Interesting pattern in the number of steps in the Collatz conjecture.

Question

I was playing around with the Collatz conjecture and plotting random things when I came across this.

The x axis is the natural numbers and the y axis is the number of steps it takes for each number to reach one.

This looked interesting to me, I mean, there is definitely a pattern here, don't you agree? :)

Does this look similar to anything?

Yes, I manually generated a bunch of numbers and plotted them using Mathematica. — kasra, Jan 15 '21 at 13:08
numbers and the number of steps it took them to reach one up to 50000. — kasra, Jan 15 '21 at 13:09
The x axis has been cut out in the screenshot. It goes up to 50000. — kasra, Jan 15 '21 at 13:10
The picture also appears in this article (https://arxiv.org/abs/1911.01229), so maybe there's also some information in it that you consider interesting. — Jake28, Jan 15 '21 at 13:13
the lower chart at https://oeis.org/A006577/graph is similar — Henry, Jan 15 '21 at 13:13
https://en.wikipedia.org/wiki/Collatz_conjecture and https://math.stackexchange.com/questions/687416/are-there-any-explanations-for-these-patterns-in-the-collatz-sequences — , Jan 15 '21 at 15:44

Math777 · Accepted Answer · 2021-01-15T19:57:09.573

You asked if your plot looks similar to anything. The simple answer is that the plot is not new as similar ones can be found on Collatz conjecture Wikipedia page.

Your other question ask if anyone agrees that there is a pattern. Yes, I do agree there is a general pattern. The graph shows that at different depths on the Collatz tree with root 1, there are clusters of numbers whose distance apart is small; hence the appearance of dashed lines. Additionally, the curved downward patterns are an indication that some small natural numbers require more steps to reach 1 than the larger natural numbers that they ascend to before the trajectories descend.

pbmath · Answer 2 · 2024-02-08T02:59:11.577

There's two ways to answer what it looks like. As for what other "graph" or "spiral" it looks like, what comes to mind is the Sacks Spiral which I think has to do with prime numbers. Sacks Spiral image (The Sacks spiral is not the closest lookalike I admit.)

Your picture also looks like the intersection of a bunch of logarithms (green) and negative logarithms (red). I'm not saying that's what it is, just what it looks like. collatz steps image

It also looks to me like there's some kind of 3D thing going on, even though it's a 2D image. This is the same graph as yours, but from 1 to a 100 million, taken from Wikipedia. Cylindrical shape. It looks like a cylinder to me.

Interesting pattern in the number of steps in the Collatz conjecture.

2 Answers2