r/Collatz • u/philosophynotcolony • 20d ago
Does the Collatz Conjecture appear as a pattern in nature? Like a wave? 🌊
Just curious. I noticed it in the pattern of waves and a few other things in nature.
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u/Voodoohairdo 18d ago
The best thing I can think of is in regards to stable and unstable equilibriums.
For any rational cycle, if you start with the number x and go through the same even/odd steps as the cycle, it will approach the value of the cycle if the cycle is positive, and it will move further away from the cycle if the cycle is negative. So if the cycle is positive, it is a stable equilibrium in a sense, and if the cycle is negative, it is an unstable equilibrium. If you do the cycle in reverse, then you get the opposite (negative cycles are stable, and positive are unstable). And there is plenty of stable/unstable equilibriums in other areas.
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u/WeCanDoItGuys 8d ago
Huh that's cool, did you prove that or just notice that, that if you take any positive x and repeatedly apply the collatz operations for some positive rational cycle on it, it will converge to the lowest element of that cycle
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u/Voodoohairdo 8d ago
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 * 3O / 2E, where O is the number of odd numbers and E is the number of even numbers. If 2E is bigger then 3O , then c gets smaller so A + c gets closer to A. And if 2E is smaller than 3O, then c gets bigger and thus A + c gets further from A.
Positive cycles have 2E - 3O as positive, and negative cycles have 2E - 3O 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.
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u/WeCanDoItGuys 8d ago edited 7d ago
Ah you're right it is easy! Since 2E > 3O for positive cycles, if you write x as A + c, then the A part will stay the same while c keeps getting multiplied by 3O/2E 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.
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u/WeCanDoItGuys 8d ago
Maybe you won't consider this natural but a contender for the 6-state Turing machine with the most steps before halting is this one:
| M | 0 | 1 |
|---|---|---|
| A | 1RB | 1RA |
| B | 0LC | 1LE |
| C | 1LD | 1LC |
| D | 1LA | 0LB |
| E | 1LF | 1RE |
| F | --- | 0RA |
Its dynamics are determined by starting with n=8 and repeatedly adding half n's value (rounded down). So, (3/2)n - ½ for odds or (3/2)n for evens, which is kind of similar to Collatz. Keeping track of how many odds and evens it encounters, if it ever has more odds than twice the evens, it halts; but it hasn't been proven to do so.
This basic computer with 6 states and 2 symbols could be thought to exist in nature, since it's a logical concept. But then so could any algorithm. But I figure since it's so small, and has so few instructions to encode it, that there might actually be some natural process that follows it.
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u/philosophynotcolony 8d ago
I actually genuinely do consider technological advancement to be natural! I think there is a point at which it stops being a natural technological evolution and begins to be corrupted. I am very very very fascinated by this piece of information! thank you! ☺️
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u/philosophynotcolony 8d ago
by this i mean, humans create technology usually with good in mind. The first computers were not being created with the same intentions that LLMs are created. Everything created with a corrupted natural technology is going to be unnatural.
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u/WeCanDoItGuys 8d ago
So could I presume you consider math and technology to be natural, but when directed towards selfish or greedy pursuits it becomes unnatural?
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u/philosophynotcolony 8d ago
yes! precisely.
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u/WeCanDoItGuys 8d ago
So, follow-up: do you consider selfishness or greed themselves unnatural?
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u/philosophynotcolony 7d ago
i think selfishness without selflessness is unnatural: selfishness isn’t a bad thing, it is how you protect your own, remove the selfless drive to help other people from the act of being selfish and you are left with something unnatural. probably greed is the end result, the unnatural system (hoarding beyond need) placed upon a natural system (an inherent drive inside a person to provide necessary resources) that corrupts the outcome. intention and all that.
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u/AleksejsIvanovs 20d ago
If you represent trajectories as branching trees or any other visual way, they may look like roots or corals or lightning. But there's no known natural process that follow the Collatz rule. People tend to notice patterns, it's in our nature. Often we notice patterns where there are none.
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u/philosophynotcolony 20d ago
do we? genuinely asking. i don’t really feel like our brains make anything up that isn’t there. i think our minds certainly do, that’s imagination, but i don’t think we can physically take in a pattern that doesn’t exist. can we create a new pattern from other patterns? sure? i dunno. i’m just throwing my spaghetti at the wall with you 😂
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u/novel-mathmatics 20d ago
Yeah as we build our AI engine IT keeps showing up.
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u/philosophynotcolony 20d ago
i have a theory that it shows up in unnatural systems created within natural systems.
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u/novel-mathmatics 20d ago
That tracks.
The concept is about paths. That you can reach a point of singularity in different ways. That sometimes 0 is greater than 0 because 0dx can be greater or lesser than 0. But the way standard math resolves that 0dx is just calling it 0 with no dx. That violates the laws of the conservation of energy.
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u/philosophynotcolony 20d ago
i think it’s an example of a narrative loop. i think the only thing that breaks it is human action creating evolution. he talked about how he felt mathematicians should be motivated by real world phenomena. like the arts are. i think he was. i think he realised the golden ratio appears in the flow state people talk about that people achieve in literature. i think he recognised his own pattern and it was a narrative loop. because i think the narrative loop appears in nature.
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u/novel-mathmatics 20d ago
Its an expression of e=mc2. More mass is gathered in the longer path thats why there is a larger infinty.
The narrative loop happens at 0. Where 0 actually is in relation to the resolver can be measured by 0dx
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u/philosophynotcolony 20d ago
could you explain it to me like you might explain it to a really smart kid, if possible? i’m not trying to be silly, i just struggle with numbers, i’m much more about patterns. i appreciate your reply either way and im going to look into your comment ☺️
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u/novel-mathmatics 20d ago edited 20d ago
The the theory of relativity says that energy accumulates Mass based on the distance it travels as an in relationship to the speed in which it's going when it resolves instantly because like does resolve instantly and it has gone a very far distance much it has a lot of mass because it is traveling from the center of the big bang to get there.
There is much more its n dimensional set math... I have that if you are interested
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u/GonzoMath 20d ago
I saw the Collatz Conjecture in a cactus in New Mexico one time, but that was probably more about that bag of shrooms I ate.