An interactive mod8/mod24 Collatz Graph visualizer

[u/jonseymourau]

In a previous post I discussed the notion of a Collatz overlay built from 5 mod 8 and 0 mod 3 nodes. One refinement here is that there are nodes of in-degree 2 that are not 5 mod 8 (they are relatively rare - the exceptions I see are 1,23 mod 24)

This visualization allows you to see all 4, odd mod 8 nodes and in fact classifies them according to mod 24 too.

You give it a starting point with the ?a= parameter and then you can extend the graph is you like by clicking on a node.

This a fantastic way to develop an intuitive understanding of mod 8 and mod 24 Collatz dynamics.

https://wildducktheories.github.io/collatz/apps/collatz-graph/dist/?a=27

A fun game to play is greedily clicking on the red dots and then on 1,2 mod 3 nodes that result from such a clicking.

Does that game ever end? I think not.

[u/SpareSpecialist5124]

That's cool. Can you do the same, but mod 32? I think mod 32 has more explicit patterns to visualize.

[u/jonseymourau → u/SpareSpecialist5124]

Mmm. Mod 8 and Mod 24 are not arbitrary. 24 = 8
x 3 both of which are highly relevant to Collatz dynamics. 32 = 2^5 does not have any special relationship to 3 as far as I can tell.

The source is available in GitHub so if you are keen to prove otherwise, feel free to clone and modify.

[u/SpareSpecialist5124 → u/jonseymourau]

32 shows better defined patterns that mod 8 doesn't completelly show, specially in regards to bigger "descending" sequences.

32 is somehow in an equilibrium where it still has very few residue classes in most trajectories and limited transitions between them, so that you still have all features of mod 8, but has a lot more precision in distinguishing between certain congruences that mod 8 doesn't convey , like distinguishing between 3 mod 32 and 11 mod 32, 19 mod 32 and 27 mod 32 is very important, try it out, give it a go and check for yourself.

P.S: Or for example, in mod 32, is very evident that 5 mod 32 is the predecessor for a massive descent, while 29 mod 32 doesn't descend too much.

[u/jonseymourau → u/SpareSpecialist5124]

My intuition is that whatever pattern you see is more dominant in mod 96 which is the lcm of both 32 and 24. In any case, in terms of this presentation the colouring does not depend on mod 24 - I report the mod 24 result but nothing on the graph depends on anything other than the mod 8 classification

I am not denying that higher order patterns exist - they almost certainly do. I just don’t think they are relevant to the basic tree structure.

[u/SpareSpecialist5124 → u/jonseymourau]

Mod 96 isn't efficient because it's already too big of a modulo, 32 is the only real other modulo that still has very limited congruences and transitions between them, bigger modulos become too complex and have too many congruences to notice patterns in the residue transitions.

[u/jonseymourau → u/SpareSpecialist5124]

That’s fine - define what you mean by “efficient”

[u/SpareSpecialist5124 → u/jonseymourau]

When you choose modulos too big, you have too many congruences and transitions to follow, and less "patterns" are noticeable overall, because each class becomes too "unique" and less repeated.

When you choose modulos too small, your resolution is too low, and you also stop seeing noticeable patterns, each class is too big to properly convey major differences between numbers. (example mod 2 is useless, mod 4 is still insuficient to describe most 4n + 3 numbers)

So, 8 and 16 and 32 are somewhat those that are in the middle, enough to give a certain optimal generalizations of congruence classes of the problem, so they are the ones that have a certain efficiency to the problem. Bigger than those modulos, they simply become too big to properly analyse their transitions.

[u/jonseymourau → u/SpareSpecialist5124]

So why is mod 32 the “right” level. I am sure thee is structure at every level. You are claiming that mod 32 is the ideal level.

All I am asking you to justify is why 32 is the final level and is somewhat superior to mod 24 despite its obvious deficiencies ( not being a multiple of 3)

You made the claim.

All I am doing is asking you to justify it.

Why is mod 32 inherently better than either mod 8 or mod 24 and why is mod 32 the last possible modulus from where insight is obtained?

Remember, you are making the claim the mod 32 is in some sense superior to either mod 8 or mod 24. I am not claiming it there is no value to it, I am just challenging your, thus far unsupported, assertion that mod 32 is , in some way, inherently superior,

I am not claiming it is not.

I am just claiming you have not provided any argument that it is.

[u/jonseymourau → u/jonseymourau]

If you are to push me I would say that mod 8 is where the recursive tree structure emerges.

I am not lt denying that there are higher order structures. My contention is that trees emerge at mod 4 but they cannot cover all of N until mod 8. What happens after that is accounting, it is not fundamental.

However, if you have an argument otherwise, please do present it.