Conjecture about Octagonal Disk Chaos

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brad klee

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Jun 25, 2026, 11:18:50 AMJun 25
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Hi Seqfans, 


It occurred to me that the 4.8.8 tiling could imbue a square plane section with a 
sort of coordinate map that would be useful for preparing physics experiments. 

The experiment is about octagonal disks that fly around a square container and 
collide with each other and with walls while obeying most of the laws of physics. 

Angular momentum is ignored or the octagonal bodies have infinite inertial moments. 
This forces simple edge-to-edge collisions that resolve by transposition. 

The 4.8.8 coordinate system gives us all allowable initial positions for octagonal 
disks, they will all have the same mass, and unit velocity along a cardinal direction. 

A first experiment uses eight disks to construct a C4 symmetric clock on a 6x6 +
5x5 region of the 4.8.8 tiling. It can be seen in the background here:  


The foreground is the chaotic solution obtained by dropping the fewest disks 
from the clock, only two. Actually we don't know how to prove that it has no
finite recurrence time, but exponential growth of the state space is already
very convincing. 

Harm.On.ica found that 5 parts deletions led to chaotic behavior, and then made
images for the initial conditions here: 


Then we searched smaller spaces for more simple examples and found three 
body cases in a 4x4 + 3x3 container, more than ten cases here: 


You can see from the complexity sidebar that the three body experiment has the 
same chaos-producing exponential growth feature because bit count is growing
about linearly with time: 


That bit count applies to the entire 4N dimensional state space and the time coordinate.

In the three body case we've already filtered out initial conditions with indeterminacy 
around highly ordered initial conditions. Exact point-to-point collisions should become
increasingly rare as bitcount increases, as should indeterminate sandwiching states.  

We have some amount of confidence that the three body evolutions will go on forever
without breaking, and then can extract from the event sequence the sequencing of 
A+B, B+C, C+A collisions. Map this sequence to a lex-earliest ternary sequence, 
and we have something that could go into OEIS. 

ex. 0 1 1 1 2 1 1 1 0 1 0 1 0 1 1 0 2 2 2 0 2 2 1 2 2 1 1 1 2 0 2 0 1 1 2 0 0 1 1 2 2 0 2 1 1 2 0 0 1 

It might be interesting to put one of these into the OEIS with a conjecture to see if 
anyone could actually prove it, I don't know how to: 

Conjecture.
Consider the entire class of three-body hard octagon collision experiments showing 
roughly linear bitwise or roughly exponential exploration of state space. If one of these 
experiments continues indefinitely without breaking, its ternary pair-collision sequence
is conjectured to determine the digits of a //transcendental// number. It's seemingly 
less obvious that the number would not be the result of a period integral. That might
also be true if smaller orders of magnitude are reached chaotically instead of through 
smooth convergence (which is ultimately needed for an algebraic base model). 
[  ]

It seems really unlikely that this experiment would produce an algebraic number, but 
who knows, maybe sqrt(2)? The chaos is obviously generated through sqrt(2), but 
since the disks get so far off clock tracks, as time goes on, it becomes increasingly 
impossible to tell that sqrt(2) was ever involved. Periodic solutions, when they stay 
on well-defined clock tracks, retain fairly simple, low bit-complexity coefficients in 
the Q[[sqrt(2)]] field. 

I don't know if physicists will care at all about this experiment, other than observing 
the edge of chaos creeping into strongly-confined initial condition regimes where most
good periodic solutions can be expected to live. 

If a math physics student wants to do more on this project, I would suggest re-doing 
small box searches and making 4N dimensional phase diagrams over the discrete 
set of valid initial conditions. What happens: periodic, broken, chaotic, anything else? 

The simplification question is kind of obvious here: can we find any chaos generation 
using a smaller geometry? I don't think two discs could do it, and as container size
decreases there's a limit to how easily we can evade bad break conditions. 




All the best, 









--Brad























  
  


 



brad klee

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Jun 25, 2026, 7:11:42 PMJun 25
to SeqFan

Another Harm.On.ica found these from a suggestion to search disks with half unit edge:

image.png
I'm not sure these are any better than the first batch, 
    but here are the (unverified) collision digit sequences: 


28: 0,1,1,0,1,2,0,2,2,0,1,0,1,1,0,2,1,2,1,2
44: 0,0,0,0,1,1,0,2,0,1,0,0,1,0,1,0,1,2,0,2
46: 0,1,0,1,0,0,2,0,2,1,2,1,1,2,1,1,2,2,1,1
65: 0,1,1,0,0,1,0,2,1,2,1,2,2,0,2,1,2,1,1,2
78: 0,0,1,0,1,0,1,0,1,2,1,1,2,2,2,1,1,1,2,2
80: 0,1,0,1,2,0,1,0,1,1,0,2,2,0,2,0,2,0,2,0
84: 0,1,0,1,2,0,1,0,1,0,0,2,1,2,0,1,0,0,1,0

I searched briefly and didn't find any of them in the OEIS. Assuming this is really 
chaotogenesis, there's no reason to expect to find these sequences. 

Claude has obliged checked certificates despite an annoying rate limits. This 
data should be easy to verify in an hour or two when my limit resets, ha ha. 

We're using a mixture of Go, Python, and even did an extra C++ check, so 
I'm not that worried about rigor for the time being. 

Cheers, 





--Brad













On Thursday, June 25th, 2026 at 10:18 AM, brad klee <brad...@proton.me> wrote:

brad klee

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Jun 25, 2026, 10:09:17 PMJun 25
to SeqFan
Another video "Ɐpex ChⱯotogenesis 8 & 31" : :


This seems to be a minimum. I don't know how we could get 
more simple than filling 3/4 quadrants. There were only 
two good results in this space with cardinal-direction velocity 
vectors: 

Class 8:
0,1,2,0,0,2,0,0,0,1,0,1,0,1,2,1,2,1,1,2

Class 31:
0,1,0,2,1,1,2,1,0,1,1,2,1,2,0,0,2,0,2,2



Claude only allows me one double-check every five or six
hours. I can't afford more, so the github data won't be updated 
until tomorrow sometime. 
 

Happy spelunking, 
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