No matter where you are... there be STRANGE ATTRACTORS.... those lovely gems of chaos...

in mathematics •  2 years ago

That is not dead which can eternal lie,
And with strange aeons even death may die.. - H.P. Lovecraft

That quote may not seem immediately relevant. Yet I have a tingling little seed of an idea that some day I may play upon that quote and use the term strange attractors instead of strange aeons.

Strange Attractors


Source: www.chaos-math.org

My initial approach to discussing strange attractors with you the reader will be from a non-technical aspect. Instead I want to give you examples of them occurring in your life. Recently, in watching @andrarchy post Journalist Fears For Life After Clinton Post, Wave Theory, and The Game Update where he mentioned Wave Theory, it inspired me to think that was similar in many ways to Strange Attractors.

You can literally pick any discipline. You could work at a restaurant, you could be an IT firm, etc. There will be predictable periods where business is expected to be busy such as breakfast, lunch, and dinner at a restaurant. Those predictable times are not what this article is about.


Source: ocean.nationalgeographic.com

This article is about those Rogue Waves. The busy spikes that seem to come out of nowhere. It wasn't that a single bus drove up and disgorged it's passengers to come eat at your restaurant. Such events would also make sense. Instead it is those moments where you suddenly get very busy and the different customers don't appear to have any relationship to each other. Why did they all suddenly pick that time period to come?

This happens in all other disciplines I am aware of. Something breaks over in one place, and in another place something completely unrelated to the first place also breaks. Why do these things occur in groups, clusters, or waves?

This is the term I refer to when I refer to strange attractors. For some strange unknown reason these clusters occur. They seem to occur across every discipline I have ever had any interaction with.

I am fairly certain that you the reader know exactly what I am talking about. It has likely happened to you many times and you've thought "this is really weird." Am I wrong? Are you strange attractor free?

My first encounter with the term



Source: www.amazon.com

In the early 1990s I took a special topic class that is not normally offered at the University I was attending. It was in the mathematics department. It was called Special Topic: Stochastics. Stochastics is another name for Chaos Theory. It is a fun sub-topic of mathematics. It is where the famous butterfly effect statement comes from, and where fractals, and many other things come from. It has gone beyond theoretical and entered the world of practical in a lot of cases. Your cell phone antenna that makes it possible to have the range it does is because the antenna is actually inside of the case of your phone in a complex fractal shape to increase it's surface area while being in a small space. Stochastics and it's study lead to the possibility of your cell phone being as small and portable as it is.

So what are strange attractors in Stochastics?



Source: www.chaos-math.org

In studying fractals and chaos as we graph out things we notice these clustering patterns. There are points upon which the information and graphs converge. It is almost a mystical order out of chaos moment. The technical explanation for strange attractors is in itself interesting, yet it may not be everyone's cup of tea. I am going to share a few good sources to begin your journey if you want to give it a deeper dive.

Mandelbrot Set


Source: stackoverflow.com

The mandelbrot set is one of my favorite fractals. I have spent a lot of time exploring it. I have written many programs to explore it and I am absolutely certain I will write more. The mandelbrot set is a great chance to explore the concept of strange attractor.

The main immediately identifiable bulbous shape that has become the symbol for the mandelbrot set can be infinitely zoomed in on at points and it will get more and more detail as you dive deep into it. Yet during this journey you will continue to see that bulbous shape popping out of the "chaos" and appearing amidst points in places you could not see before. This shows the order that is hidden in the complexity of chaos.


Source: ibc9.net


Source: old youtube video that I made with some of my experimentation

This shape in of itself is a strange attractor.

Feast or Famine.

I simply love to look at the events around me and see the strange attractors in the form of events in our lives.

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My first Mandelbrot set was generated by a program I typed off from a magazine. On a Commodore 64. It ran the whole day. I did not know what I was looking at, but it was beautiful and captivating in all its strange attractiveness. Pun not intended at all.

Then came Fractint...

Two decades later, and Mandelbulber generates complex 3D representations within seconds.

I showed the stuff to a friend from school, who is deeply religious and orthodox. "It is only math?" he asked, and I confirmed it: only math, as much as a parabola is only math. "No human did anything to make this?" I confirmed again: if you accept the circle as a perfect idea of which a drawing with a compass is an imperfect, approximate representation, then no human ever did anything to make this fractal except to make it visible, by drawing it with a very complex compass.

He sat in awe, looked at the picture and said "God is awesome".

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Hehehe.... I might have typed some of the same code into a Commodore 64. I do know that was the first place I ever made a mandelbrot. Since it was interpreted basic it definitely was slow. You could watch each pixel slowly appear.

I love this stuff and haven't had the time to full dive into it for awhile. I will admit on this subject these days I kind of live in a bubble. I do my own experiments without seeing if someone has already done them. About the only fractal like things I've used from outside of mandelbrot and myself for awhile are perlin and simplex noise functions.

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I'll look into those again!

I started, but gave up temporarily on it, writing/porting an L-System engine for POV-Ray and came to appreciate Hilbert Curves and such things... :)

I like to think of crypto currencies as a mandelbrot set. As we zoom in, or progress over time, we will see the limitless possibilities that they open up.

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Yeah if you go google images for stochastics your going to find a lot of images that look just like all the crypto-currency and stock trading charts. :)

Fibbonacci Set is also fractal in nature and I see people using that for charts often.