Here is another fractal / confetti hybrid. This works better because the fractal part is deemphasized. You probably recognized the dark potion as a tendril in a typical fractal with smaller branches off to the sides.

Although, in this case, while an improvement over the previous, I think it just adds unnecessary complexity. Layering the confetti mixing algorithm over a simple collection of circles, squares and lines works equally as well, if not better.

I tried to mix the confetti algorithm with the fractal algorithm. I am not pleased with the result. But I am posting it anyway if anyone else was wondering what a (failed) mashup might look like.

The main problem is the confetti algorithm and fractal algorithm have equal influence. The image cannot make up its mind what it is. Often that type of ambiguity is desirable, but it does not work here.

Here I am playing some more with varying the amount of confetti mixing across the image. The function that controls the mixing has become more complex.

A few days ago I called this abstract art and algorithmic art, and implied that it is not fractal art. What is the difference? Why do we need definitions? It is what it is. Feel free to appreciate it (or despise it) without defining it. Classifying it does not change it. Still, classification is an interesting exercise, so let's go there, gently, with that understanding that it does not actually matter.

Fractal art is algorithmic art since it is generated by an algorithm. Fractal art however is limited to a very specific algorithm. For many fractal artists, that algorithm is written by someone else. Even though I write the program(s) for my fractal art, I follow a recipe that is shared by thousands of fellow fractal artists.

Algorithmic art puts no restrictions on the algorithm. Everything is fair game. And generally every image is generated by a different algorithm. The algorithm itself sits on equal footing with all the other artistic stuff. So I like to reserve algorithmic art for the cases where creating the algorithm itself is a key part process.

I am paranoid that someone will vehemently disagree, and that paranoia drives me to deal with the minutia. First, I in no way intend to diminish the work of fractal artists that use someone else's program. Discovery and selection an interesting fractal region, then framing it and coloring it is the artistic input. That takes a lot of skill and practice and creativity.

I should say something similar for using a computer rather than a brush and acrylics. Maybe I will save that for another day.

Second, there are no clear boundaries. Many fractal artists soon get bored with z^2+c and start experimenting with other, and increasing exotic, formulas. Most fractal software allow users to write formulas, so there is no limit, it crosses over into general algorithmic art.

Some may consider this image a fractal. I don't but someone else might. Fractals are sometimes defined as having fractional dimension. I doubt that anyone could compute a Hausdorff dimension for today's image from the underlying formula/algorithm. But by appearance, it certainly seems to fit the definition. Also the process of repeatedly chopping and shuffling pixels seems fractal in nature.

In today's image, different "entropy" levels are assigned to different areas. Recall (yesterday's post Confetti #2,) the algorithm generates different distorted squares, which are used to shuffle around the colors in the starting image. The algorithm has parameters that control the size of the distortion. Rather than setting these parameters universally for the whole image. Another "set up" function sets them differently for different areas of the image.

Previously I mentioned that making these distorted too small or too large, or too variable or too uniform produced uninteresting results. But there are interesting results when the more extreme parameter values are mixed in the same image.

Today's image has the same basic starting point as yesterday. Here the typical "square" is an actual square. Yesterday they were elongated rectangles.

If you are not a daily reader, the algorithm is described in the previous post Confetti #1. The final algorithm is quite different, and much more complex than where I started.

Sometimes algorithmic art incorporates randomness into the algorithm. Despite what the appearance may suggest, this image and the rest are not random. The algorithm will produce this image, and only this image on every run. And in response to the cynics, no, that consistency is not achieved by setting a seed for a random number generator. For example there is a reference "square", that gets modified. Width, height, orientation angle are changes. The changes are determined by a function based on the x,y screen coordinates. There is a mix of different distorted squares. The mix has a mean and standard deviation like a random sample in statistics. But it is deterministic, not random.

Today starts a new series called Confetti. I am leaving fractals behind for now. Although I am sure I will return to fractals, I always do.

This is abstract art and algorithmic art. In the world of mathematics and computers definitions are precise. You can look at something and definitely decide if it fits the definition or not. The real world is not so clean, and the above linked Wikipedia articles are trying to classify things that are difficult to classify. I am sure someday I will be compelled to return to these definitions and add my two cents. But, you know, one day at a time. I will avoid that rabbit hole today, and simply assert that this is that type of art.

This image is generated by an algorithm that I wrote. The original idea was to start with a simple minimalist geometric abstraction, in this case white and black with a small splotch of blue and cyan, with smooth color transitions. Then chop the image up into small squares, displace the squares by varying amounts and then repeat until the original was suitably mixed up. When I implemented this algorithm in a straight forward manner, as suggested by the description, I was disappointed in the results. It was not want I expected, and not in a serendipitous way.

It works better to leave the square in place and pull the colors from elsewhere into the square. The squares are not uniform, they have different width, height and orientation. So they are actually tilted rectangles. The simple repeat loop did not work well, the last set of cuts were too obvious and a distraction. The repeat steps needed a kind of memory which informed the size and displacement for the next step.

There is a narrow range of rectangle sizes that works. If too large then it just looks like random rectangles. If too small, then the confetti effect disappears and it looks too much like the original smooth color image. Also, it takes a lot of iterations to get the squares chopped up and messy. This one has 80 iterations, although usually 20 to 50 is enough.

