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).

The final step in this zoom set. This is the long filament extending from the nose of the mini in the previous post. Zoom is 6.67x10^-9.

Here is a magnification of an area of the lower part of the previous post, a little left of center. Coordinates -0.74910617+0.06947989i. The post has a zoom factor of 4.27x10^-7. My program defines zoom differently than conventional usage. Perhaps I should call it something else. The number is the distance from the center to the left and right sides of the view port. If zoom is 1, and center is 0, then the horizontal display area ranges from -1 to +1. I find this convention much easier to work with. (And I will not explain it every time I use it.)

Here is the top left of yesterday's image. If you are keeping score, the center is at -0.7491014 + 0.0694844i.

Here is a closer look at one of the spirals in the upper left of yesterday's fractal.,

This is a zoom of one of the fireworks explosions on the right side of yesterday's image.

Today's fractal image is from deep in the Seahorse Valley

This is a zoom into the lower right of the previous image. Coordinates are -0.0821507304089322 + 0.8799895735168i. The zoom is 10^8, the iteration bailout is set for 30720. That may seem like a deep zoom, but many people have dug much deeper into the Mandelbrot set.

Fractal art, such as in the previous post, often resembles Psychedelic art. It may be interesting to ruminate about the similarity in a future post. But let's leave the deep thinking for later, and for now just say "it's the colors".

When I am building a fractal I am always tempted to add more and brighter colors. In my program, that is an easy knob to turn. There are so many colors to choose from, I do not want to leave any out.

It takes more effort to work with a limited palette. Here is the same fractal as the previous post, but with a blue and brown color palette.

This is a magnification of one of the spirals in the lower left of the previous image. Coordinates are -0.0821502555 + 0.8799897640i.

Here is a zoom into the area between the top bulb and the main body of the minibrot in the previous post. For those with a map, the center is at -0.0821489887 + 0.8799892298i

One fun feature of many escape time fractals is that they are self-similar. By zooming in on the tendrils surrounding the top bulb in the previous image we find a minibrot.

OK, time for another reboot of my web site. I will showcase computer generated art. I plan to include many types of computer art. My interested in computer art solidified when mathematics, art, and computer programming came together in a Scientific American article in 1985. And then with the 1986 book The Beauty of Fractals by Heinz-Otto Peitgen and Peter Richter.

Fractals will be the starting point for this blog. In particular escape time fractals. Wikipedia knows everything, and explains Fractals better than I can. Today's image is of a specific type of escape time fractal, the Mandelbrot set. Again, I will let Wikipedia link provide the details.