In my three magnets pendulum post I described that system and showed some results, including its basin of attraction. The basin is a “map” where each pixel indicates a starting position and pixel color tells which of the three magnets the pendulum bob will stop at. The image below shows simulated results, with the three magnet positions indicated by black circles. See my earlier post to learn more about this.
The “oil spill” pattern is an example of a fractal nature. The fractal definition is not completely set in stone, but generally one expects to find feature details at every scale within the fractal. Most people know the Mandelbrot set fractal, or have at least seen pictures of it. There exists also many so called “deep zoom” videos, taking the viewer deeper and deeper into this geometry. See for instance this zoom.
A Zoom Into The Basin Of Attraction
In similar spirit I wanted to make a zoom video for the three magnets pendulum basin of attraction. This is something I have not seen anywhere else. The zoom consists of simulated images over smaller and smaller dimensions, all spanning across the same center point. Of course, each pixel in an image always represents a pendulum starting position.
Zooming in further and further requires more and more decimal numbers to describe positions. Eventually one runs into a numerical limit where the computer does not store enough decimals to continue. For example, assume we can only store 4 decimals. Then a position 0.12345 stores as either 0.1234 or 0.1235, which results in a loss of detail. I don’t want to go much deeper into this topic, but most computers of today allow you to work with 15 or so decimals. This follows from the double-precision (64 bit) number format in wide use.
The video below shows a zoom into the basin of attraction. It homes in on an area to the upper right of the red “heart”. I was able to scale the zoom down to about 1E-14 of the original size before the numerical limit begins to show. This manifests most obviously as pixels don’t resolve anymore, but gangs into larger and larger squares. To get a sense of scale for 1E-14, it compares as reading the text of this blog to distinguishing virus sized features on the surface of the moon.
On A Side Note
Mandelbrot zooms exist to incredibly large depths. Some of these videos are several hours long, such as this one. Enthusiasts are able to do this by using clever tricks to bypass the absolute number precision limitation. I do not at the moment see a way to apply something similar for the three magnets pendulum case.
Philosophical Take On The Results
For those philosophically inclined, an interesting take on the above result relates to the question “what can we know?”. I will only scratch the surface here and start with Newton. His thinking in physics, not the least his laws of motion, played a big role in promoting determinism. This is the idea that given complete knowledge of all physical matter and the laws that governs it, it is possible to calculate (predict) all future events.
This view has been challenged over the years. For instance, quantum mechanics introduced uncertainty to how well we could pinpoint matter properties, and theory of relativity played with our sense of simultaneity, etc. However, Newton’s theory, the mathematical description as used with great success in everyday life, is usually considered deterministic. Given positions, velocities and acceleration (forces acting) we can calculate future events.
Well, in principle this is true. But as the basin of attraction zoom video shows, there exists limited regions within it where new patterns keep resolving. A pixel in the image shows a certain color, but as we zoom into it, it may turn out to have complicated internal structure. So, you may need to zoom to infinity to get certainty. Some argue that this is still within determinism. But in practice you can never know the outcome with infinite accuracy.
A Chaotic Result
As mentioned in my first post on this topic, the three magnets pendulum movement is chaotic. Among other things, this means an extreme sensitivity to starting position of the pendulum bob. This is evidenced by the zoom video above. Towards the end, at a magnification of around 1E-14 (see image below), there are still fine details in three colors.
Again, consider the scale differences here. I have indicated three different starting positions in the image, one in each of the colored regions. If we assume the pendulum size (magnet separation) to be 10 cm or so, these starting positions are some 1E-15 m apart. This is way smaller than the diameter of an atom, in fact it is of the size of a proton or neutron. We can measure position to this level, but it is not possible to actually position our pendulum bob that accurate.
Using the three starting positions above, I simulated their pendulum paths and overlaid them on the basin of attraction in the video below. The traces follow one another closely for a while before starting to separate and finally coming to rest over each of the three magnets.