Sequence Fractals Part V #21


s0=1.0, si+1=(0.8+0.6i)*si
Center: 0+0i; Zoom = 0.004

Here is the original starting point for this series. The previous formula, z^{2^{s_i}}+c, was my intent. I messed up the order of operations in the program code. I have a few images with this formula, and I felt compelled to first show the big picture view.

You may be familiar with the exponent rule (xa)b = x(ab). The rule works for real numbers, but not for complex numbers. The multi-value nature of complex exponentiation gets in the way. Try x=-i, a=2, b=1/2 as an example.

The image is here only to introduce the formula. It is not very pretty, and I am not going to talk about mathematical significance, so there is nothing more to say.

Sequence Fractals Part V #20


s0=1.0, si+1=(0.8+0.6i)*si
Center: -0.207+0.211; Zoom = 16

Compared to yesterday’s picture Sequence Fractals Part V #19, the sequence still moves around the unit circle, but at a slightly smaller step. The palette has been greatly reduced.

Continuing yesterday’s discussion, this image is messy, and lacks mathematical significance. Well, I like messy, so I am going to push on. But no more searching for mathematical significance. I will however continue to include information on the underlying formula in case someone wants to try this at home.

The colors in the limited palettes create a mood. The smaller transitions are less jarring. A few places of sharp contrast tease the viewer’s focus, but does not overwhelm as the last image does.

Sequence Fractals Part V #19


s0=1.0, si+1=(0.6+0.8i)*si
Center: -1.141+1.458; Zoom = 4

Shallow zoom into the trash heap in the upper left of yesterday’s Sequence Fractals Part V #18 picture.

Complex exponentiation is a multi-valued function. Yes, by definition a function must have be single valued. But sometimes in complex analysis that requirement is ignored. (One work around is the consider the function that returns sets of complex numbers. The set, i.e. the function value, may consist of more than one element, but the set itself is a singular, specific, entity.)

You are already familiar with this. 1 and -1 are both reasonable values for 10.5. If w is a rational number, say a/b in lowest terms, then there are b candidates for zw. If w is irrational, or if w is complex and either the real or imaginary part is irrational, then zw has infinite values. Of course the computer does not deal very well with sets and infinity. So we just pick one value and ignore the rest.

In summary, we cannot find critical points, we cannot find a meaningful escape radius. On top of that each of thousands of iteration steps involves an arbitrarily pick of one number out of an infinity of choices for the of exponentiation calculation. I think one could say that even though the pictures are generated with numbers, they are mathematically worthless.

Sequence Fractals Part V #18


s0=1.0, si+1=(0.6+0.8i)*si
Center: 0+0i; Zoom = 0.5

Here is the usual home base view, horizontal range from -2 to 2, of yesterday’s setup, Sequence Fractals Part V #17. Inside / capture coloring is turned off. The right side is in the capture set. The usual complex exponent cutting and chopping has totally shredded the upper right.

The start point for iteration is 0. I have no idea whether this is a critical point. Does the concept of a critical point even make sense here? Each iteration step is a different function. Each of those functions have a different set of critical points. Anyway, different starting points do not seem to make much difference in the result.

I have the escape radius set very large, 10000. Many orbits never leave this radius. But of the ones that do leave, do they stay gone? Or do they return as with rational functions. I suspect that they eventually return.

Sequence Fractals Part V #17


s0=1.0, si+1=(0.6+0.8i)*si
Center: 0+0i; Zoom = 0.004

The start of this series Sequence Fractals Part V #1 was actually the third starting point I considered. Here is the second candidate. The stacked exponents seemed like a natural place to add an exponent to the traditional z2+c. After queuing up a few, I decided that z^{s_i}+c was a simpler place to start.

While no longer the starting point, these are still part of the story. The most noticeable traits are shared by the different placement of the exponent. In fact given the sequence si, we could create a new sequence ri = 2^{s_i}, and then use the previous formula, z^{r_i}+c. Like I said, I did not think about these things when I started.

One of the favorite sequences, where si jumps around the unit circle is back. There is a large area of interest. This image is zoomed way out. The horizontal range is -250,250.

Sequence Fractals Part V #16


si = 2.5, 0.5, -1.5 … (three step repeating)
Center:1.359+0.409i; Zoom = 6.4

The same formula as the previous two pictures. But with experimental coloring.

Orbit-escape detection is turned off. The negative exponent creates some similarities to Rational Fractions. See the discussion in Sequence Fractals Part V #5 and Sequence Fractals Part V #6. For the same reasons, iteration count coloring does not work here.

The coloring algorithm counts how many times the orbit gets close to zero. “Count” is inaccurate, it actually generates a weighted average of all the values that get close. The average value is a complex number. Each of the real and imaginary components are used to generate a color, and then the two colors are blended together.

For the record, here “close to 0” is defined as “|z|<0.9". That parameter is subject to much trial and error tweaking. Finding two palettes that blend nicely also takes some effort. The bottom line, today's picture is more about my aesthetic choices than mathematics.

Sequence Fractals Part V #15


si = 2.5, 0.5, -1.5 … (three step repeating)
Center:-1.11937+0.07167i; Zoom = 1600

Same formula as yesterday’s Sequence Fractals Part V #14, different neighborhood.

The orange blob looks like a distorted cubic multibrot. But there is nothing cubic in the formula. The exponents are 5/2, 1/2, -3/2. Seems unlikely, but perhaps the 3 in the numerator in the third exponent is doing it. I think the area surrounding the orange spot looks like ice crystals forming on glass on a cold winter night.

Sequence Fractals Part V #14


si = 2.5, 0.5, -1.5 … (three step repeating)
Center:-1.16588+0.14008i; Zoom = 410

Now three repeating exponents. Each involving a square root, one is negative.

I really like this one. It is something that I would put into my abstract art collection. There is a range of densities, and levels of complexity. The viewer is challenged to enter the more complex and chaotic blue regions. But when that gets uncomfortable, focus snaps back to the larger features.