Algorithms as fractals
Blum, Shub, and Smale proved that membership in the Mandelbrot set is undecidable in the Real RAM model of computation (known in some upstart circles as the BSS model).  

The high-level argument is one sentence long: Any Real RAM computable set is the countable union of semi-algebraic sets, so its boundary has Hausdorff dimension 1, but the boundary of the Mandelbrot set has Hausdorff dimension 2.  By the same argument, almost every interesting fractal is uncomputable in the real-RAM model.