Plank's Length

In this movie, I am displaying the real and imaginary components of the dynamic as it is falling into singularity. This should give us a good idea as to what is happening to these points during the iteration process.

So, what we see here is that the digits on the left begin to stabilize as the digits on the right continue to change. As we get closer to singularity, more digits on the left have stabilized and less digits on the right are changing. Eventually, the whole thing stops.

Why does it stop? Because it runs out of digits. Why does it run out of digits? In general, a 32-bit computer can only represent 17 digits of precision at any one time. That is why I am only displaying 17 digits of precision in this video.

So, it turns out, loop-singularities ARE in fact an artifact of the computer and its limitations. If I were use a 64-bit computer instead of a 32-bit computer, then I should end up with double the number of digits or 34 digits of precision instead of 17. This would allow me to travel twice as far down the wormhole before I reach loop-singularity. You see, the loop-singularity still exists, it's just farther down the hole.

The other thing I want to point out here is that, if we imagine that we had an infinite number of bits and therefore an infinite number of digits, then it becomes clear that these "fractal dynamic fields" are falling into a point or singularity represented by two irrational numbers, a real one and an imaginary one.

In other words, the singularity of a fractal dynamic field is a complex irrational number.
Why is this important? Earlier I suggested that all real fractals should have irrational fractal dimensions and now, we have our fractal vorticies orbiting around irrational points in the complex plane. There seems to be some mysterious connection between fractals and irrational numbers that should be investigated further.