Symmetry in Chaos | Hacker News Reader

Symmetry in Chaos

paulbourke.net

218 pointslioeters51 comments

Paul Bourke's website is great; don't forget to check his other articles!

Over 10 years ago, it inspired me to play with strange attractors, which eventually ended with me writing https://github.com/chaoskit/chaoskit.

It was fun and I learned a lot, but it's definitely a deep rabbit hole. I've moved on since then.

Marty Golubitsky's Homepage https://www.asc.ohio-state.edu/golubitsky.4/

Books by same: https://www.asc.ohio-state.edu/golubitsky.4/mgbooks.html

https://en.wikipedia.org/wiki/Marty_Golubitsky

Fearful Symmetry: Is God a Geometer? Ian Stewart + Martin Golubitsky

is a good read, Ian Stewart is well known to UK recreational math fans and frequently hosts STEM material on th BBC.

https://www.goodreads.com/en/book/show/1651085

    This sequel to the bestselling *Does God Play Dice?* will open your eyes to the broken symmetries that lie all around you, from the shapes of clouds to the drops of dew on a spider's web, from centipedes to corn circles. It will take you to the farthest reaches of the universe and bring you face-to-face with some of the deepest questions of modern physics.

What's interesting about physics is almost everything us symmetrical at the most fundamental levels. An interesting violation here is the charge parity symmetry, which may explain why matter exists in large quantities and anti matter does not. Symmetry violations like this may also explain why we have an arrow of time that runs in one direction.

This looks like a fractal that's been around for a long time in Fractint and UltraFractal with the name 'Icons'.

I believe it's like a generalisation of the well-known 'bifurcation diagram' of the logistic map such that it's got more dimensions, and these images are like slices of it 'through the page'.

It makes gorgeous soft and subtle details! Unfortunately since it's essentially a histogram it's tricky to produce high res renderings. Paul Bourke is a total hero of this realm! Great stuff!

These are awesome, but it would be nice to understand, even at some heuristic level, why they're periodic with the period they do have.

Alternatively, what is a change to a completely periodic orbit to chaos with the same "periodic symmetry" that gives some enlightenment of where the chaos is being inserted and why it isn't destroying the periodic orbit.

Does anyone have an idea of whats' going on, or references that I can look at?

James Gleick's "Chaos: Making a new science" will answer your questions and is a fun read!

Could you point to the chapter or page?

I have a hard time pointing you somewhere, I think you should start with the first chapter and see from there, it's a good primer.

I've skimmed Gleick's book before and my feeling is that it's a high level overview of the subject without much content in either explaining the underlying math, the motivation behind it or giving some deeper insight into the subject, at least at a level that I would consider valuable.

I'm probably being too pejorative, but these books (like GEB or the like) are something I consider "feel good" books that give the illusion of understanding rather than any real insight. They're great for motivating people to learn more and popularizing mathematics as something to be valued but I find them to be very bad for actual understanding. As a litmus test, can you name any prediction that people can make after reading the book? Are there any falsifiable experiments that people can run?

Contrast this with John Baez's post on roots of polynomials with integer roots [0]. Not only are there pretty pictures but there's an in depth explanation of what the structures are and how they show up (IFS, connectivity, etc.).

Something along the line of Baez's treatment of roots of polynomials with integer coefficients is what I was looking for on how these structures show up in these chaotic systems.

[0] https://math.ucr.edu/home/baez/roots/

EDIT: "GEB" not "GED"

It's been a while, but I remember many experiments being presented in Gleick's book ( The double pendulum being a chaotic system, or the paper showing how it was impossible to predict the distance between two points in a structure after steching and folding it)

The gist being that some systems are much more sensitive than others to initial variables - and these are what we call chaotic.

For the double pendulum, for example, you try to release the pendulum from the same point, with 0 force - no matter how precise you are, there will be variability in the position, air pressure, temperature, turbulence and whatnot that will induce a small variability. This is unavoidable - but in chaotic systems, the effects are impossible to predict i.e. the pendulum always swings differently.

The pretty pictures come from what are called "strange attractors" which are all dependent on the system you are studying because they flow from the systems attributes. It's not black magic - a simple example from the book is that heat distribution can be chaotic in liquid systems like coffee cups - so it's impossible to predict the precise temperature inside any cup given it's initial state, but we all know it's cold after an hour. This is not the best explanation for this concept, I believe the last third of the book is on the subject.

Hopefully this could help out.

Edit: if you were to plot position of the pendulum for many thousands of drops, some patterns will emerge - the pendulum moves in a "finite" set of positions, because it physically can't go in some positions after some others, for example. Or it will have the same period in rising and falling in all graphs from having close to the same initial energy but not orientation.

So it'll look neat - but it doesn't necessarily mean much.

OK, that's fair, I was being too critical of Gleick's book.

