Numbers Are Leaves
christo.sh
Numbers Are Leaves
1–10 of 74 posts
Re: Numbers Are Leaves
#2This talks about representing numbers as graphs whilst showing that the graphs can appear to be reminiscent of leaves, not that the numbers are leaf nodes of graphs, which is where I thought it was going at first.
Re: Numbers Are Leaves
#3Beautiful!
(And now we all go cross eyed trying to see if there's a pattern for factoring in there somewhere...)
Re: Numbers Are Leaves
#4> I would like to understand why numbers looks like leaves and not some other fractal like snowflakes for example (the inverse question, why biological leaves have this specific self-similar structure may yield more fruit).
Snowflakes are dendrite formations highly influenced by how nucleation happens, the graph layout choices highly impact what you will see;
Von Neumann Ordinals are possibly better approached as hierarchical branching processes, and being deterministic, Tokunaga self-similarity is one fairly elegant path to follow in the deterministic case such as this. (IMHO)
Re: Numbers Are Leaves
#5> I think a fair thing to ask is if the force-directed layout engine is uniquely responsible for the leaf-like structure and this has nothing to do with the Von Neumann ordinals themselves.
> To check this I generated some Collatz trees and they ended up looking like microbes. I think it's safe to say the answer is no.
"Uniquely responsible" seems to be doing too much work. These structures are reminiscent of dendritic fractals coming about from diffusion-limited aggregation in electrochemical deposition[1], which is a transitional phenomenon with a phase diagram (molar concentration of the electrolyte vs voltage) There is a sweet spot in the middle where you get intricate dendritic crystals: outside of that you get smooth layers or big blobs / spikes without much internal structure.
I suspect it is primarily the modeling of the forces which is responsible for the behavior, with the topological structure of the tree somewhat indirectly corresponding to the chemical and electrical parameters. I am by no means an expert but it seems likely to me that these von Neumann dendrites vs the structure of Collatz trees is a fairly shallow relationship: lots of trees with similar graph-theoretic properties to the von Neumann trees would also demonstrate the leaf-like fractals, but with no meaningful relationship to the natural numbers. But it would be interesting to make this more precise.
[1] https://en.wikipedia.org/wiki/Diffusion-limited_aggregation
Re: Numbers Are Leaves
#6> why biological leaves have this specific self-similar structure may yield more fruit
I hope the pun was intentional :)
That said, I think the actual answer is in part due to the choice of presentation rather than the mathematical structure — Force-Directed graph layout — despite the line immediately after this quote.
The outer surface of a leaf doesn't physically blend into other leaves that it touches, so it gets repelled mechanically during growth, and it is also connected mechanically to the parts it grew from.
Consider the failure of Collatz trees to look like leaves as a counterpoint to the failure of the `dot` graphs to look like leaves.
Re: Numbers Are Leaves
#7Too few examples. Get to 2039484 and report back if it looks like a leaf or a blob.
Re: Numbers Are Leaves
#8I went into it thinking they meant numbers are leaf nodes of a graph, but no: they meant the entire graph looks like a leaf.
Re: Numbers Are Leaves
#9> I would like to understand why numbers looks like leaves
1st of all they don't. The graph doesn't look like pinnatids or palmatids. There is some resemblance of an alternating disposition of leaves, but that's not the shape of the leaf itself but the distribution of them, and it's a stretch.
Secondly, I'll take the generous interpretation of the question which is, why the graph looks mathematically like leaves, and not a question of biological impact, whatever resemblance is a coincidence which brings us to
Third, OP is playing a game with numbers with no objective goal (yes all maths is this, but this is straight up chmess), there is no ultimate meaning to be derived of it.
Re: Numbers Are Leaves
#10This has almost nothing to do with numbers or ZFC and almost everything to do with how graph layout algo's produce their outputs...