# Complex infinity at a point and division by zero

Edit: I’ve found out that the book was written for Mathematica 7, which was a pretty long time ago. It boils down to changes in syntax most probably, but simple renaming to lower lettercase does not work.

Following Stan Wagon’s Mathematica in Action, chapter 19, subsection 19.2, I’ve run into problems.

1. I cannot redefine ReIm[z] as is done on p.496, Mathematica just states that Tag ReIm in ReIm[z_] is protected.
2. Trying the same procedure on that page and over the course of the following two pages with a function that I’ve called reim[z], it’s not possible to get the hyperbolic triangle.
3. I’m then left to use ReIm[z] for the triangle, which will work regardless of whether I “redefine” or not.
4. Those four definitions of LFT functions and turning off the division-by-zero message off, nothing happens once again.
5. And then, regardless of what I do in the previous 4 steps, I can not get the tessellation shown on p.498. Instead, I get one of two shown below.

What I think happens is that at the time of writing the book, ReIm was not a legitimate function in Mathematica. It was probably implemented sometime afterwards and now it inadvertently affects this code as well. Is it possible to “add on” to a predefined definition in Mathematica? Or to somehow bypass these errors with a new function?

The problematic ReIm[z] part:

ReIm[z_]:=N[{Re[z], Im[z]}];  ReIm[ComplexInfinity]={0,1000};  Attributes[ReIm]=Listable; 

The LFT (Linear Fractional Transformation) involving ReIm[z] which seems to do nothing, together with the turning off of the errors:

LFT[mat_List][z_?NumericQ] := reim[Divide @@ (mat - {z, 1})]; Off[Power::infy, General::dbyz, Divide::infy]; 

The most problematic part of the code:

polys = Table[{FaceForm[Hue[Random[], 0.6]],      Polygon[LFT[w][triangle[]]]}, {w, G}]; 

^This lists out errors of the type: “\emph{Indeterminate expression $$\frac{0}{0}$$ encountered}.” While thisĖ gives one of the two attached pictures:

Graphics[{EdgeForm[Black], polys}, PlotRange -> {{-3, 4}, {0, 2.4}},   Frame -> True, FrameTicks -> False]