Metamagical Themas: Structure and Strangeness, Part 2
Attractors, Strange Attractors, and Butterflies causing Tornados
This is a brain dump about Section Four of Hofstadter’s collection of essays - Metamagical Themas. For an introduction to the book and hyperlinks to all the sections, see HERE. This section is broken up into two parts. This is Part 2. Part 1 is found HERE. I’ll probably take a break from this series after this for a little bit, as I have half a dozen partially hashed-out posts related to things other than this series that have been put on the back burner.
Things that appear strange and chaotic perhaps have more structure and order to them than is apparent at first glance. This was visually demonstrated with the Rubik’s cube in Part 1, but now we’re going to be more… well I’m not sure if it’s more practical or more esoteric.
First, the concept of an attractor.
Attractors emerge when we recursively feed the output of a function back into the function over and over again, and a pattern emerges. The output hones in on a stable answer, regardless of where you started at.
An example:
“In this sentence, the number of occurrences of 0 is ___, of 1 is ___, of 2 is ___, of 3 is ___, of 4 is ___, of 5 is ___, of 6 is ___, of 7 is ___, of 8 is___,and of 9 is ___.”
One way to search for a solution to this puzzle is to fill in the blanks with an arbitrary sequence of ten numbers, such as <0,1,2,3,4,5,6,7,8,9>, and see what happens when you check out the truth of the resulting sentence.
It turns out actually to have two occurrences of each digit. Thus the vector <0,1,2,3,4,5,6,7,8,9> leads to the vector <2,2,2,2,2,2,2,2,2,2> by the process we'll call "Robinsonizing". Where does that vector lead?
Clearly to <1,1,11,1,1,1,1,1,1,1>, which leads to <1,12,1,1,1,1,1,1,1,1>.
which leads to <1,11,2,1,1,1,1,1,1,1>, which leads to <1,11,2,1,1,1,1,1,1,1> -and lo and behold, we've entered a closed loop!
This vector <1,11,2,1,1,1,1,1,1,1> is like a whirlpool or a vacuum cleaner: it sucks things near to it into its vortex. It is a trap, a fixed point - an attractor.
We can graphically depict this concept too.
Imagine an upside-down parabola on a graph with a line with a slope of 1 running through it. Now imagine you are drawing in a 3rd line, and the way you draw that line is governed by two rules: (1) Move vertically until you hit the curve; then (2) Move horizontally until you hit the diagonal line. Repeat steps (1) and (2) over and over again until the end of time.
Actually, that’s hard to imagine. Just look at the picture below.
What is happening in this exercise?
We are led in a merry chase 'round and 'round the point whose x-coordinate and y-coordinate are x. Gradually we close down on that point. Thus x is a special kind of fixed point, because it attracts iterated values of f(x). It is the simplest example of an attractor: every possible seed (except 0) is drawn, through iteration of f, to this stable x-value. This x is therefore called an attractive or stable fixed point. By contrast, 0 is a repellent or unstable fixed point, since the orbit of any initial x-value, even one infinitesimally removed from 0, will proceed to move away from 0 and toward x. Note that sometimes the iterates off will overshoot x, sometimes they will fall short-but they inexorably draw closer to x', zeroing in on it.
We are always drawn to the attractor in this mathematical example- no matter where we start. X = .13, X = 0.0000000001, etc. It doesn’t matter. Run the iteration enough, and you will always begin to close in on the attractor. That magic point seems to be... attracting everything towards it. A more intuitive example is a marble sitting at the bottom of a round dish. If you push it lightly with your finger, it will oscillate for a while but eventually will come to rest again where it was before: at the attractor state. It’s easy to visualize this as an attractor state because it is still. But attractors don’t have to be a fixed point in space; a planet that stably orbits another has found an attractor state in a way.
A third example of an attractor state out in the wild is found when looking at a population of wild rabbits. This video1 by Veritasium will do better than I can type in words, so take a break and watch the first 5 minutes. In short, wild populations tend to stabilize at attractor points. Too many rabbits→ not enough food → decrease population; conversely, Too few rabbits → excess food → increase in the population.2 Eventually, a stable attractor state is reached, regardless of how many rabbits you start with. As you watch, you’ll notice that our parabola from earlier reappears. When he begins to speak about equilibrium rates, at the 3-minute mark, substitute that word for attractor states. I think things will begin to click if you do this.
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Ok. Moving on from Attractors3 to StRaNgE AtTRaCtORs.
Strange attractors
Attractors make intuitive sense. Marbles are attracted to the bottom of the bowl - big deal. How do they become STraNGe?
Back to our friend Chaos. In the physics and math world, it became more and more clear in the 1970’s that the tiniest of changes in the set of conditions led to large discrepancies the farther out you look. This was dubbed the butterfly effect - even tiny, seemingly insignificant changes make can have huge consequences the longer it plays out. The joke is that a butterfly flapping its wings in Brazil leads to a tornado in Texas as a downstream consequence. More creatively, maybe it leads to someone falling in love. That sounds equally predictable and much more optimistic. Regardless, the tiniest of changes in the set of initial conditions of a system leads to larger and larger discrepancies the farther out you project.
