Right, the picture does not match the title. In fact it is the opposite of the title.

You likely have seen fractal images that look like the above. Especially in older books or on old websites. I cannot say why the pictures were produced like this. Perhaps the authors liked the strong contrast between colors, or maybe it was a limitation of their software, or they may have been demonstrating a mathematical property. Whatever the reason, many older fractals look like this.

I created the image to provide a starting point for a series of technical posts on Smooth coloring in fractals. There is something artistically intriguing about the hard contrast of black and white. I want to explore that in the future. But today’s post is about some technical aspects of creating fractal images.

The next few post will be somewhat technical. They should still be accessible to everyone. But beginner level knowledge of escape time fractals, of which the Mandelbrot set is the most famous, is assumed. The introduction in Mandelbrot set and the section on Escape Time Coloring will provide sufficient background.

The output of the fractal calculation is an integer, the number of iterations before the orbit exceeds a threshold escape value. Ignore the details in that definition, all that really matters is that the result is an integer. You get discrete, non-continuous values. Not continuous values like real numbers.

The coloring algorithm assigns a color to each value, since the input is discrete, the output, the colors are also discrete. In this case even numbers are black and odd numbers are white (or maybe vice-a-versa, I do not remember).