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.
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).
This site is getting ten non-search hits a day, with three to five spam comments in Russian (deleted).
So I am probably all alone here, and I am writing this post to myself. I will go ahead and be self-indulgent, after all, no one will notice. If you happen to read this, and you are not me, go ahead and skip to the next post. I will not be insulted.
The low hit rate is
OK, I do not expect more at this time. I have not started promoting the site
yet. I really need to be consistent and regularly put up new posts. My last
posts were two months ago. Interestingly, the hit rate was slightly higher for
a short time.
So I need to make posting a habit. This is the usual “I have been busy / I will get better” post that you see in blogs that are dying or never get started. I know, I have been there myself several times. I do not want that to happen again.
So, starting now, at least one post a day. Even if I have nothing to say I will write something. Even if it is solipsistic introspection like this. Things improve with practice. Hopefully, both the quality and quantity of my posts will improve with practice. Hopefully this public commitment keeps me going.
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 6.67×10-9.
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 4.27×10-7. 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.