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).