Complex Unsnarling

I want to explain a little more what was happening in the two videos – Snarl Dance and Complex Snarl – from the last post.

(Graph 1: Left: y1=c^x1. Right: y2=c^2^x2. For x1=0, x2=-1.)

It’s a little hard to see, so I apologize. I spent several hours trying to get Excel to format the plots correctly because it couldn’t deal with 64K worth of data points, and I don’t really want to go through that again to make the lines thicker. If you want to get a better image, just click on the picture to get to the source file. Anyway, I specifically chose a+ib at (0.5+i*0.86) to create a spiral that decays very slowly. It’s pretty close to a circle, but after x1 reaches 65,536, y1 is (-1*10^-149 – i*7*10^-150), which is more or less 0.

We have two plots in Graph 1. The left plot is for y1=c^x1, where x1=0. By definition, this has a value of 1. The right plot is for y2=c^2^x2, which is the basic formula for the Mandelbrot fractal space. I won’t start plotting y2 until the first iteration of the left-hand formula.

(Graph 2: Left, x1=1; right, x2=0)

In Graph 2, the left plot shows x1=1, and we’ve started on our death spiral. On the right, we have the data point we’d start with for the fractal, which is (0.5+i0.86). Note that here, both plots are using the same value, but y1=c^1 already has one calculation completed.

(Graph 3: x1=2; x2=1)

Excel wants to draw a curve between points, but doesn’t have enough data yet to get the curve correct. It’s starting to look like a circle, though (maybe). y1=c^2 and y2= c^2^1 = c^2. The only real difference between the two curves at this stage is that the one on the right didn’t start at 1+i0.

(Graph 4: x1=4; x2=2)

Ok, what’s happening now is that we’re essentially sampling our curve on the left to get the data points for the curve on the right. When x2=2, c^2^x2 equals c^2^2, equals c^4. That is, we’ve taken 4 steps along our spiral on the left, and chosen to use that 4th value for the 2nd step on the “chaotic” line on the right. This line only looks chaotic because we’re losing the underlying logic of the spiral by taking exponentially larger sampling steps.

(Graph 5: x1=8; x2=3)

The sampling step size pattern is going to follow 2^x, for x from 1 to some upper bound. In other words, 2, 4, 8, 16, 32, 64, etc. 2^3 = 8, so our third data point on the right is actually the 8th point on the left. Unfortunately, for the starting value I’ve selected, this is very close to point one. For the rotation angle we have for this spiral, we’re essentially going to be bouncing between points 1 and 2 as the magnitude decreases, but with a small rotational offset.

(Graph 6: x1=16; x2=4)

Again, because the spiral takes 6 steps to complete one rotation, and our next sample is at point 16, we’re hitting almost the same two points, to give a “bouncing effect” to the curve on the right. If I’d picked some other value for c, the results on the right would be different (ref. Graph 8 below.)

(Graph 7: x1=65,536; x2=16)

I’ve kind of decided to jump forward a bit to where x2 is 16. This gives me my 16th sample, which is going to be data point 2^16=65,536 from the plot on the left. The death spiral has lived up to its name, and y1 has gone to 0, requiring that y2 does the same thing. Looking at both plots, it’s pretty obvious that the magnitude of y2 (sqrt(a^2 + b^2)) never exceeds 2. If we were drawing a fractal, we’d time out at x2=50 and color the point 0.5+i*0.86 black.

(Graph 8: c=0.0954 + i*0.99542)

As promised above, Graph 8 shows what happens if you use a different starting value for c (in this case, a+ib = 0.0954 + i*0.99542). This point is very close to lying on a circle. y=c^x still decays to 0, but even after x1 reaches 65,536, y1 isn’t at 0 yet. The plot for y2=c^2^x2 is much more chaotic because this is the kind of image I wanted this time. On the other hand, the magnitude of y again never exceeds 2, so in a fractal, this point would still be colored black. If you’re drawing fractals, then all the really interesting action takes place when the starting value of c lies on a spiral that just barely tends to infinity. The closer that spiral is to a circle, while remaining unbounded, the prettier the resulting fractal images are.

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