Re: Drawing a 3D representation of complex Fermat curve x^n + y^n = 1

• To: mathgroup at smc.vnet.net
• Subject: [mg74794] Re: Drawing a 3D representation of complex Fermat curve x^n + y^n = 1
• From: "Sebastian Meznaric" <meznaric at gmail.com>
• Date: Thu, 5 Apr 2007 04:14:24 -0400 (EDT)
• References: <euvn6c\$f1k\$1@smc.vnet.net>

On Apr 4, 1:21 am, "Ben" <benlo... at gmail.com> wrote:
> On page 233 of my edition of Simon Singh's book Fermat's Enigma, there
> is a figure (Figure 20) of two surfaces.  The caption says that they
> are "geometrical representations of equation x^n + y^n = 1, where n=3
> for the first image and n=5 for the second.  here, x and y are
> regarded as complex variables."  My question is what projection was
> used to represent a complex surface in complex 2-space as a surface in
> real 3-dimensional space.  If anyone knows some Mathematica code that
> generated these images, I'd love to see it.
>
> Many thanks,
> -Ben Lotto

I have not seen the image in the book. But here is what I would do: If
you treat one variable's real and imaginary part and put them on the
cartesian x and y axes and put say the other variable's real part on
the z axis and treat imaginary part as time then you can create a
parametric curve in 3D space that moves with time. If you collect all
the curves together then what you get is a surface in 3D. To plot the
moving curve in Mathematica you would wrap Do around the
ParametricPlot3D where ParametricPlot3D is called with one parameter
(say z) and time is evaluated externally. If you, instead of using it
as a parameter to Do, put time as the second parameter to the
ParametricPlot3D you will obtain the surface representing the
collection of points. This works because the equation actually
represents a curve in the complex-2 space rather than a surface. So
reducing the dimension of the space increases the dimension of the
geometric object creating a surface. In order to do this you will need
to express your equation in an explicit form so for n>1 you will have
to produce n functions that solve your equation, plot each separately
and then combine the plots with Show.

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