Lattès Example

I gave a DynaLite on the Lattès example last week and given the shortage of information on the web I'm posting my notes. This is a more algebraic geometry perspective than those offered in Beardon and Carleson/Gamelin, it somewhat follows insight from Milnor. Here are my notes which are somewhat out of order, and here is a helpful picture of a torus generated with the following tikz code. The clip function is really handy, and you can put things outside the clipped box by just inserting the code for them prior to the clip command.

\begin{tikzpicture}[scale=1.5]

      %Clip

      \clip (-3,-3) rectangle (3,3);

      

      %Horizontals

      \foreach \x in {-2,-1,...,2}

 \draw[-] (3.000000,\x) -- (-3.000000,\x);

    

      %Slants slope = .61825

      \foreach \x in {-5,-4,...,5}

 \draw[-] (2+\x,2*1.61825) -- (-2+\x,2*-1.61825);

 

      %Fills

      \fill[green!20!white] (0,0) -- (1,0) --   (1.61825,1) --(.61825,1) --cycle;

      \fill[blue!20!white] (0,0) -- (-1,0) --   (-1.61825,-1) --(-.61825,-1) --cycle;

      %axis

      \draw[<->][red] (3,0) -- (-3,0);

      \draw[<->][red] (0,3) -- (0,-3);

\end{tikzpicture}