As for the randomness, I have wondered if Collatz sequences are somehow related to the properties of a common prng with multiplier 3/2, infinite length state vector, and mod 2 on the output with this formula: https://en.m.wikipedia.org/wiki/Linear_congruential_generato.... I assume this could be part of what makes the conjecture both interesting and difficult and beautiful.
Very cool to see there is some patterns hiding in the randomness too!
Talking about shower thoughts on Collatz visualizations..
A while ago I though of a way of structuring the collatz orbits by arranging integers in a 2d grid with odd numbers being arranged along the X axis and multiples of the power of two along the Y axis.
https://gist.githubusercontent.com/ginkgo/13121db56b65b1237e...
So essentially any odd number n and all numbers n * 2^m belong to the same group of numbers that eventually reduces to n. All that's left is the 3n+1 orbits which are shown as lines from the odd numbers.
This reveals quite a bit of structure (IMO) especially only every second odd number goes to an orbit reducing to an odd number larger than it (and it's always in the form n * 2^1) all the other orbits every 4th, 8th, 16th odd integer immediately reduce to an odd number that's lower.
Anyone seen an arrangement like this for the Collatz orbits?
Here is a python script that plots on the Z axis - each number through an entire Collatz cycle until the result of 1, on the X axis - the number of cycles needed to get to 1, on the Y axis - add 1 for odd numbers, subtract 1 for even numbers. https://gist.github.com/bwanaaa/4c77b33311916b230c8b1891bab4...
You can open it in colab to visualize it. You can change the range of integers by modifying line 33 in the function def generate_all_sequences():
Interestingly, it seems there are more odd numbers than even ones in a collatz sequence as all graphs tend to the positive Y axis. All numbers tend to generate 3x as many odd results as even and they all seem to do this at this same rate.
In the first 750 integers, the number 703 reaches as high a collatz result as 250504.
Interesting take. The visualization of the inverse tree highlights just how sparse the “preimage space” is under Collatz iterations. The idea that this sparsity contributes to the apparent randomness is compelling. I’m curious whether modeling the process modulo powers of 2 and 3, or via 2-adic analysis, could formalize some of these heuristic observations. Also, the assumption that most numbers “fall off” rapidly aligns with empirical behavior, but it’s still not clear how to bound exceptional trajectories.
> I've been telling people for years if businesses want employees to have better ideas, they should have more showers in their offices. So far everyone seems to think I'm joking. I'm not.
I have definitely noticed that some of my best ideas or breakthroughs come to me when showering, or sleeping, or eating, or driving, or doing the dishes, or basically any mundane autopilot task where my mind is free to wander. But yeah no, having a shower room in the office is both gross and weird. Maybe offices should.. encourage you to... wash some dishes?
Awesome! Now, what about plotting in 3D? with the coordinates of (f_n, f_{n+1}, f_{n+2}) ?
> The points look quite uniformly distributed to me. If I squint, then maybe I can see some structure, but it's hard to describe and I could be imagining it.
It doesn't, these points look like what happens if you ask someone who doesn't know what a uniform distribution looks like to generate a uniformly distributed set of points though.
Here's what an actual uniform distribution looks like... much less "uniform": https://claude.ai/public/artifacts/00549caf-2ec1-4803-b909-6...
Credit to the book "Struck By Lightning" for making me aware of this fact, many years ago now. Disclaimer that the author is a family friend.
Edit: I misunderstood what was being plotted in the article, and as a result had claude plot random instead of evenly spaced X coordinates. It doesn't change my point, but this version has the appropriate distribution to compare to (evenly spaced x, uniformly randomly y coordinates): https://claude.ai/public/artifacts/a04a3023-25d3-4d99-889d-a...