and exponential/logarithm functions. The complex exponential/logarithm is periodic, unlike their real analogues.

Second, yes, 4D is required to plot a complex function, because each complex number has two degrees of freedom (the real part and the imaginary part). So to fully represent a function F, you need 2 dimensions to represent the original number, and another 2 dimensions to represent the resulting number after F acts on it. So that's 4D. And that's one of the reasons 4D visualization is so useful: it lets you see complex functions in their full glory. :-) (This does require being able to visualize 2-manifolds in 4D, though, which I find actually a lot harder than visualizing 3-manifolds because of the way 2D sheets can knot themselves in 4D. Or should I say ways -- 2-manifolds in 4D can knot in really strange ways... like the Klein bottle and the real projective plane.)
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