[HN Gopher] Symmetry in Chaos
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Symmetry in Chaos
Author : lioeters
Score : 162 points
Date : 2023-08-13 12:42 UTC (10 hours ago)
(HTM) web link (paulbourke.net)
(TXT) w3m dump (paulbourke.net)
| progrus wrote:
| Wow this is cool. I thought it was was going to be the latest
| cosmology from the stringcels.
| ykonstant wrote:
| Those are gorgeous; I would enjoy some explanation on how the
| authors came to investigate these particular recurrences.
| Wolfenstein98k wrote:
| Can't be read on mobile (Brave browser)
| alkyon wrote:
| Horizontally only;) But the article is well worth it.
| [deleted]
| Horffupolde wrote:
| I would argue that position-symmetry is not unexpected in a
| chaotic system. Symmetry means that one of your dimensions is
| redundant.
| ironSkillet wrote:
| Why does it mean that?
| jfoutz wrote:
| 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.
| [deleted]
| [deleted]
| defrost wrote:
| _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.
| euroderf wrote:
| Please stop feeding drugs to my pet Spirograph.
| [deleted]
| cossatot wrote:
| 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.
| IAmGraydon wrote:
| 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.
| viesauvage wrote:
| > 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.
| DonHopkins wrote:
| No, Chaos is the enemy of Control!
|
| https://www.youtube.com/watch?v=3KF5NfzmIvU
|
| "This is KAOS. We don't shoosh here!"
| talkingtab wrote:
| "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.
| ilaksh wrote:
| 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...
| fellowmartian wrote:
| 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.
| defrost wrote:
| http://bactra.org/reviews/wolfram/
| VHRanger wrote:
| 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
| abetusk wrote:
| 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?
| lanstin wrote:
| 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.
| dmbche wrote:
| James Gleick's "Chaos: Making a new science" will answer your
| questions and is a fun read!
| abetusk wrote:
| Could you point to the chapter or page?
| dmbche wrote:
| 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.
| abetusk wrote:
| 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"
| dmbche wrote:
| 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.
| abetusk wrote:
| 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.
| dmbche wrote:
| 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!
| abetusk wrote:
| See alimw's response [0].
|
| [0] https://news.ycombinator.com/item?id=37114898
| alimw wrote:
| All of these are aperiodic; it says as much in the first
| paragraph. Are you talking about the rotational symmetry?
| abetusk wrote:
| 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?
| alimw wrote:
| They iterate f where f(z) = (a0 + a1 z z + a2 Re(zn) + a3
| i) z + a4 zn-1. You can easily check that f(z exp(i 2 p /
| n)) = exp(i 2 p / n) f(z).
| bannedbybros wrote:
| [dead]
| DonHopkins wrote:
| 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
| [deleted]
| HDevo wrote:
| 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.
| jes wrote:
| This reminds me of an Alan Watt's talk on alternating levels of
| order and chaos in nature.
| passion__desire wrote:
| 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.
| jes wrote:
| Please say more!
| andybak wrote:
| Ok. I need to know more.
| passion__desire wrote:
| 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.
| sandman1906 wrote:
| Do you remember the title of it?
| pluijzer wrote:
| 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...
| jes wrote:
| Perfect. Appreciate the ref.
| andybak wrote:
| 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.
| jes wrote:
| Will check it out! Ty!
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