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On Nov 16, 2016, at 12:57 PM, James Propp <jamespropp@gmail.com> wrote:

http://pyrigan.com/wp-content/uploads/2016/10/CelebrationOfMind.pdf <http://pyrigan.com/wp-content/uploads/2016/10/CelebrationOfMind.pdf>

First of all he claims that the graph of the function z = xy has constant negative curvature. It does not. Besides the fact that direct calculation proves otherwise, it's a celebrated theorem of Hilbert that there does not exist any complete surface of constant negative curvature in 3-dimensional space, but the graph of that function would be just that if defined over all x and y. And because the function is polynomial, its Gaussian curvature function K(x, y) (for the curvature at the point (x, y, xy) on the graph) must be real analytic. So it can't be constant for some of the surface and non-constant for other parts. ((( Direct calculation gives K(x, y) = -1/(1 + x^2 + y^2)^2 . ))) His manipulation of 3D pieces seems fascinating; I wish I could follow what he's doing, but I don't understand it. —Dan P.S. Can someone please explain this Mathematica output? I cannot. The -uv^2 term ought to be -(uv)^2 and cancel the u^2 v^2 term. ----- In[11]:= mI = {{1+v^2, uv}, {uv, 1+u^2}} 2 2 Out[11]= {{1 + v , uv}, {uv, 1 + u }} In[12]:= Inverse[mI] 2 1 + u uv Out[12]= {{-------------------------, -(-------------------------)}, 2 2 2 2 2 2 2 2 2 2 1 + u - uv + v + u v 1 + u - uv + v + u v 2 uv 1 + v

{-(-------------------------), -------------------------}} 2 2 2 2 2 2 2 2 2 2 1 + u - uv + v + u v 1 + u - uv + v + u v

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