The buddhabrot images in these galleries are from 'islands' in the 'tail' of the Mandelbrot set..

Some of the Islands in the image above have arrows pointing to them. There are an infinite number of these islands, each one similar in shape to the largest Mandelbrot set shape. However no two islands are exactly the same, both the mandelbrot set itself, shown in black, and the region around the Mandelbrot set.

These islands are unique as they straddle the real axis of the complex plane, and are thus symmetrical about the real axis.

The process used to create Mandelbrot images, both the classic and the buddha, in these galleries is quite simple. Describing it is somewhat difficult however.

The entire mandelbrot set lies within a radius of 2 from the origin of the complex plane. To determine whether or not a point in the complex plane is part of the Mandelbrot set is not completely possible. Thus all of these images represent approximations of a certain ideal.

To approximate the Mandelbot set a computer program uses a very simple feedback loop. The mathematical way of describing this is Z(new) = Z^2 + C. Z and C are both complex numbers. Z is initialized to zero. C is the point of inquiry in the complex plane. With each pass, or iteration, through this feedback loop a new Z value is created. If the magnitude of Z ever exceeds 2 it is said to have 'escaped' or 'exploded' as its value rapidly grows thereafter. If so this point is not part of the Mandelbrot set

A rectangular region of the complex plane is examined. If the magnitude of Z ever exceeds two that point is not part of the Mandelbrot set. For the classic images this point would be given a color based on how many iterations through the feedback loop were required for it to escape. When sampling a region around an island the resulting classic image will look like the one below:

Once a grid of points are examined an image emerges. Nearly all the complex numbers in the black part of the image, comprising the island, are part of the Mandelbrot set. These points can be iterated forever and their magnitude will never exceed 2. Each new Z created by the feedback loop will remain within two units of the origin. It's the boundary between the black and colored regions which cannot be accurately determined. Some points will take infinity-1 iterations before escaping. Therefore the computer program which creates the image will stop after a certain number of times through the feedback loop. If a point being queried reaches this amount it's colored black and assumed to be part of the Mandelbrot set (classic images). Of course the point may not be part of the Mandelbrot set, simply more itertations were required to determine this.

To create the buddhabrot images a different technique is used. The same feedback loop is used, and all the Z components created during the iteration process are accumulated. There are two components to an imaginary number (real, imaginary) and thus a 2D plane can be used to plot them. Each time through the loop a new Z (x, y point) is created. If that point lies within the region being examined its value increases by one. Only points which escape, and are not part of the mandelbrot set, contribute to the buddhabrot images

An apt analogy would be to consider the escaping points Mandelbrot radiation and as more and more points are accumulated a digital film is slowly exposed. Below we see the buddhabrot develop. It corresponds to the same region of the complex plane as the classic image above. The first image (left)we have sampled 1/32 of the points in the region, then 1/16, 1/8, 1/4 and finally all the points - the far right and brightest image.

What I call the Mandelbrot heart is the brightest region of the buddhabrot along the real axis. These amazing images offer much to the imagination - faces, skulls and spooky aliens. Below is the heart from the buddhabrot above.

I want to emphasizse only a trifling difference exists between a program which creates classic pictures and one which creates buddhabrot pictures.

As you'll see each island's buddhabrot or fingerprint is unique and recognizable. Enoy the galleries!

-Eric Bazan, September 1, 2011

boofreak@gmail.com