replaces the black and white bands with different colors. Other than the
colors, this is the same as yesterday’s R=10 image.
You probably would
have chosen different colors. Perhaps opposite or adjacent colors from a color
wheel. But it is still trying to find an interesting way to arrange colors in
concentric, warped circles.
The focus should be
the Mandelbrot set in the middle, not the surrounding color bands. OK, a
mathematician may be interested in the equipotential line. But
for the rest of us, it is a distraction.
The palette used
here is actually my favorite palette, I will use it for the rest of the
pictures in this series.
In the previous
image, the escape threshold was set as tight as possible (R=2.0). Notice how
all of the black and white bands meet at the left most tip of the mandelbrot
set. This image has a larger threshold, it uses R=10.0. Now there is always a
gap between the bands.
Not that we are any
closer to smooth coloring. This is just another example of non-continuous
diversion: A normal blogger would combine these multiple posts into a single
article. I am trying to get into the habit of making daily posts. Trying to
write a large post a day will result in immediate failure. The article never
feels complete, and often never gets posted. Whereas, adding one small
increment to the story each day is possible, and feels like success. I can find
the motivation to add one more picture to the story each day.)
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.