Mandelbrot & CoBlog

Maths

Collatz fractal - an unespected connection

This is about how I figured out a connection between the "take-off time" and the "stopping time" of the Collatz sequence.

I came across an excellent article on Xander's blog a few weeks ago about the Collatz conjecture. The article dates back to 2012, but caught my attention. So I decided to replicate the results.

Lothar Collatz was a German mathematician who made the following conjoncture in 1937 :

it means that for any integer, the sequence will finally reaches 1.

There is abundant literature on this conjonture that still not has been demonstrated yet.

In his article, Xander proposes to extend the series into the complex plane by the formula :

which is equivalent to :

and can be simplified as :

The proposed formula is definitly correct... but, I was not able to get the same images !

After many tingings, I finally got it !

I had made the assumption that the sequence should be stopped as soon as ||zn|| ≤ 1, which is the translation of un=1 in the complex plane.

However, in order to reproduce the images of the article, I had to change the condition of the stop to ||zn|| >100. I was very perplexe about the choice of the author so I have tried to contact him and I look forward to his reply.

But where it gets really interesting is to see that there is an obvious correspondence between the two sets as shown in following examples for points 10+i0 and 5+i0 :

point 10+i0 for ||zn|| ≤ 1
point 10+i0 for ||zn|| ≥ 100
point 5+i0 for ||zn|| ≤ 1
point 5+i0 for ||zn|| ≥ 100

You will be able to do your own matching explorations in the interface for the Collatz set where I have implanted both options (just click on one of the modes at the bottom right).

This could mean that there is a link between the "stopping time" and the behavior of the sequence in the early steps i.e. the "take-off time".

I will be curious to know if this result was known. Please write to me if you ever know relevant results.

Here are some links to go further :