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Feather Boa type 2,1 with p1 = (1.5,0.0) and p2 = 0.5, 0.0)

"Feather Boa" type fractal image



For a long time, I have been fascinated by some of the more obscure branches of mathematics, and by the way that simple mathematical relationships can produce complex and beautiful images. When it became possible to use computers to generate fractal images, I was hooked, and have spent many hours exploring their possibilities.

I use "fractint.exe" as my preferred fractal image generation program. Click here for the Fractint home page where you can download the latest version and view the work of many others with an interest in this area: Fractint Home Page.

One of the options in Fractint is to write your own fractal generation formula. I have developed a set of 12 formulae that I call "Feather Boas". If you want to download this formula file, click here. feboa.frm

The Feather Boas are based on a formula set that has not, so far as I can determine, been explored for fractal images before, (I can find no reference in any area accessible from the fractint home page). The formula set has the general form:

z = a:
z = (b +/- z) ^ (c +/- z)
0.5 <= |z| <= 2

  • 'a', 'b' and 'c' take the various combinations of 'pixel', 'p1' and 'p2'.
  • For each such combination, there are four equations with the different permutations of '+' and '-', ie: '+ +', '+ -', '- +' and '- -'.
The resulting images have a complexity and beauty which is unexpected for such simple formulae. They have the characteristics of fractals, (similarity with infinite magnification), but also dramatically change shape and form with very small parameter changes.

I have called these fractals 'Feather Boas' because many of the fractal images contain long frothy, frondy, repeating elements that are reminiscent of the ostrich plumes so beloved of Edwardian ladies.

In Fractint, I use the 'integer' solution method, as the 'floating point' version seems to produce errors. Quite small iteration limits, (eg 64), often produce highly complex images, however I tend to use 256, and set the screen driver for a 256 colour option so as to maximise the colour usage.

Most of the fractals are symmetrical about the x-axis as long as both p1 and p2 are real; if either are imaginary, assymmetries appear.

The fractals tend to feature islands, and many have a high degree of fragmentation so 'boundary-seeking' techniques are not generally appropriate.

It is worth zooming out on many of the fractal images as some unexpected large-scale effects occur.

As with many fractals, very small changes in the values of p1 and p2 often produce dramatic changes in the image. Some values that produce interesting images are shown here; please try others for I have by no means explored the whole domain.

Each thumbnail is a quarter of the actual image size; click on it to see the full-sized version:

For Feather Boa 1,1: z = pixel: z = (p1 + z) ^ (p2 + z):

p1 = (1.5, 0.0); p2 = (-0.48, 0.3) p1 = (1.5, 0.0); p2 = (-0.5, 0)

For Feather Boa 1,2: z = pixel: z = (p1 + z) ^ (p2 - z):

p1 = (-0.3, 0.0); p2 = (-0.1, 0.0) p1 = (1.5, 0.5); p2 = (1.5, 0.5)

For Feather Boa 1,3: z = pixel: z = (p1 - z) ^ (p2 + z):

p1 = (1.31, 0.01); p2 = (-1.14, 0.1) p1 = (2.0, 0.0); p2 = (0.0, 0.0)

For Feather Boa 1,4: z = pixel: z = (p1 - z) ^ (p2 - z):

p1 = (1.0, 0.0); p2 = (0.5, 0.0) p1 = (1.8, 0.0); p2 = (1.6, 0.0) p1 = (2.0, 0.0); p2 = (1.0, 0.0)

For Feather Boa 2,1: z = p1: z = (p2 + z) ^ (pixel + z):

p1 = (1.4, 0.01); p2 = (-0.67, 0.0) p1 = (1.0, 0.0); p2 = (0.5, 0.0) p1 = (-1.0, 0.0); p2 = (-2.0, 0.0) p1 = (0.0, 0.45); p2 = (0.0, 1.93) p1 = (0.0, 0.45); p2 = (0.0, 1.93)

For Feather Boa 2,2: z = p1: z = (p2 + z) ^ (pixel - z):

p1 = (1.3, 0.0); p2 = (-0.3, 0.44) p1 = (0.75, 0.0); p2 = (0.5, 0.0)

For Feather Boa 2,3: z = p1: z = (p2 - z) ^ (pixel + z):

p1 = (0.95, 0.1); p2 = (0.53, 0.1) p1 = (0.75, 0.0); p2 = (0.5, 0.0)

For Feather Boa 2,4: z = p1: z = (p2 - z) ^ (pixel - z):

p1 = (0.75, 0.0); p2 = (1.92, 0.0) p1 = (0.25, 0.0); p2 = (0.7, 0.0)

For Feather Boa 3,1: z = p1: z = (pixel + z) ^ (p2 + z):

p1 = (-0.49, 0.0); p2 = (0.18, 0.0) p1 = (-0.5, 0.0); p2 = (-0.96, 0.0) p1 = (1.1, 0.0); p2 = (-1.0, 0.0)

For Feather Boa 3,2: z = p1: z = (pixel + z) ^ (p2 - z):

p1 = (0.5, 0.0); p2 = (0.98, 0.0) p1 = (0.5, 0.0); p2 = (-0.5, 0.0) p1 = (-0.3, 0.0); p2 = (-0.3, 0.0)

For Feather Boa 3,3: z = p1: z = (pixel - z) ^ (p2 + z):

p1 = (0.55, 0.0); p2 = (-2.55, 0.0) p1 = (0.0, 0.45); p2 = (0.0, 0.63) p1 = (0.97, 0.0); p2 = (0.23, 0.0) p1 = (1.85, 0.0); p2 = (0.6, 0.0)

For Feather Boa 3,4: z = p1: z = (pixel - z) ^ (p2 - z):

p1 = (0.75, 0.0); p2 = (0.0, 0.0) p1 = (0.75, 0.0); p2 = (0.95, 0.0) p1 = (-0.25, 0.0); p2 = (0.0, 0.0) p1 = (-.25, 0.0); p2 = (0.25, 0.0)



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This page last updated on 2024-06-22 JGM