Mountains are not cones, clouds are not spheres, trees are not cylinders, neither does lightning travel in a straight line.—Benoit Mandelbrot, who died in October

He coined the term “fractal” to describe the complexity of forms we see in nature. The salient feature is self-similarity, which means that each small part of the structure has the same overall form as the structure itself—scale invariance. Keep zooming in or out and it looks largely the same. We see this in everything from coastlines to blood vessels.

Non-linear dynamic systems, which have inherently unpredictable behavior known as chaos, because the structure of their so-called attractors in phase space, which describe their behavior, have fractal structure.

This discovery dashed the hopes of those who previously believed that, given enough detailed data about the present state of a system, one could predict the future of the system with arbitrary precision. This is now know to be impossible. That’s why we can’t solve a 3-body problem or other seemingly simple aspects of nature. That’s why things like weather forecasts will always be problematic.

Technically “chaotic” isn’t the same as “random” (the former is deterministic, the latter is not) but functionally they amount to the same thing.

The book to read is Chaos by James Gleick.

Not sure what you mean by “opportunity across time.”

It depends on what you mean by “random complexity.” Fractals in nature are not random, but…oh bugger…gasman just gave a better answer.

Fractals occur because that’s the state of energy that is the lowest for them to occur. It’s “easier” for leaves to make fractal patterns because if they didn’t, they’d use more energy. Similarly, honeycombs, tree bark, and soap bubbles and beer bubbles often share a hexagonal pattern because it’s the most efficient use of space.

My favorite naturally occurring fractal

@ninjacolin Great question. You might take a look at this question and some of the answers to it. Also see this video. This is just part one, but it’s worth watching all the parts.

As you can see, nature has used fractals to design since the very foundations of the Universe. Our galaxies are fractals. The Universe is a fractal. Wherever we find dynamical systems pushed far from equilibrium, we see them being ordered by fractal geometry.

@gasman said:

“Non-linear dynamic systems [...] have fractal structure. This discovery dashed the hopes of those who previously believed that [...] one could predict the future of the system with arbitrary precision.”

I don’t understand this set of sentences. Could you rephrase?

“Technically “chaotic” isn’t the same as “random” (the former is deterministic, the latter is not) but functionally they amount to the same thing”

Since non-determinism has never been observed so they must mean the same thing until some evidence develops. :)

As for “opportunity across time”.. I meant to compare “chaos across time” and “opportunity across time.” Since, it’s the iterations of Moments that produce the variations we’re speaking of. Know what I mean?

I’m certain that this math describes the “shape” of human intuition.

@ninjacolin There are purely stochastic “white noise” events observed in nature. It requires a very large number of samples to allow for statistical analysis distinguish them from seemingly random events which are actually the ordered chaos produced by a non-linear dynamical system with multiple strange attractors driven far from equilibrium. Both exist, and they are not one and the same.

By deterministic, I mean that the equations of motion exactly determine the future state of the system. Nonetheless the system displays “sensitive dependence on initial conditions.” So if you change the initial state by a seemingly trivial amount (in the 8th decimal place, let’s say) then the calculated final state will be completely different due to this small initial change. Since nothing can be measured with infinite precision, its behavior is unpredictable.

This is in contrast to truly random processes, such as radioactive decay or other quantum effects, where the probabilities of events can be calculated precisely from equations, but their actual occurrence is nonetheless random and statistical. This is empirically true despite Einstein’s protest that “God does not play dice.”

@ETpro there are many things that occur that I don’t know the origins of but there isn’t anything I can think of that exists without being required to do so. I believe everything that exists necessarily does so.

@ninjacolin Aha, That gives me a better understanding of what you mean by deterministic. THat, I agree with.

@ninjacolin I don’t understand this set of sentences. Could you rephrase?
I’ll try—I have a background in physics but this stuff wasn’t even known by the time I said goodbye to physics & moved on to other fields…

Once we had Newton’s laws of motion and a full analytical approach to describing physical systems—i.e., by the close of the 19th Century—there was great confidence among physicists in the “clockwork universe”—if you know the position and momentum of every particle, then (in principle) you could work out how the future will unfold. This is the epitome of determinism.

Sure, you can’t predict (let’s say) how billiard balls will fly apart on the break shot, but that’s just because details are lacking. If you could somehow measure everything with great precision then you could predict every ball’s trajectory. Small deviations from initial conditions give rise only to small deviations from future behavior, so approximate knowledge of a system’s present state is sufficient at estimating future states.

Eventually, however, it became obvious that non-linear dynamics—the rules that govern the behavior of most real-world physical systems—often give rise to chaotic behavior, characterized by (as described above) sensitive dependence on initial conditions. This changes the entire view. Now you have to know exactly—with infinite precision—what the initial state is before you can predict future behavior. Otherwise the system is inherently unpredictable, even though it still qualifies as deterministic.

Physicists speak of “phase space” as a multi-dimensional description of a many-body system. In this mathematical view, as a system evolves it traces a path through phase space. In the old view this path was simple and smooth. With chaos theory, however, this path is itself a fractal that defies detailed description. This is the connection between fractals and chaos.

I defer to others to explain it more clearly. If I knew more I could probably do better!