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.
A shower thought turned into a Collatz visualization
11–20 of 32 posts
Re: A shower thought turned into a Collatz visualization
#12> 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 y…
Sinks where you wash dishes tend to be far more dirty than any shower.
Having showers at work is awesome, it means you can take a proper bike ride to work, and freshen up.
Re: A shower thought turned into a Collatz visualization
#13> 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": htt…
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Re: A shower thought turned into a Collatz visualization
#14> 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": htt…
For what it's worth, I thought "uniform" was a fine description - as you zoom out, the pattern looks more organised and less random. That is a property of uniform distributions.
https://bookdown.org/kevin_davisross/probsim-book/sec-linear...
Re: A shower thought turned into a Collatz visualization
#15> 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": htt…
Irrational rotations of a torus are uniformly distributed and closely resemble the image from the blog. The images you linked, on the other hand, are random sequences with positive entropy (which are also uniformly distributed). Confusing these two things is what happens when someone without the necessary expertise tries to sound smart.
Re: A shower thought turned into a Collatz visualization
#16[flagged]
Re: A shower thought turned into a Collatz visualization
#17> 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": htt…
The author invented a new low discrepancy sequence generator
Re: A shower thought turned into a Collatz visualization
#18Talking 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?
Re: A shower thought turned into a Collatz visualization
#19> 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": htt…
"Uniformly distributed" doesn't just mean sampled from a uniform distribution.
It also means evenly spaced, as is the case here. It reminds me of Poisson disk sampling. Here's an article about Poisson disk sampling that uses "uniform" in the sense the author is, and also compares to the uniform probability distribution: https://medium.com/@hemalatha.psna/implementation-of-poisson...
Re: A shower thought turned into a Collatz visualization
#20Awesome! Now, what about plotting in 3D? with the coordinates of (f_n, f_{n+1}, f_{n+2}) ?