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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