After all these adjustments I came up with something close to my original intent.

And here the the second alternate coloring of the image in Vines #11. There are more colors to help the fractal fireworks standout. The vines now appear as shadows or waves around the main fractal shape.

I was going to move on to another topic today. But my obsessive compulsive disorder prevents me from leaving this topic with "yeah, that one could be better if I worked on it more". So I went back to work on it. Here are two alternate views of the same fractal space, with better foreground / background separation. I muted the background, removed the corners on the vines and added gold / brown to highlight the high spots in the fractal shape.

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...

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.

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.

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.

No words, just a picture today.

Another location, still coloring with vines.

The r₁ 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.

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.

You are in luck, much less text today...

Today the vine coloring formula parameters have a smaller r_2 value. The color gradient is steeper from center to the edge of the arcs or vines, making them more pronounced.

Two days ago I promised to shift gears. So here is a new series called vines. The title may not obvious with today's image. More on that later.

Today's image is the same area as the previous two, Smooth 1D Color #18 and Smooth 1D Colors #17. It is also a continuation of the accumulator colors method. However, now the parameters are set in such a way to create shapes in the background, rather than just smoothing out the color bands. The colors are still continuous, we are not going to bring back the banding in the early fractals. The goal now is to put interesting things in the background.

The previous images were generated by overlaying many fuzzy disks. This one is generated by overlaying many fuzzy circles. By strict definition, if not by common usage, disks and circles are not the same thing. A circle is the edge or border and a disk is the inside of that round thing you are thinking about.

Some gentle mathematics follow, I promise it is gentle, please give it a chance even if you do not consider yourself a "math person". It is very easy.

I introduced the math for accumulator colors in Smooth 1D Colors #10. The simple formula $ 1 - |z|/r$ played a key role. Actually many of the pictures use $ 1 - |z-a|/r$. Which is the same thing when a = 0.

Today's image uses $ 1 - ||z-a|-r_1|/r_2$

First a recap on the old formula. a and r are arbitrary parameters. a is complex and r is real. (A complex number is a pair of real numbers.) Different values of a and r produce different colorings, and a lot of opportunities to experiment.

z has many values, in an academic paper it would be called $ z_i$. z tracks a point's orbit in the fractal calculation. The mathematician's definition of orbit is different from the astronomer's. For now, just think of it as a big bucket of numbers that you get from the fractal calculation.

$ 1 - |z-a|/r$ is a disk centered at a with radius r.

People with some mathematical sophistication will object, for you folks, the set of z's for which this function is positive is a disk centered at a with radius r.

Let's say C = the circle centered at a with radius r. This is just to make my typing and your reading a bit easier. The aforementioned disk is the interior of C.

Let's break it down the formula.

If z = a then $ 1 - |z-a|/r = 1$

If z is inside C, then |z-a| < r and $ 0 < 1 - |z-a|/r <= 1$

If z is on C, then |z-a| = r and $ 1 - |z-a|/r = 0$

If z is outside C, then |z-a| > r and $ 1 - |z-a|/r < 0$

I called this a fuzzy disk. Imagine we are painting, and this formula tells us how much paint to use. 1 is full coverage, 0 or less is no coverage, and the numbers in-between indicate partial coverage. 1/2 is a mix of 1/2 the new color and 1/2 the previous color. If we put black paint on a white background with this recipe we get a blurred disk, solid black at the center, and a smooth transitions to white at the edge.

Now, back to our bucket of numbers, the z's. Apply this formula to each one, and add up all of the positive values. The resulting sum is used to look up a color in a color palette. Paint the screen pixel that color. This is done for every pixel on the screen. (Each pixel gets a different bucket of z numbers, and so a different color.)

Wow, that is just recap. On to today's picture and

$ 1 - ||z-a|-r_1|/r_2$

If z is distance $ r_1$ from a, then $ |z-a|-r_1 = 0$, call this circle C. On C the full expression $ 1 - ||z-a|-r_1|/r_2 = 1$

If z is distance $ r_1 + r_2$ from a, then $ ||z-a|-r_1| = r_2$ and $ 1 - ||z-a|-r_1|/r_2 = 0$. Call this circle C+.

Also if z is distance $ r_1 - r_2$ from a, then $ ||z-a|-r_1| = r_2$ and $ 1 - ||z-a|-r_1|/r_2 = 0$. Call this circle C-. If  $ r_1 < r_2$ then C- does not exist. It would have negative radius. That is not a problem.

Summarizing, we have three circles, all centered at a, with different radii.

If z inside C- or outside C+ then $ 1 - ||z-a|-r_1|/r_2 < 0$

If z outside C- and inside C+ then $ 1 - ||z-a|-r_1|/r_2 > 0$

If z is on C then $ 1 - ||z-a|-r_1|/r_2 = 0$

And that is the fuzzy circle. The formula is one on the circle C and smoothly fades to 0, in both directions, in and out.

This formula has all the important properties for smooth coloring. It is a suitable candidate for accumulator coloring algorithm. Since the "hot spot" is a circle rather than a point, lines (actually arcs of distorted circles) begin to appear in the image.

(2019/06/07 edit note: Substantial revision incorporating suggestions from someone who considers herself not a "math person".)

OK, one more repost from my Smooth Colors series on Fractal Forums.