Six Collatz tiles

[cosmo]

Hello,

I have been very passionate about studying the Collatz process symbolically, i.e. looking at the action of Collatz on numbers written in some bases, in particular bases 2, 3 and 6.

Part of my PhD thesis (Chapter 1) was to realise that these dynamics can be encompassed using the following set of 6 Wang tiles:

Each of these tiles corresponds to a base 6 digit, vertical sides speak in base 3 and horizontal sides speak in base 2. All corners but top-left are total and deterministic:

• Top-right corner is Euclidean division by 3: tile_name = 3*top + right (e.g. 5 = 3*1 + 2)
• Bottom-left is Eculidean division by 2: tile_name = 2*left + bottom (e.g. 5 = 2*2 +1)
• Bottom-right is Chinese remainder theorem for 2x3 = 6, e.g. 5 is the only number < 6 such that modulo 2 is 1 (bottom) and modulo 3 is 2 (right)

Collatz sequences can be constructed from parity vectors in the following way:

• Use tile 4 for odd entries of the vector (diagonal blue arrows below)
• Use horizontal arrow labelled 0 for even entries of the vector (horizontal blue arrows below)

For instance, the parity vector 1, 0, 0, 1, 0, 1 is encoded as follows:

Then, since the bottom-right corner of the tiles is deterministic, there is a unique assembly to reconstruct:

Which corresponds to the Collatz sequence 20 = T^6(45) where 45 is encoded in base-2 on the top of the assembly and 20 in base-3 on the left of the assembly:

Here is a bigger example:

These images are extracted of a longer blog post I wrote on the subject . The above focused on base 2 and 3 but these tilings also feature base 6 (top-left-going diagonal), base 3/2 (top-right-going diagonal), and more generally any base of the form 2^a3^b > 1 with a,b\in\mathbb{Z} by considering “macro-tiles”.

If you make the assembly cyclical (i.e. making the first and last point of the parity vector be the same) then, will appear 2-adic, 3-adic and 6-adic integers (i.e. base-n strings with infinitely many digits on the most significant side). See Appendix B for a survival guides on the p-adics.

In my thesis, I applied the tiles to thinking about Collatz cycles (§1.5) and ancestors (§1.6).

These tiles are also more general than Collatz and allow to represent other problems such as:

• Erdős’ conjecture on powers of two (Link outlined in §2.2.2)
• Mahler 3/2 problem (Appendix B of my thesis)
• The Hydra sequence
• You can construct similar tilesets for any p \times q with gcd(p,q) = 1. Hence you can represent other Collatz-like problems with similar assemblies (e.g. 5x+1).

I particularly like the tiles for thinking about Collatz cycles because the question reformulates into a algorithmico-geometric question: “Why on earth, only the trivial parity vector results in an assembly with the same number in base-2 on top and in base-3 to the left”:

Tiles are also interesting because they allow to state simply phenomenons that would otherwise difficult to express, such as forbidden patterns in assemblies (always coming from the non-deterministic top-left corner):

My lines of research with the tiles include:

• representing known results about the Collatz conjecture using the tiles (I would love to be able to understand this result with the tiles)
• thinking about cycles and their parity vectors (in a similar way to the questions asked in Parity sequences and cycles or in Elementary proof of no circuits? )
• thinking a bout “generalised parity vectors” which are arbitrary tiles borders instead of the very specific Collatz ones (using only tile 4 and horizontal 0 moves), looking for what types of constraints empirically prevents positive cycles to happen
• inverse reconstructions: a positive cycle must eventually reach a translation of the parity vector where all values are 0. We can start from a 0-valued party vector and reconstruct (non-deterministically) in the bottom-right direction: may we detect some patterns or invariants that Collatz parity vectors do not follow? i.e. Why are Collatz parity vectors not in the pre-image of the all-0 parity vector?

