Vines #4

Vines Fractal 34

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 zk = xk + i * yk, then d = mink (min |xk|, |yk |) 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 zk 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(|xk|,|yk |) < 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

Vines Fractal #32

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.

Vines #1

Fractal Vines #30

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”.)