Even so, this doesn't really get at the high level symmetry. Gleick's book might give some motivation for the chaotic points being restricted to a compact domain (space? manifold? area/volume?) but I don't see how to make the leap to the highly structured gross level symmetry in the OP.

I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is one part of the book I didn't integrate as well, you might be satisfied by the book - but OP also lists some references on the symmetry of chaotic systems, maybe that's a better way forward!

See alimw's response [0].

[0] https://news.ycombinator.com/item?id=37114898

All of these are aperiodic; it says as much in the first paragraph. Are you talking about the rotational symmetry?

Yes, the high level rotational symmetry. There's some gross level structure that's preserved while the individual points are (presumably) completely aperiodic/chaotic.

What's some insight into the gross level symmetric features appearing? How do you convert something that's completely periodic to an aperiodic/chaotic system with gross level periodic structure? What are the operations that allow the gross level periodic structure to remain while making the fine grained structure aperiodic/chaotic?

They iterate f where f(z) = (a₀ + a₁ z ź + a₂ Re(zⁿ) + a₃ i) z + a₄ źⁿ⁻¹. You can easily check that f(z exp(i 2 π / n)) = exp(i 2 π / n) f(z).

Thank you!

These systems do not have periodic orbits they have symmetric strange attractors.

So tracing a particle thru these systems will not result in periodicity but will trace out these symmetric structures. It is unclear from the website if these are proven symmetries or observed.

A good book that is a bit more technical than Gleick (which I found not wrong in the details it does have) is “Introduction to Applied Nonlinear Dynamical Systems and Chaos” by Stephen Wiggins. It requires basic under grad maths but is a graduate text in that very close reading is needed to follow. The first 17 chapters are intended as a semester class.

Strange attractors are by definition aperiodic and do not self-intersect, and I'm not sure I entirely understand your question(s). For example, what do you mean by "where the chaos is being inserted"? Chaotic systems are completely deterministic and are sensitive to any change in initial conditions. The chaos is there from the start. But it doesn't mean that once initial conditions are given that the system behaves chaotically for the time evolution of the system since, again, the system is deterministic.

I'm also not sure if the book answers your question(s) on this apparent aymmetry aspect, but Chaos and Fractals by David Feldman is a fantastic introduction to these topics with lots of good pictures. It is not a proof-based book, but it does include mathematics at manageable level. The book gives a great introduction to strange attractors, namely the Henon map, Rossler attractor, and the Lorenz attractor.

Do yourself a favor and get lost on this website. There's so much great material — for years I've been ending up there after looking for implementation ideas on things like metaballs and isosurfaces, to fractal noise.

I love the brutalist simplicity of the site, and the extremely high density of quality information. For those two young to know — this is a taste of what made the early web so exciting.

I would argue that position-symmetry is not unexpected in a chaotic system. Symmetry means that one of your dimensions is redundant.

Why does it mean that?

I'm not an expert, but think about linear algebra.

you can write each dimension value in terms of time t(a) -> t(b)

x = sin(t) y = 2cos(t)

or whatever. if there is a symmetry, one of those equations should cancel out.

another way to look at it is, if you can always get x, and know y "for free", or the other way around, then x and y aren't really independent. there's some other thing, like sqrt(t) that's generating both values.

Those are gorgeous; I would enjoy some explanation on how the authors came to investigate these particular recurrences.

I love how the periodicity of this video spins, locks, then finally lands in chaos:

https://twitter.com/crazyclipsonly/status/169086531377083596...

This reminds me of an Alan Watt's talk on alternating levels of order and chaos in nature.

To add some spice to your observation. I read somewhere that "This statement is false" can act as a oscillator, a "clock" going tick-tock, at the deepest level of reality.

Ok. I need to know more.

Well actually, it was a comment on a blogpost (I don't know what it was about). It said since if our world is a simulation, then it needs a computer to run on. And every computer needs a clock. So this statement changing its truth value could act as a clock.

Please say more!

Do you remember the title of it?

Web of Life

Here is the youtube video and transcript. I think the parent poster refered to section IV, around 19:20 in the video.

https://www.organism.earth/library/document/out-of-your-mind...

Perfect. Appreciate the ref.

On the theme of "Unexpected Alan Watts" I'd like to recommend the game "Everything" which after a slightly bizarre start develops into a "achieve subgoal, get an Alan Watts audio clip as a reward" gameplay loop. It's fantastic.

Will check it out! Ty!

Does anyone know where to find some code examples in which RGB images are generated from the equations? I don't really understand how the mathematical solution is translated into a raster/pixel image and what the colors correspond to. But I would love to start making playing with these.

It's right there in the article:

> The images here are generated by iterating the series typically 1,000 million times. This is performed in two passes, the first pass with many fewer iterations is used to find the bounds of the attractor on the complex plane, the second pass actually "draws" the attractor points. The process of drawing involves treating the bounded region of the complex plane as a 2D histogram, each time the series passes through a pixel region on the plane the histogram at that location is incremented. One might imagine the 2D histogram as a height field, a larger values at a point indicate that the complex series passed through that pixel more often than a point with a smaller value. At the end of the process the histogram is mapped onto colours depending on the histogram values, there are many ways to do this based on aesthetic grounds.

