Complexity of patterns

April 4, 2018

:: complexity of patterns

I wanted to talk about something I've been thinking about recently and quite often.

You look at some complex shape, or pattern, drawing, whatever it may be. And if you know how it was generated, you see the actual simplicity of the rules governing the final output that you see. Imagine that you're looking at some very very complex caustics, that seems almost random. And think of a glass that is tiled with bevels in diamond shape that is casting that caustics. You see cylindiral nature of the glass, that diamond pattern repeated across its surface and remember the refraction law, and that is all there is to produce such complicated results of the caustics for the given light position relative to the glass.

And so the question is, what's the minimal degree of complexity of some system system that can produce the same pattern as the results of it's expresion?

In a case of a glass, is there some even more simple shape and even simpler tile on its surface that can produce identical pattern? Do we need a glass, light and refraction at all, or can we, dunno, for example, put just a few points in space whose gravitational effects would produce identical pattern of projected fields around it on some carefully chosen slice of space?

Looking at it another way, to produce a square pattern, you can either draw a square, or have a cube and slice it (paraller to the faces of it) or project it to get the same square. But it feels like the latter is an overkill, why would you deal with a cube if you can have the square, if you are interested in dealing with squares? So it seems that projected/sliced cube is more complex way to describe the square pattern, so for the sake of simplicity and optimization we would always prefer the less complex version that describes certain behavior. Now imagine that you are observing a pattern that is a square and it starts getting more romboidal as time passes and you want to describe that behavior in the most simple way possible. And square is just not a good fit anymore, but slicing the cube at the angle depending on cosine of the time passed for example is now doing the job perfectly.

So, what's the minimal set of laws and initial conditions for some system that can produce a given pattern? How we measure such dimensionality of a "degree of complexity"? How do we determine what's more complex, e.g. refraction law of tiled bevels on the glass? And maybe, just maybe, is it the case that for a given pattern, no matter which system you choose and how it's set up to produce a given pattern, all those systems share the same "complexity level"? So then the pattern itself has some intrisic property (or something like that) that is bound to it. And maybe, every pattern has unique complexity level, and to get certain patterns, your system must be unique for it to produce those patterns. So instead of observing the system and measuring what kind of patterns it produces, you actually observe patterns and conclude about the system?

Once again, to sum it up, the question was - for a given pattern (recorded outcome), what's the minimal degree of variables (initial conditions and/or laws of the system) that can produce that pattern?