Mandelbrot's Bug

Chaos describes the way most of us see nature. It could be the shape of clouds, the profiles of mountain ranges, or the delicate patterns of crystalline snowflakes. Long ago, mathematicians had given up describing these because they seemed to possess no order.

Then along came Benoit Mandelbrot. An immigrant mathematician working for IBM, he had long been obsessed with patterns of disorganization. Problems with computer communication at IBM mirrored the patterns he saw elsewhere. He set out to bring order to chaos.

The Bug

This is a graphical representation of the Mandelbrot Set. The color at each particular point in the plane represent the level of difficulty in determining whether that point is in the set or not.

Points colored black repesents points included in the set. On the edges, it takes fewer tries to conclude that these points are outside the set; Less tries are first violet, then blue, then green, etc,etc to orange. Here, at the far reaches, we skip to light blue tones to make clear the contours of the profile.

Note how the primary shape is repeated in various incarnations. Click on the picture to see an expanded version. Even in the greatest detail, the patterns repeat.

Notch

This is the notch area between the two half’s of the bug. Once again, note how the frill is repeated in ever decreasing size. Yet the general shape is similar.

Cusp

This is another view of the notch in greater detail. Here we can see the shades of blue; draker blue representing relative ease in determining that these points are outside the Mandlebrot Set. Black is clearly within the set. We color the pixel black if, after 256 passes, the orbit is still contained.

(-.75,.25j),.500