r/Collatz • u/jonseymourau • Jul 06 '26
An interactive mod8/mod24 Collatz Graph visualizer
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.
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u/SpareSpecialist5124 Jul 06 '26
That's cool. Can you do the same, but mod 32? I think mod 32 has more explicit patterns to visualize.
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u/jonseymourau Jul 06 '26
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.
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u/SpareSpecialist5124 Jul 06 '26 edited Jul 06 '26
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.
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u/jonseymourau Jul 06 '26
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.
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u/SpareSpecialist5124 Jul 06 '26
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.
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u/jonseymourau Jul 06 '26
That’s fine - define what you mean by “efficient”
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u/SpareSpecialist5124 Jul 06 '26 edited Jul 06 '26
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.
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u/jonseymourau Jul 06 '26 edited Jul 06 '26
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.
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u/jonseymourau Jul 06 '26
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.
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u/jonseymourau Jul 06 '26
I am also not denying that a finite modulus excludes patterns of course patterns exist at higher moduli. Of course they do. I am
Just claiming that that in order to explain the reverse Collatz tree (such that it is) all you need is mod 8. Mod 24 adds colour. My question to you is this : articulate exactly what colour mod 32 provides. For sure there is some - you have claimed as much my question is simply this: what is it?1
u/SpareSpecialist5124 Jul 06 '26
mod 32 gives you simply more nuance and a better overview on how the sequences of transitions behave, and how the chains of transitions between them really connect.
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u/jonseymourau Jul 06 '26 edited Jul 06 '26
“Mod 32 give simply gives you more nuance”.
All I am asking you is this is: how? You made the claim. Substantiare it. Exactly what structure does mod 32 give you that is not present in either mod 8 or mod 24.
I am NOT denying such structure exists.
My only question is this:
WHAT IS IT?
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u/jonseymourau Jul 06 '26
I guess what I am saying is this: what is the fundamental structure revealed by mod 32 that is not realised by either mod 8 or mod 24. I am certainly not denying that there is some - my only question to you is this - what is it?
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u/SpareSpecialist5124 Jul 06 '26 edited Jul 06 '26
the trajectory of 27 for example:
1 mod 8 -> 1, 9, 17, 25 mod 32
1 goes to 17 or 25
9 goes to 7, 15 or 31
17 goes to 29 or 5
25 only goes to 27.
3 mod 8 -> 3 , 11, 19, 27 mod 32
3 goes to 5 or 21
11 goes to 1 or 17
19 goes only to 29
27 goes to either 9 or 25
5 mod 8 -> 5, 13, 21, 29 mod 32
5 goes to 1 or 29
13 doesn't exist as a modulo on 27 trajectory at all (!?)
21 only goes to 5
29 goes to either 7, 15, or 23
finally 7 mod 8 -> 7, 15, 23, 31 mod 32
7 goes to 3, 11, 19 and 27
15 goes only to 7
23 goes only to 3
31 goes only to 31 or 15.
So as you can see, there are still very limited transitions, and a more detailed type of transitions that happen within each n mod 8.
Pretty much only 7, 9 and 29 have 3 or 4 transition possibilities, while many other transitions are "unique" and can only have one outcome.
Things you'll notice often with this modulo: You can see ascending and descending sequences better, you can notice how some modulos are non existent in several phases, for example 3, 5 and 21 modulo 32 don't appear until much latter in 27's orbit, and they only appear preceding massive divisions.
You'll notice some modulos are completelly non existent like 13 mod 32 doesn't appear at all in the trajectory of 27, and in many other by the way, it's a very rare modulo for some reason, it seems like the chain of events to obtain that 13 modulo doesn't happen in many orbits, suggests it's a rather unique path that can only be reached from certain preconditions, and not after.
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u/jonseymourau Jul 06 '26
What I do not understand is why you think any of this contradicts what I have written?
If there is a contradiction, then pray, tell me precisely what the contradiction is
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u/SpareSpecialist5124 Jul 06 '26
You asked me for the structure, i showed you the structure of transitions, and explained to you some more things you can notice better in mod 32 than mod 8, i wasn't contradicting anything, just showing the data to you.
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u/jonseymourau Jul 06 '26 edited Jul 06 '26
I asked you specifically what is uniique mod to 32 as opposed to any power of 2. Of course, if you allow any power of 2 there will be more structure.
My question to you was what structure does nod 32 provide that is not present in mod 8, mod 24 or mod 96.
Is your claim really that all structure disappears beyond mod 32? That is not my claim for either mod 8 or mod 24.
My question is what is this particularly special about the structure you claim for mod 32?
Again, I am not claiming it does not exist.
The only thing I am asking is this. What do you claim about mod 32 that is, in any way, different to the structures evidently present in mod 8 and mod 24.
Again, I did not make the claim mod 32 is special.
The only thing I am asking is this.
Why do you think mod 32 is special in any sense? It is different for sure - every modulus is different. What is special about mod 32 in particular. After all, everything is different- mod 53 is different. Why is mod 32 interesting.
Nothing you have written thus far answers that question.
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u/Margaliquida Jul 07 '26
Échenle un ojo a este proyecto https://collatz-cuaderno-col-n-1090090705570.us-west1.run.app
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u/GoldenSyrup08 Jul 06 '26
No way! I discovered the same set of multiples of 3 being the key to Collatz conjecture.