I have only seen the first two images, but Paul Bourne + something that looks like non -Abelian sand piles!

These are beautiful, but I personally prefer, in an aesthetic sense, the strange attractors that have some asymmetry to them. However, I feel a read-through of the equations and techniques used here will be helpful.

Also check out Jim "Chaos" Crutchfield's work, and the mesmerizing video feedback he made with his analog video processing computer that he built at the University of California, Santa Cruz for his Ph.D. in physics in 1984.

Jim Crutchfield

https://en.wikipedia.org/wiki/James_P._Crutchfield

https://csc.ucdavis.edu/~chaos/index.html

Space-Time Dynamics in Video Feedback

https://www.youtube.com/watch?v=B4Kn3djJMCE

A film by Jim Crutchfield, Entropy Productions, Santa Cruz (1984). Original U-matic video transferred to digital video. 16 minutes.

See:

https://csc.ucdavis.edu/~chaos/chaos/films.htm

Citation: J. P. Crutchfield, "Space-Time Dynamics in Video Feedback". Physica 10D (1984) 229-245.

https://csc.ucdavis.edu/~chaos/chaos/pubs/stdvf-title.html

Chaotic Attractors of Driven Oscillators

https://www.youtube.com/watch?v=Sq8Vu40Bw1g

A film by Jim Crutchfield, Entropy Productions, Santa Cruz (1982). Original 16mm transferred to U-matic video and then to digital video. 13 minutes.

In 2022, Crutchfield and his graduate student Kyle Ray described a way to bring the heat production of conventional circuits below the theoretical limit of Landauer's principle by encoding information not as pulses of charge but in the momentum of moving particles.

https://www.scientificamerican.com/article/lsquo-momentum-co...

While a graduate student, Crutchfield and students from the University of California, Santa Cruz (including Doyne Farmer) built a series of computers that were capable of calculating the motion of a moving roulette ball, predicting which numbers could be excluded from the outcome:

The Eudaemonic Pie / Newton's Casino: The Bizarre True Story of How a Band of Physicists and Computer Wizards Took on Las Vegas

https://archive.org/details/eudaemonicpie00bass_0/mode/2up

Thanks for all the references!

Can't be read on mobile (Brave browser)

Horizontally only;) But the article is well worth it.

Please stop feeding drugs to my pet Spirograph.

> Chaos

I think there's more like a semantic slipping over this term, so that chaos ends up passing for another kind of order.

Chaos is chaos. It's a bunch of really random stuff happening within an undefined, undetermined space, with randomly variable means. Like a landslide is chaos. The arrangement of stuff post-landslide will be a complete, incoherent, unpredictable mess.

Today's normie "scientific" definition of "chaos" is what stands for "weird" or more accurately, ASYMMETRICAL.

More philosophy, please.

No, Chaos is the enemy of Control!

https://www.youtube.com/watch?v=3KF5NfzmIvU

"This is KAOS. We don't shoosh here!"

"A new kind of Science". Wolfram. Don't leave home without it.

Seriously though. We talk about AI, but In My Opinion, the most powerful and interesting game changer is "Complex Adaptive Systems". I call them gestalts - the sum is greater than the parts because of how the parts interact to produce behaviors. Gestalts are all around us and we seem bizarrely unaware of them. Do you have some money? That means you are participating in a complex adaptive system or gestalt. Got cells? Perhaps you are a gestalt? Are you self aware? Does that mean you are a self aware gestalt, participating in other gestalts. Etc.

The difficult question is whether you (yes you personally) can build a gestalt. Can you take some pieces - entities - give them a set of rules and get them to do some accomplish a purpose. That is the problem Wolfram was asking

The RIP router is a defunct example that has all the pieces. Entities + Ability to interact + rules => function.

Oh, and don't forget to vote.

http://bactra.org/reviews/wolfram/

Thanks, whenever Wolframs stuff comes up that bactra piece and this one:

singlelunch.com/2020/04/23/why-stephen-wolframs-research-program-is-a-dead-end

Should be mentioned since people should have the context that he's considered a crank byoat everyone in the communities he's doing research in

I used to believe in the “sum is greater than the parts” bit, in the context of emergence etc, but then I’d realized it’s at odds with basic information theory: where does the information go? what computes these behaviors? To me the answer is pretty clear: you’re outsourcing some of your storage and compute into the environment. Without a suitable substrate none of this would be possible.

In other words, metasystems via metasystem transitions.

See Cosmic Evolutionary Philosophy and a Dialectical Approach to Technological Singularity

https://www.semanticscholar.org/paper/Cosmic-Evolutionary-Ph...

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