I’d be happy to discuss, answer questions or anything if this framework interests you

P.S: @mathkook made a cool video about the tiles

[cosmo]

Here is how to read base 3/2 with the tiles:

The mapping from tiles to base 3/2 digits is as follows:

• Tiles 0 and 1 map to digit 0/2
• Tile 2 maps to digit 2/2
• Tile 3 maps to digit -1/2
• Tiles 4 and 5 map to digit 1/2

Here: ‘2515524’ (lowest significant digit at the end) becomes:

1/2 + 2/2 (3/2) + 1/2 (3/2)^2 + 1/2 (3/2)^3 + 0/2 (3/2)^4 + 1/2 (3/2)^5 + 2/2 (3/2)^6 = 20

[Perryman]

Does the base 2,3,6,3o2 conversion extend to the p-adics as well? Does it hold arbitrarily in any shape of a tiling?

motivation: wondered if I could draw this base-conversion out for 4-2-1 cycle. Yet when naively looking I can get 4 in base 3, but the corresponding base 2 yields 1 (and if I went further one down, it’d be 110_3 and 100_2). But I can read 1242_3o2 and can read off a 4 in base-6. Or, if I consider just the ternary 1 from the 4-tile, I get 10, and the corresponding 100 in binary. The base-6 would be 4, but that 3/2 conversion of 42 would be…. 7/4? Or if the parity vector went down left left instead of downleft left left… not comfortable enough with the p-adics to do the math, but this sort of thing is making me wonder if there has been meaning explored in the 4-2-1 loop and base conversion from your thesis.

I have wanted to consider a geometric proof more deeply since going through your paper, but it seems there is something fundamental I am misunderstanding about when the rule is “provably true”.

My intuition on approaching a proof may not be fruitful if the implication is that if the base-3 of some collatz term will correspond to its conversion to base, then the base conversion applies. But the general geometric intuition I was seeing was that, some sort of composition rule might be seen that could possibly deny existence of a nontrivial cycle. Especially if there is a proven/provable base-conversion rule of some sort, that could make statements about relations of large sections of tilings be made or provably not makable. I don’t have a much clearer idea than this yet - just been bouncing around in my head for the past year or so and curious to hear your thoughts!

[Perryman]

@cosmo I realized when I tried to update Coreli for new drawsvg I messed up the rendering, so I dug out my tiles to show what I mean. Haven’t fully re-read the paper yet, but here’s an annotated version of what I’m seeing - when tiling out 4-2-1 on loop, I’m not sure how to find the proper base 3/2 term- and I need to follow along with the rules directly from the paper again.

But I’m getting for base 3/2 conversion of “1242”=4, and “124”=2. Trying to understand if these can be used as constraints so I can approach this reasoning angle.

Or using the stop tokens… 12RS… taken as 12 in base 3/2 converted to 1, and the 1 tile fits the rest. But if I look at 100 in binary I get 1S in ternary - if the S corresponded to a 1 that would match 11=4
Or if in base 3/2 the “12RS” corresponded to “1242” or “1252” then it would convert to 4 in decimal, so if I assign R=4 S=2 then I could take base 3/2 “1242” by arbitrarily choosing to stop there and taking the base 6 would work there.

[cosmo]

Hi @Perryman, thank you for your questions!

Does the base 2,3,6,3o2 conversion extend to the p-adics as well? Does it hold arbitrarily in any shape of a tiling?

Yes it does extend to the adics! Take a (random) parity vector, repeat it infinitely in both directions, then you will see appear the adic representations of all the elements of this rational cycle by looking in the appropriate direction!

I realized when I tried to update Coreli for new drawsvg I messed up the rendering

Yes… they changed coordinate system lol… I have just updated the github repo requirements.txt to force the use of drawSvg 1.9, starting from fresh the following should work:

pip install coreli==0.0.9

Or from coreli’s root for dev:

python3.12 -m venv venv
source venv/bin/activate
pip install -r requirements.txt

This post may help for base 3/2: Tristan Stérin | 6 Collatz tiles - Base 3/2 follow up

Thank you again!!