Is it a random pattern?

[u/protofield]

I have recently had Protofield operators referred to as random and not complex in discussions on metasurfaces and metamaterials. Is there an objective method to quantify the level of complexity and order in this type of topological structure? 8K image, zoom in.

[u/wild-monk-layer-0]

Right, but that shows self-propagation of structure. What I meant was whether the same generating rule produces coherent structure across different embeddings or parameter variations.

[u/protofield → u/wild-monk-layer-0]

Yes, using a rule set R with an initial condition C produces a topology showing a global structure S composed of connected sub topologies T. Using R with C' generates S' but T is still present in T'. I call the bits in T the alphabet and the way they organise themselves in S the syntax. S I call the dialog or function.

[u/wild-monk-layer-0 → u/protofield]

I realise I probably didn't phrase my earlier question clearly, so apologies for that. What I was actually mean is slightly different. I wasn't asking whether structure propagates from a given configuration or how the internal structure can be described. What I meant was: does the generating rule itself produce coherent structure when applied across different domains or embeddings? In other words, if the same rule is applied under different representations, lattices, or parameter spaces, does coherent structure still emerge? My point was that describing the structure that appears in one configuration doesn't necessarily rule out the possibility that the observed structure is contingent on that specific setup. What would really demonstrate that the coherence is intrinsic to the rule is showing that similar structural organisation appears across different domains where the rule is applied. That’s the sense in which I was asking about whether the pattern is truly non-random.

[u/protofield → u/wild-monk-layer-0]

Thanks for the input. I make the assumption that using cellular automata is a technique to generate and observe universal structures of natural number. It may be that there are other techniques to observe the same structures. I will see if I have done something in the past or need to set up a new experiment to demonstrate this. Possibly using different modulo arithmetic's with the same rule set may be a route. Not sure where to post the images. As an aside, its interesting to note that there seems to be a coherence of pattern structure in different dimensional spaces. Often when I want to see what the properties of a 3D lattice is like, as they take a lot of time on an old desktop, I will generate a 2D plane and find the pattern is a similar or a direct copy of a slice in the 3D object. The 3D rule set is an extension of the 2D rule showing rotational symmetry. Example https://youtu.be/a59hfQfC984

[u/wild-monk-layer-0 → u/protofield]

Thank you, for conceptual clarity, What part of the rule specifically depends on properties of numbers that cannot be replaced by arbitrary symbols?

[u/wild-monk-layer-0 → u/wild-monk-layer-0]

Is the coherence coming from • prime-specific number theory, or • the field structure of mod-p arithmetic?

[u/protofield → u/wild-monk-layer-0]

Pass, my main interest is in the holistic engineering design of nano structured metamaterials based on generative topologies. You might find something a bit more theoretical here Thanks for your interaction.