> 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.
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...
mkl · 1h ago
"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...
willrftaylor · 5h ago
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.
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.
WithinReason · 4h ago
The author invented a new low discrepancy sequence generator
manwe150 · 8h ago
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!
willmarquis · 6h ago
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.
ginko · 3h ago
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.
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?
spocchio · 1h ago
Awesome! Now, what about plotting in 3D? with the coordinates of (f_n, f_{n+1}, f_{n+2}) ?
90s_dev · 7h ago
> 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?
coolcase · 33m ago
Most places I work have a shower - albeit not just off the main office. Usually part of the building infra. It's great if you cycle to work for example or play sport at lunchtime. It's a good thing if well placed.
dotancohen · 32m ago
What's the trick to finding it?
gpm · 7h ago
I've worked in an office with a very nice (clean, private, towels and soap dispensers provided) shower. Actually two (in separate rooms). And taken showers in them. It definitely was not gross, nor do I think anyone thought it weird, nor was I the only person who used them.
After a bicycle commute to work though, not randomly during the day.
xigency · 6h ago
Taking a wild guess, was this at Fog Creek / Stack Overflow?
philjohn · 23m ago
That last 3 companies I've worked for have had this. 2 FAANGs and 1 mid size, household name, tech company.
gpm · 6h ago
Nope, Bay Area not NYC.
90s_dev · 7h ago
Depends. In skyscrapers with multifloor offices, sure. Regular small-ish offices, the showers would be too proximal for comfort.
gpm · 6h ago
Still going to disagree... these are effectively equivalent to the bathrooms you would see in the average house. Designed for purpose of course (e.g. no tub, and a rack of rolled towels instead of a place to hang them).
If they were communal ones like you'd see in many gyms I'd see your point. Or if they weren't very well cleaned. But this was just... convenient and nice. Apart from enabling more active transport to work, I don't think anyone thought twice about them.
The office was technically multifloor, 2. Probably a few hundred people in the office on an average day (no clients, just employees). Solely in use by the company I was working for.
amszmidt · 6h ago
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.
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...
https://bookdown.org/kevin_davisross/probsim-book/sec-linear...
Very cool to see there is some patterns hiding in the randomness too!
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?
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?
After a bicycle commute to work though, not randomly during the day.
If they were communal ones like you'd see in many gyms I'd see your point. Or if they weren't very well cleaned. But this was just... convenient and nice. Apart from enabling more active transport to work, I don't think anyone thought twice about them.
The office was technically multifloor, 2. Probably a few hundred people in the office on an average day (no clients, just employees). Solely in use by the company I was working for.
Having showers at work is awesome, it means you can take a proper bike ride to work, and freshen up.