Life on the complex plane


The image of the Mandelbrot set is a map — a 3rd person perspective survey — of Julia sets. But each image of the Julia Set is a 1st person perspective on the same space as that described by the image of the Mandelbrot set.

And each neighboring point in the Mandelbrot set describes the whole differently, sometimes subtly but sometimes drastically. This difference is unpredictable but somehow in retrospect unfailingly intuitively perfect.

Each Julia set is a subjective impression of the whole, processed according to an accident of birthplace, which resembles the whole to some degree, contains the whole, overlaps with it, but fails to trace it out with reliable accuracy.

What is the space in which the Julia and Mandelbrot sets are situated? It is called “the complex plane” — a two-dimensional space, with a continuum of real numbers extending horizontally, and a continuum imaginary numbers extending vertically.

The heart of the process that generates both sets is Zn+1=Zn2+C — with C being a real plus imaginary coordinates of the point in question. So, if the starting point is 0.1011 on the horizontal axis (the real numbers) and 0.9563 on the vertical axis (the imaginary numbers), C would be 0.1011+0.9563i.

In the generation of the Julia set, the Z jumps all over the complex plane painting a whole like a skillful painter developing a composition. In the generation of the Mandelbrot set, the image proceeds systematically, point by point —  a sociologist doing a study on how long painters take to complete their respective work. The plotter of the Mandelbrot set walks from painter to neighboring painter (from C to C, for instance from 0.1011+0.9563i to 0.1011+0.95630000001i ), stopwatch in hand, timing how long it takes for the painter to walk away from his canvas dripping paint into the infinite corners of the universe-heaven complex, or, alternatively descends into apparently interminable frittering refinement.

Depending on where the process starts, not roughly but infinitely precisely, the picture of the whole is potentially radically divergent, and it impossible to know where it will go and how it will conclude except by patiently tracing it out, much as it is impossible to know how we will be changed from an experience of learning except by living it out.

I’ve been thinking this thought for more than a decade, and occasionally saying bits of it here and there, but today I just needed to get it out.

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