(discovered around 1989)
Normally the Mandelbrot Set is displayed by plotting a color at each pixel of the image, where the
color of the pixel is a mapping of the output of the equation iterated at a location in the
complex plane represented by that pixel, with pixel x and y representing the real and imaginary
numbers. After the equation is iterated, the color plotted represents the final value of Z
(either the magnitude, or whether the magnitude is larger than or less than 2 - or some variation
on this information). But there are other variations on what could be plotted. Take for instance
the amazing life-history that the value takes as it bounces around a region
of the complex plane - forming a jig-jaggy squiggle - before it's magnitude it taken. The
software that I wrote to generate Mandelbrot-like images was set up to display these
squiggles - which transformed as one moved the mouse cursor over regions of the Mandelbrot Set.
I noticed something about these squiggles - each one was uniquely different.
I asked myself what was unique about the qualities of these squiggles that was
not captured in the mere value of the magnitude of Z after some number of iterations. The answer had to
do with the
qualitative visual effect of the squiggles. Each squiggle could be somewhat characterized by examining the
angles (changes in direction) of the value as it bounced around. By averaging all the angles in a squiggle,
I was able to extract a value which was, to some extent, characteristic of that squiggle (though not perfectly unique).
It was a number which was more about the character of the iteration
as it was happening, before the "conclusion" (the resulting magnitude of Z).
Because this way of displaying the Mandelbrot Set showed something about the life-histories of the values as
the equation was iterated at each location, and because the
resuting images seemed
to evoke a hidden something underneath, I called them "X Rays" of the Mandelbrot Set. The images shown
above are all variations on X-rays of the Mandelbrot Set.
(for similar treatments to Mandelbrot-like images, see Clifford Pickover's book "Computers, Pattern, Chaos, and Beauty",
and other books by him about math and art. His web page is:
(c) copyright 2001 Jeffrey Ventrella