Mapping Collatz function iterations for all and any N into one Graph...it is possible? Maybe!

Feb 2020
27
4
Australia
I have devised a graph in Mod 9 to illustrate the deterministic path that all iterations of N follow under the Collatz function of n/2 if even and 3n+1 if n is odd. Significant or not? Let’s discuss…

Collatz Chart.jpg
A010878a(n) = n mod 9.
0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0,
1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0,….towards ∞
  • Convert n to n mod 9
  • Locate corresponding node on graph (NB - 0 has been replaced with 9)
  • If n mod 9 is derived from an odd number follow broken edge to next state (lands on iterated state)
  • If n mod 9 is derived from an even number follow unbroken line to next state (lands on iterated state)
  • Ex. n =13 so 4 mod 9 (uneven). Follow broken edge back to 4 which now represents 4 mod 9 even (40)
  • 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
  • (4) (4) (2) (1) (5)(7)(8) (4)(2) (1)
  • ANY iterated sequence of n converted to n mod 9 will follow exact paths predetermined by graph

 

skipjack

Forum Staff
Dec 2006
21,479
2,470
The image looks as though it needs to be tidied up near 1 and 8.
 
Feb 2020
27
4
Australia
TheAnsweris42 said:
I have looked at your post as well, about the collatz function. Is it your intent to prove the conjecture by using this map? It is a clever approach, I would like to see if a mapping in 3 dimensions would be possible.
Thanks for looking at my post. The graph is designed as a 1D State Diagram/Directed Graph to reveal the hidden deterministic process that can explain why n always has returned to 1. It's not offered as a proof but as a study of the regulation and structure that underlies Collatz Conjecture . By using the equivalence class of mod 9/base 10 the graph also represents the results of all N under 3n+1 and n/2 iteration. This all needs to be discussed but this Forum does not seem to be appropriate for the subject matter.

I would imagine in 3D it would look like a protein folding animation (biology).
 

SDK

Sep 2016
804
544
USA
You haven't really said why you have drawn this so its hard to discuss it. The fact that this graph is strongly connected tells you from a dynamical systems point of view that looking at orbits mod 9 does not provide any meaningful information. So this can't possibly shed any light on the conjecture. On the other hand it looks cool.
 
Feb 2020
27
4
Australia
You haven't really said why you have drawn this so its hard to discuss it. The fact that this graph is strongly connected tells you from a dynamical systems point of view that looking at orbits mod 9 does not provide any meaningful information. So this can't possibly shed any light on the conjecture. On the other hand it looks cool.
The graph reproduces the exact path of any n under Collatz function. The graph is achieved by converting collatz sequence results to mod 9 so as to represent all N. Try it by selecting n and converting the results of each step into mod 9. (use Digit Sum for speed) Locate n''s mod 9 node and compare your results with the trajectory along the predetermined edges. Broken and unbroken edges are based on whether the mod 9 result has been derived from an odd or even number.
Please just give this a shot before you put me in the 'cranky' department
 

SDK

Sep 2016
804
544
USA
The graph reproduces the exact path of any n under Collatz function. The graph is achieved by converting collatz sequence results to mod 9 so as to represent all N. Try it by selecting n and converting the results of each step into mod 9. (use Digit Sum for speed) Locate n''s mod 9 node and compare your results with the trajectory along the predetermined edges. Broken and unbroken edges are based on whether the mod 9 result has been derived from an odd or even number.
Please just give this a shot before you put me in the 'cranky' department
I understand what the graph means and I'm not claiming its cranky. In fact I think its cool. However, I'm pointing out that it can't contain any nontrivial information about orbits because the graph is strongly connected. In other words, it doesn't reveal any structure or rule out any possibilities. The Collatz conjecture is about ruling out certain kinds of orbits so the point is that reducing mod 9 doesn't really provide any insight into the conjecture. That doesn't mean that what you did is cranky.
 
Feb 2020
27
4
Australia
....the graph is strongly connected. In other words, it doesn't reveal any structure or rule out any possibilities. The Collatz conjecture is about ruling out certain kinds of orbits so the point is that reducing mod 9 doesn't really provide any insight into the conjecture.
Can you explain your reservations concerning the fact that graph is strongly connected? From my interpretation the graph contains all possible orbits under the Collatz iterative process - there are no surprises or inescapable loops except when 1 ≅ 1(mod 9) and then 4, 2,1 loop follows. And it reveals the deterministic and mechanistic structure driving n to 1.

Anyway I'm so pleased at least you think the graph is cool.
 
Feb 2020
62
1
St Louis
I think a good question would be why is it that when we convert any n to mod9 it can be mapped this way. Nothing happens without reason.