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Two Curious Cases Of Fractals

Here is a spooky case of two similar and seemingly disconnected concepts both appearing independendtly both in physical experements and mathematical dynamics.

Magnet Maps

The Relationship Between Chaos, Fractals, & Physics by Hiro Shimoyama (7 minutes)

This video displays an interesting and deceptively simple physical demonstration using 3 magnets and a pendulum used to show how where the pendulum ends up is entirely dependent on its initial starting place. This shows something that we refer to as 'sensitivity to initial conditions' as well as suprising geometric structures that govern our physics and statistics.

A pendulum is released above 3 magnets and you see which magnet the pendulum goes to depending on where you initially released it. The results are suprisingly complex and looks like someone had too much fun with feathering their paint.

Map of what magnet the pendulum will end up at based on starting condition

As you can see, it looks a lot like 3 different colors of paint mixing together, forming these feathered bands within bands within bands of color. And between those bands are even more color bands, pretty much forever. Whats the deal??

Some observations to keep in your mind

The pendulum MUST stop on one of the three colored magnets, there is no possible positions where the pendulum gets stuck in some intermediate zone or cycles forever between them.

at areas of high sensitivity to initial conditions (places where there are A LOT of different colored bands in close proximity to eachother) all 3 colors are always present

The map is made from tracking where the pendulum goes after many different starting points

Well, I didn't really understand this too well until I came across another video by 3Blue1Brown discussing Newtons Fractal.

Newton's method, and the fractal it creates that Newton knew nothing about

Newton's Fractal trifold [ (Zn) - p(zn)/p'(Zn) ]

The newtons fractal is a kind of structure that emerges when we apply newtons method to many points on the complex plane, and map out which points go to which root numbers.

The newton's fractal has this sort of 'blobs within blobs within blobs' structure or maybe braids within braids within braids.

The moment I saw this video alarm bells started ringing off in my head as near schitzophrentic mental connections were being made between these two. It took a lot of time for me to realize why.

On first glance these two don't seem very related to eachother, the magnets thing is like a feathered paintjob while the newtons fractal is like someone decided to put braids within braids forever. Lets look at the similarities though

both are structures generated from tracking many different initial positions and seeing how that position influences trajectories over time

both contain areas of high sensitivity where all of the colors are in one small area

I SUSPECT that much like how all the colors in newtons fractal share the same boundary at sensitive spots, so is the same with the physical magnet map. Im not sure on that though, just speculation.

There seems to be an analogy between physical magnets and mathematical root numbers. Both serve as attractors, the magnets literally attracting the pendulum and the roots attracting numbers/points processed under newtons method.

Its kind of spooky that two seemingly disconnected processes share such commonalities. The closer you look into these kinds of things the more you will find these spooky connections that are too specific to just be coincidences. Theres something fundimental going on that links these two phenomenon together, and our language and understanding are still not at a high enough level to grasp that more fundimental concept which trancesnd physicality and abstraction.

Difference between physical and abstract

There is one difference between these two that I think is interesting. There are actually some equations where newtons method can in fact cycle forever without ever approaching a root. Its a very small likelyhood of finding one of these cycling zones but it is definitely possible

Beyond the Mandelbrot set, an intro to holomorphic dynamics by 3b1b

This video demonstrates just how to find these small cyclical spots in the various newtons method fractals and a very spooky way the mandelbrot set pops up when doing this.

In physical reality, a pendulum almost certainly can never cycle forever because of friction, but perhaps there are near-cyclic trajectories that preserve momentum/negate friction as much as physically possible (much like an euler disc toy) that are incredibly hard to find but still exist as a possibility.

Eulers Disk Science Toy (Gemipedia Article)

Eulers Disk by Grand Illusions (2 minutes)

I think thats why these examples reasonate with me so much. I am a big supporter of the idea that mathematics is not invented but discovered, that purely logical abstractions are more 'real' than they are given credit for, that ethereal objects exist. This notion pisses off materialist, followers scientism, and even mathematicians themselves who believe mathematics is just a created tool. Good, fuck em.

There are so many invisible threads and connections between mathematics and physical science that those connections become lines and those lines give way to a structure we cannot percieve or scarcely appreciate in its totality. Reality is a multi-layered tapestry of logic, structure, emotion, spirit, and experience, overlapping the same way these fractals do. By focusing on the boundaries of only one part, we fail to adknowledge the whole. As above, so below? Something like that anyways.