Pi is NOT infinite! (and neither is the mandelbrot)
Is pi infinite? If it is, is it special in this regard? There are literally countless formulae you could use to construct "infinite" numbers, but what does it mean? Surely there is no more information there than is contained by the algorithm for generating it (this applies to fractals too). It's when you try to express it as decimal that you go into an infinite loop.
Here's the idea: Develop a theory of how hard it is to construct patterns that never repeat and in what ways you can do this, in order to gain insight into the supposed infinity of pi. This would *not* be a theory pertaining specifically to generating numbers that expand to infinite places; that would defeat the purpose because irrational numbers are already explored and also the conversion to decimal (or any other kind of number) merely complicates the reasoning behind pi supposedly being "infinite." It's to be merely a theory about the ease or difficulty of constructing infinite, non-repeating patterns in general.