Smooth 1D Colors #3

Fractal 14

Today’s image 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.

Smooth 1D Colors #2

Fractal  13

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 colorings.

(Introspective 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.)

Smooth 1D Colors #1

Fractal 12

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).


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.