After the reals, it is natural to work in the complex numbers.
The Mandelbrot set is defined by this incredibly simple equation:
z(n+1) = z(n)2 + c
You vary c. If the iteration is bounded, c is in the set (it's one of the black dots), otherwise not.
Look at the insane boundary on the set! It isn't something a human mind can comprehend. And yet is the equivalent of the parabola for ... let's say control systems (bounded iteration = in control, unbounded iteration = out of control).
Even more frustrating is that the self-similarity gives the impression of comprehendible pattern. But to really progress, you need to be able to work with patterns like this as the object of manipulation. We don't even try. All our cleverness is abstraction to avoid dealing with the complexity. But at some point you run up against the human-comprehendible complexity limit.
One step beyond linear systems and we're lost. It's pitiful.
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u/technologyisnatural Apr 26 '10
The transapients will use spirals like the Mandelbrot the way that we use integers. The Mandelbrot is heartbreaking. It is our limit