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        general >> complex analysis >> Complex lines in three dimensions
        
(Message started by: Benoit_Mandelbrot on Jan 15^th, 2004, 9:56am)

Title: Complex lines in three dimensions
Post by Benoit_Mandelbrot on Jan 15^th, 2004, 9:56am

I've been anylizing the connection between complex numbers and functions, and I've found a way to plot complex numbers in three dimensions. This would be a 3d parametric function [t,real(f(t)),imag(f(t))]. The z-axis would introduce the imaginary part of the output number, adding three dimensions to functions. This would be a curved line, not a surface.

Update: OOps! my bad! I ment to put in the functions, but I must have like not somehow.

Title: Re: Complex lines in three dimensions
Post by towr on Jan 15^th, 2004, 1:21pm

I don't really see much of an advantage over simply plotting the real and imaginary part seperately in a 2D plot. Not in the last place because our screens are still not 3D (unlike what sci-fi has been promissing us for years)

Going into the 3D dimension would be more helpfull for plotting [bbc] to [bbc] functions though, since I'd be hard pressed to put the input on one axis in this case.
so we'd have [Re(x), Im(x), Re(f(x))] and [Re(x), Im(x), Im(f(x))] in the same plot. This however would give two surfaces.. (But that's to be expected with a two dimensional input)

Title: Re: Complex lines in three dimensions
Post by Icarus on Jan 15^th, 2004, 4:41pm

Perhaps I am misunderstanding, but I don't see what your point is here. If I understand you correctly (if t has real and imaginary parts, you can't plot it as one of your three coordinates like you show), what you are actually talking about is graphing a function
f : [bbr] [to] [bbc] : t [mapsto] f(t)
as the 3D-curve {(t, re(f(t)), im(f(t))) | t [in] [bbr] }.

Despite towr's doubts, there are times when it is handy to plot curves into [bbc] this way rather than as a parametric curve in the plane. But this is hardly new. People have been doing exactly this practically from the days of DeCartes.

Title: Re: Complex lines in three dimensions
Post by Benoit_Mandelbrot on Jan 16^th, 2004, 8:47am

Yea, I ment to do [t,real(f(t)),imag(f(t))], not to do the silly one I did before. :o

Hey, I'm only 18 and still in high school. I don't have a phd in mathematics just yet, so I don't know everything about math.

Title: Re: Complex lines in three dimensions
Post by Icarus on Jan 16^th, 2004, 3:26pm

Unfortunately it's true that just about every simple good idea someone else has already had some time before. Math grows at the edges. It's extremely rare to come across something undone in the middle.

But I asked because I was unsure that I had interpreted your comments correctly.