Intricate visualizations of the Collatz Conjecture

Written By: Ishaana Rao

The Collatz conjecture was introduced by German Mathematician Lothar Collatz, in 1973, and is considered one of the ‘hardest maths questions’ ever solved. It looks like an inviting piecewise function split into two simple sections, where you would only have to divide the inputted number by two, or multiply by 3 and add one. However, this is far from the case. The conjecture never ends: you keep going round, and round, and round… until this is disproved.

Although the Collatz conjecture, also known as the 3n+1 conjecture, is considered ‘unsolvable’ — it is still worth exploring and understanding its visualisations, approaches, and essentially how it works. Despite it being simple to look at and understand, the conjecture is hard to disprove using simple algebra.

A brief look at the conjecture at first glance:

Image from Wikipedia

For example, you take the number 12. As 12 is even, we would divide it by two, as n/2 states, and get the number 6. Since 6 is even, we again divide it by 2 to get 3. 3 is an odd number, and so we now apply the 3n + 1 rule. 3(3)+1 = 10. 10 is even, so therefore we divide it by 2. We get 5, which is odd, so we multiply it by 3 and add 1. We get 16. 16 divided by 2 is 8. 8 divided by 2 is 4. 4 by 2 is 2, and 2 is divided by 1. Great, we get one! But, is getting a final answer as easy as proving the conjecture?

From this simple example, it is clear to see that it isn’t just as simple to look at the conjecture and try to plug in numbers by brute force to get a final answer. We get stuck in a loop, which even to this day has been unsolvable, but has been proved until 268.

However, even if we can’t solve conjecture, we can admire its beautiful visualizations, and analyze how it works.

The first visualization was inspired by Roy John, where he used the mathematical sequence to run the conjecture on python. This was the result:

Visualization of conjecture in Python (see citations)

We can see visually that it looks very much like a feather. However, in reality, it is not a feather; it is the Collatz conjecture as modeled by the number of numbers it has been proven to work for so far. The blue line shows the multiples of the ‘2’ strand, where once you hit a multiple of 2, the conjecture only follows the n/2 mod 0 pieces of the piecewise function. For example, once you hit 32, 32/2 = 16, 16/2= 8, 8/2= 4, 4/2 =1. This proves that it does, in fact, work. The small strands branching off of the main blue one are the powers of 2.

In fact, here is a simplified version of the feather:

This describes the conjecture in a simpler way.

It should be noted that once the conjecture reaches a final, terminal value of 1, it begins to cycle again through 1,2, and 4 indefinitely.

Eg) n=6 : 6,3,10,5,16,8,4,2,1,4,2,1,4,2,1,4,2,1,4,2,1… etc

Eg) n= 976 : 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1….

If we observe this sequence closely, we see why an alternative name for the conjecture is a ‘hailstone sequence’, as the number fluctuates up and down values.

Hail Stones & Their relation to the Collatz Conjecture (anti-grele.fr/en/anatomy/)

Applications of the conjecture:

Interestingly, the conjecture can’t be solved through induction, transvections, or cohomology. (Cohomology is a general term for a sequence of abelian groups) Moreover, it can be assumed that the prime questions being answered from solving this conjecture would be “how does prime factorisation of an affect prime factorisation of a+1?” as the conjecture is modeled as n/2, and 3n+ 1. It also gives a hint to what sorts of operations affect the conjecture as a whole, this being the difference between adding 1 to a number, resulting in a potentially large change to the factorisation of such a number, or the factoring out a power of 2 from a number- which may have a smaller change as opposed to adding 1.

What is Cohomology and its general term for a sequence of abelian groups?

Cohomology is associated with abelian groups — in particular, ones associated with topological space. Abelian groups, otherwise known as commutative groups, in particular, are groups in which the order that group elements are communicated are not necessarily followed — like Collatz. Moreover, homomorphisms in algebra are according to Mathematics Stack Exchange, give “ways to relate different algebraic objects” as they are “a structure-preserving map between two algebraic structures of the same type” according to Wikipedia.

Group Homomorphisms according to Wikipedia

Interestingly, there is also a possibility that the solution to the conjecture may even be used in cryptography and computers. At the very least, solving the conjecture will benefit mathematicians, and in turn ‘sharpen our minds.’

In more recent news, Terence Tow, a mathematician, came close to deducing the conjecture by further providing information that the conjecture starting with the number n will almost always end up with a final number that is less than n, n/2, ln n, or even root n. According to Tow’s talk, he claimed that this would be the “closest he would get to solving the conjecture”

For fellow Singaporeans, it may be interesting to hear that even Prime Minister Lee of Singapore spent some of his time working on the Collatz Conjecture during his holidays.

Works Cited

“Abelian Group.” Wikipedia, the Free Encyclopedia, Wikimedia Foundation, Inc, 21 Dec. 2001, en.wikipedia.org/wiki/Abelian_group. Accessed 1 Mar. 2022.

“Applications for Collatz’s Conjecture?” The Museum of HP Calculators, 29 Sept. 2014, www.hpmuseum.org/forum/thread-2202.html.

Auto, Hermes. “PM Lee Spending Some Vacation Time on the Collatz Conjecture: 5 Things About the Unsolved Maths Problem.” The Straits Times, 21 Dec. 2019, www.straitstimes.com/singapore/pm-lee-spending-some-vacation-time-on-the-collatz-conjecture-5-things-about-the-unsolved.

“Collatz Conjecture.” 5 Feb. 2002, en.wikipedia.org/wiki/Collatz_conjecture.

“Group Homomorphism.” Wikipedia, the Free Encyclopedia, Wikimedia Foundation, Inc, 9 Oct. 2001, en.wikipedia.org/wiki/Group_homomorphism. Accessed 1 Mar. 2022.

“The Simple Math Problem We Still Can’t Solve.” Quanta Magazine, 22 Sept. 2020, www.quantamagazine.org/why-mathematicians-still-cant-solve-the-collatz-conjecture-20200922/.

“Visualization of Collatz Conjecture Using Python.” Ultimate Theorem, ultimatetheorem.blogspot.com/2020/08/visualization-of-collatz-conjecture_10.html.

“What is the Importance of the Collatz Conjecture?” Mathematics Stack Exchange, math.stackexchange.com/questions/2694/what-is-the-importance-of-the-collatz-conjecture.

“What is the Importance of the Collatz Conjecture?” Mathematics Stack Exchange, math.stackexchange.com/questions/2694/what-is-the-importance-of-the-collatz-conjecture.