Today’s image has some of the same issues as yesterday’s, too noisy and not in a good way. But since I had it queued up, and I am busy on other stuff it is what I have for today…

# Month: June 2019

## Vines #10

Center: -0.7535-0.0482i, window width = .0019.

I was undecided whether to post this one or delete it. The accumulator coloring uses a square rather than a circle. The problem is that the base fractal, that is same area but with default escape-time coloring, is complex and interesting on its own. In a sense, that starting point is thick. The vine coloring is fighting with, and colliding with the base fractal. There is not enough surrounding open area for the “decorations”.

I may return to this one. A lighter touch on the enhancements, similar to Smooth 1D colors #16 or Vines #1 may produce better results.

I have moved on to other things, and there is more work to do before starting the next topic. I do not want to get in the habit of skipping a day because what I have is not perfect. So I decided to post this as-is.

## Vines #9

Location -0.718423+0.297283i. Window width = .0000076.

The coloring is similar to yesterday’s image Vines #8. Each flower / star / snowflake surrounds a period eleven Misiurewicz point. There are eleven branches coming out of each.

## Vines #8

Fractal location 0.020 + 0.818i, image width 0.016. With more a conventional coloring method, the fractal shape here is a jagged line with branches, it is very thin. It looks like lightning. But it is rather boring, as if it is calling out for some decoration.

The vines algorithm is one way to add decorations. Points in what would normally be called the actual fractal (non-escaping orbits) are removed from this image. Focus is on the path the nearby escaping points take while escaping. Like wrapping vines around a jagged invisible branch.

## Vines #7

## Vines #6

No words, just a picture today.

## Galleries

I missed a day’s post. That is just how it looks on the front page / news feed. There are several new gallery pages. You can view them here: Galleries or by clicking the “Galleries” link in the top tool bar.

So far, all of the images have been previously published as feature images in blog posts. Sections for upcoming posts (blog post planned, but words not ready yet) and previously unpublished art (no words or blog post) are planned.

If you click on a picture in a gallery you get a full resolution version, and from there you can right-click to download a copy. If you right-click in the gallery you may get a reduced-bandwidth low resolution version, so do the extra click to get a good copy. Please be careful what you do with the downloaded image. Remember that is has a copyright.

This site is built with WordPress, and the galleries are the default WordPress gallery. As with everything, there is a tradeoff. The default gallery is easy to get started. They are easy to put up, but there is lot of mindless repetition involved. I am sure I forgot to click a box here, or fill in a field there. I would like to have a link from the gallery to the original blog post. Also, each image has a, currently inaccessible, page for user comments. It would be nice to have links to those locations as well. But the default gallery widget only allows one link per picture, and I figure the high quality image is the best choice. Still, without the easy default tool, the galleries would still be empty.

The same is true for WordPress in general. It is almost easy enough to use that I can just start typing a blog post, and it takes care of the rest. But then I need to attach the feature image, set tags, categories and other easy things that become annoying with repetition. Of course if I had to DIY the user registration, comments, and database programming, the site would be single “under construction” page.

I also added a couple of widgets on the right to help navigate to older posts. Not the best, but they were built in WordPress options.

Enough talking about myself to myself. More pictures tomorrow…

## Vines #5

Another location, still coloring with vines.

## Vines #4

The r_{1} parameter in accumulator function has been reduced. In other words, here is a more viny picture for the Vines series.

What follows is a lengthy post that has almost nothing to do with the featured image. While looking for vine fractals on the internet I found some stalk fractals which reminded me of some other ways to color fractals. I do not plan to do a series on these alternatives, or create my own examples. Although that plan may change someday if I am staring at a blank post and have nothing else to write about. Nonetheless, they deserve some mention.

Previously, I stated that a fractal calculation creates a bucket of numbers that are used to determine a color. The numbers have a particular order so a list or sequence of numbers is a more accurate description than bucket. If you are curious, Robert Devaney has a good entry level article What is the Mandelbrot Set.

I will just start with this list of numbers, “the orbit”, and discuss how to create a color. The first method people used, and probably still the most common is escape time counting, which amounts to simply counting how many numbers are less than a fixed threshold value. I wrote about that method way back in Smooth 1D Colors #1, to demonstrate non-smooth coloring.

I always found iteration counting somewhat disappointing. You have this big list of numbers, and all you do is count them? Surely there is more information and other things of interest in these numbers than a count. Back in the day I really did believe we were missing some deep mathematical meaning. Now, I do not care about mathematical meaning, I am just looking to create pretty pictures. Simple counting overlooks much potential.

In addition to smoothing out the color bands in those early escape time fractals, I was motivated to develop accumulator colors because it used all of the numbers in the orbit. It provides alternate interesting ways to color the same basic fractal.

Here are a couple of “stalky” early coloring ideas that go beyond counting. First the Pickover stalk, Image, Article. (The Wikipedia article incorrectly calls this an orbit trap, and I think it overstates the mathematical significance.)

This method uses the closest distance the orbit gets to either the x or y axes. Or a little more formally, if z_{k} = x_{k} + i * y_{k}, then d = min_{k} (min |x_{k}|, |y_{k} |) is used for coloring. k is an index that ranges over all of the points in the orbit.

The second example is orbit traps, here is an example of orbit trap stalks. The orbit trap method tests each orbit point against a boolean (true/false) condition. If z_{k} is the first orbit point for which the test is true, then the index number k is used to determine the color. The exact formula is not documented for the linked image, but it would be something like min(|x_{k}|,|y_{k} |) < 0.1.

The Wikipedia article on orbit traps is incorrect. I found a couple of good references here VisMath and Softology The latter is a support page for the fractal software Visions of Chaos. I do not know anything about the software, if you try it out, let me know what you think.

So, where do these two methods fit into the fractal coloring landscape? Both methods use all of the orbit points, at least in the sense that they test each point and select one. Once selected, the color value is determined by that single orbit point. Pickover stalks uses the minimum value over all the orbit points, a real number. It turns out to be a continuous function of the orbit, and generates smooth colorings. Orbit traps use the index of the first orbit point, an integer, so it produces discrete colors. With orbit traps the focus becomes more on the trap shape than the generating fractal. Often a psuedo-3D effect is applied to the trapped region. When done right the result is artistically pleasing.

There is a key distinction between these methods. Pickover Stalks, and other min-value methods check all orbit points and finds the minimum value of some test function. They are naturally continuous. Whereas orbit traps use a boolean test condition and uses the first index (sometimes the last index) where the conditions is true. The result is a discrete value, the index. They are similar in that both apply some kind of test function to the orbit point. (As does accumulator colors.) The various min-distance methods, and accumulator colors predate orbit traps.

The Wikipedia article on orbit traps is actually describing the various min-distance methods. Some fractal programs have started calling any method that applies a test function to a point an orbit trap.

Bill Clinton might have said “It all depends on what your definition of orbit trap is”. (He did not say that, but he would if he read this post.) Definitions change and are basically whatever is the common usage. But, usually I prefer accuracy and precision. So around here, I will use the more precise definitions of the various color methods.

## Vines #3

Another vines image, same fractal location but with variation in the fractal coloring parameters.

After I decided to call this series “Vines”, I checked the internet for similar images. I found a few fractal images called vines. Those were images only, without description of method. From the appearance they were generated via a different method.

There are many fractals called “Stalks”. I had considered that, but vines are curved and stalks are straight. The methods behind the stalks fractals incorporate straight lines in various ways.

Tomorrow (hopefully, if not, then some other future date) I will talk about stalks and one or two other ways to color fractals.