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On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures of 3-dimensional amoebas. Should it look likes thick foam bubbles?

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For a moment I thought this question was about amoebas. –  Andrej Bauer Jan 30 '13 at 12:36
    
In 3-d, you have to be clear whether you're talking about the amoeba of a curve or a surface. If it's a surface, then yes, its complement is a bunch of convex bubbles. –  Allen Knutson Jan 30 '13 at 12:47

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There is now a 3-dimensional amoeba on wikipedia.

You are welcome. The main reason for the difficulty of making such amoebas is that it is the projection of a 4-dimensional surface (the zero set of the polynomial). Finding a parametrization of the zeros set of an arbitrary polynomial in 2 variables is highly non-trivial. The case on wikipedia is linear, thus we may easily parametrize it. Now, this set is projected with $Log|\cdot|$, thus one has to be a bit careful which points to include to make the picture "pretty", since there may be zeros of the polynomial "far" away, and close to 0.

I did not draw the projection of the parametrized 4-dimensional surface, but rather the projection of a lot of points on the surface, chosen in a manner that makes them somewhat evenly distributed.

EDIT: Also, rumour has it that Frank Sottile has some 3-dimensional amoebas somewhere on his web page, but I have not been able to find them.

EDIT 2: Now, amoebas never look like soap bubbles, but the connected components of their complements do. (These components are always convex, aka. soap bubbles)

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The first link is broken, should be en.wikipedia.org/wiki/Amoeba_(mathematics). –  Ketil Tveiten Jan 30 '13 at 12:39
    
Thanks for the answer and that you created a picture of it! –  Gilles Bonnet Jan 30 '13 at 22:58
    
How frustrating, that both links are broken! It is for different reasons, and neither poster is to blame. –  Allen Knutson Jan 31 '13 at 3:19

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