Where the world ceases to be the stage for personal hopes and desires, where we, as free beings, behold it in wonder, to question and contemplate, there we enter the realm of art and science. If we trace out what we behold and experience through the language of logic, we are doing science; if we show it in forms whose interrelationships are not accessible to our conscious thought but are intuitively recognized as meaningful, we are doing art. Common to both is the devotion to something beyond the personal, removed from the arbitrary. A Einstein
coastline, into which you can zoom, almost without limit, dividing and subdividing it into smaller and smaller parts, each of which is a clone of the original whole.
My post concerns fractal plots, of which the black beetle (Mandelbrot) shown above is probably the most iconic. By zooming in on the first section of the antenna protruding from its head you are rewarded with
the plot following it (above left). Zoom in on that and the next appears... and so on ad infinitum. These, then are mathematical plots, obtained from an equation by repeatedly applying a process (a
function or formula)to that equation, often hundreds, even thousands of times. The colour is applied to show how many times the function had to be applied for the plot to arrive at
that particular point. Because of the vast numbers of calculations required these plots could not be realised before the advent of computers. (Mathematicians knew the plots were there. What they had not realised was how beautiful they would prove to be.) Now anyone can produce them. Those above I produced using Chaospro, one of the many fractal generators available free on the web.
The images below are from the Julia sets. These are found by exploring the periphery of the Mandelbrot.
Thoughts on a Fractal Plot
There's something in the way infinity
evolves from repetition and goes on
to cosy up to what is local and well-loved.
It has to do with how a line repeats itself
and yet surprises you, how little gasps
of disbelief greet what was sure to be.
It has to do with laws of nature laid
in stone, and flexible as oil, though hid
in folds away from human thought and prying eyes.
We've seen it the way that men like Koch
and Mandelbrot set out to map their world,
who gave geometry its extra twist, who said:
there's something in the complex plane - a void,
a vacancy, a darkness, an abyss -
whereby infinity bleeds back into the world.
And where to us it seems most whimsical,
in sudden exit or return, it tells in truth
of discipline and order writing home.
It had to do with clouds, their borders
and their surfaces, when Gaston Julia had yet
to intuit how beauty slept
with storm and chaos in a geometric set,
how Adam's single fruit upon its single tree
would bifurcate to great complexity.
It had to do with how the new (non-linear)
would wipe the lines that Euclid drew, and how
a simple plot could crown such turbulence as queen.
There's something in oneself that echoes this,
first turns it like a wooden bowl upon a lathe,
then fills it with earth's most delicious fruits.