So how well can we predict the future? Not very well at all, at least when it comes to chaotic systems. The further into the future, the harder it becomes. Interestingly, the same is true when looking into the past of chaotic systems while trying to identify initial conditions. There is a chaotic cloud of fog that shows up when we look in either direction. But as talked about in part one, maybe this fog can dissipate away.
Well, we have covered attractors and turbulence [Substack readers, we didn’t actually cover turbulence, sorry]; what about strange attractors? We have now built up the necessary concepts to understand this idea. When a periodically driven two-dimensional (or higher-dimensional) dissipative system is modeled by a set of coupled iterations, the successive points lit up by the flashes of the periodic strobe light trace out a shape that plays the role, for this system, that a simple orbit did for our parabola. But when one is operating in a space of more than one dimension, the possibilities are richer. Certainly it is possible to have a stable fixed point (an attractor of period one). This would just mean that at every flash of the strobe, the point representing the systems state is exactly where it was last time. It is also possible to have a periodic attractor: one where after some finite number of flashes, the point has returned to a preceding position. [Substack reader’s, this would be analogous to the rabbit colonies in Veritasium’s video having a period of 2 and then 4]
But there is another option: that the point never returns to its original position in phase space, and that successive flashes suggest it to be jumping around quite erratically inside a restricted region of phase space. However, over a period of time, this region may take shape before an observer's eyes as the strobe flashes periodically.
Let’s break this down.
Imagine a firefly flying around an invisible light source that you cannot see. Every five seconds, the firefly’s light goes off and reveals its location. The first time the firefly’s light flashes, it is over there on the left. The next time it is over there on the right. Then it is back here. Then back on the left but slightly more up. Then it is over there. Repeat this over and over again. If we were to watch this happen in live time, we would have no idea where the firefly would be the next time it flashes. Even if one flash is super close to where it had been previously, that doesn’t clue us in on where it’s going to be next. The location of the fly with every flash appears chaotic and never returns to the same point. But if we tracked the location over time, a pattern would emerge. We would find that it is circling around that invisible infrared light that we didn’t know was there. Mapping these flashes over time, slowly, a sphere would emerge around this unbeknownst light source. This leaves us in a strange place: The fly’s instance-specific position is absolutely unpredictable, yet there is an underlying structure to where it appears over time.
This is what makes it a strange attractor. It never closes in on itself — i.e., it never repeats itself (non-periodic), yet has a structure to it.
One can’t help but wonder what exactly is doing the strange attracting in strange attractors4. In this case, we were told there was a light we cannot perceive, and flies have some sort of nervous system and goals in life (I think). Out in the real world, things are less obvious, though. Strange attractors occur in nonconscious systems by simply doing the recursive steps we started with at the beginning of simple attractors. That is, do something, plug the result back into the equation, and do it again. The random noise we thought was chaos when doing this turned out to have a pattern and structure to it. These patterns are quite aesthetically beautiful.
If you are ambitious, this 13-minute video (after watching it multiple times over the course of a week) made the strange attractors really click for me, but it’s not for everyone.
The take-home of strange attractors (granted, I’m stretching myself even to grasp them5) is that there are systems that are chaotic and unpredictable locally but have structure when zoomed out on a global scale. Why do they have this structure on a zoomed-out scale? I don’t know.
Hofstader ends with a clever quip.
In the Introduction, I described the space of my columns as gradually emerging as, month by month, I revealed one more dot in that space. What is this, if not a Poincaré [firefly example from above] map of my mental meanderings? During my column-writing era, my mind would light up like a monthly firefly and reveal where it was to the outside world! I just wonder: Would the shape I was thus tracing out turn out to be a strange attractor
Like the Fibonacci sequence and other weird mathematical things, attractors and strange attractors seem universally woven into lots of seemingly unrelated topics, like rabbit populations and atmospheric weather patterns. That’s weird, right? Sometimes people fantasize they lived in a world with magic instead of boring old science, but this is the type of stuff that makes me feel like you don’t need to live in a fantasy world to be feel a sense of wonder.
I originally watched this video years ago. At that time, I laughed at the top comment and clicked like. When stumbling across it again recently, I laughed again at the top comment, and clicked like again, only to realize I had already clicked like on it years ago.
This is a simplification. Predator population levels, etc. would influence the rabbit population too.
I often wonder and speculate how this concept (attractor states) could be applied to psychopathology and consciousness science. I wish someone much smarter than me had written a book on the intersection of psychiatry and attractor states. How much of the mysteries of the mind will one day be better understood in these terms? Memory recollection, sense of self, Addiction as an attractor state?
if anything at all! I’m not even sure this is a valid question to ask. It may be like asking, ‘What is the smell of yellow’? It’s an invalid question. The words that form a sentence and look like a question. Ah yes, what is the smell of yellow? But truly, the concepts are not compatible, and spending time trying to answer the question is unfruitful. Other questions like ‘What is the meaning of life’ I think fall into this trap of looking like a valid question.
I’ve elsewhere read that strange attractors are fractals, but I don’t what that even means. Making me confident that I don’t understand this all too well.