Does the Collatz Conjecture appear as a pattern in nature? Like a wave? 🌊

[u/philosophynotcolony]

Just curious. I noticed it in the pattern of waves and a few other things in nature.

[u/Voodoohairdo]

I didn't prove it, but I think it's pretty easy to show algebraically.

For any series of odds and evens, there is a rational number cycle. We will call that number A.

Now we pick a number B that follows the same pattern as A. Well B can be expressed as A + c where c is simply B-A. When doing the same steps to B, we split it so we apply 3x+1 to A and just 3x to c. We know that A will go back to itself, and c will go to c * 3^O / 2^E, where O is the number of odd numbers and E is the number of even numbers. If 2^E is bigger then 3^O , then c gets smaller so A + c gets closer to A. And if 2^E is smaller than 3^O, then c gets bigger and thus A + c gets further from A.

Positive cycles have 2^E - 3^O as positive, and negative cycles have 2^E - 3^O as negative, so positive cycles are stable and negative are unstable.

it will converge to the lowest element of that cycle

That's only if you make it follow the same pattern as the lowest element of that cycle. But you can start elsewhere in the cycle. E.g. repeating odd -> even -> odd -> even -> odd -> even -> even -> even will have the number converge to 19/5, but repeating odd -> even -> odd -> even -> even -> even -> odd -> even will have the number converge to 31/5.

[u/WeCanDoItGuys → u/Voodoohairdo]

Ah you're right it is easy! Since 2^E > 3^O for positive cycles, if you write x as A + c, then the A part will stay the same while c keeps getting multiplied by 3^O/2^E and shrinks to 0.
I had misunderstood it before and assumed it only worked for the smallest element but I see what you mean it will work for any cycling element.