New VIbE:X : : Dodecagonal pair CHAOS!

[brad klee]

Hi Seqfans, 

I hope I don't need to apologize for this, but I ended up in a Hofstadter-and-Kim

inspired musing about a reflectogram for the all-caps APEX, which is already 

very symmetric. If you just add and "I" and a ":" you get AIPE:X, which is still 

quite legible even though the meaning has shifted. 

Of course the APEX is the top of the mountain the highest you can ever get, 

while AIPE:X seems to imply that an LLM was involved somehow. 

The reflected VIbE:X has its first letter pointing down instead of up, which 

indicates that we're searching for the minimal of something, not the max. 

It's especially clever and witty because after "Vibe Coding", Matthew Schwartz 

had to write an article about "Vibe Physics" for Anthropic, which has stimulated 

the debate about legality of authorship and inspired us to consider for what 

other X can we VIbE:X? 

New discoveries are happening every day, I guess. Our previous offering with 

octagons obtained two equally good candidates, so why not keep going and 

look for something truly unique? 

The direction is obvious--just add more sides to the polygon, and look what 

we have found here with only two non-rotating, hard dodecagonal disks in 

a square container (L = 4*W):

https://www.youtube.com/shorts/JrL1aG3vkAo

(another Harm.On.ica production)

We then take the facet index mod 3 and we write out a body-body collision 

sequence as follows: 

1,0,2,1,0,2,1,2,1,2,2,2,1,2,2,2,0,2,2,0,2,1,1,0,2 2,2,1,1,2,0,0,0,1,2,2,1,2,1,

0,2,2,2,1,0,1,0,2,1,0 2,0,0,1,1,2,2,0,0,0,0,2,2,1,1,2,1,0,0,2,0,0,1,0,1 . . .

Claude checked this and a range of other results fairly easy over four or five

access-limited payment-gated sessions. Our previous conjecture also applies

to this sequence as it grows chaotically with increasing bitwise complexity.

An interesting difficult of this problem is that the time reverse sequence obtains 

a corner collision, so we do not have the bi-infinite property here. Again, the 

infinite continuation property relies on a statistical argument that if bitwise 

complexity keeps increasing, exact point-like collision or sandwiching with

a wall becomes increasingly unlikely, i.e. toward an uncountable 0% chance. 

That's still not proven, and I haven't succeeded in proving the octagonal case 

is almost always periodic for pairs. What's happening different here is that the 

two extra pairs of facets on the dodecagon allow the velocity vector to unfix 

from a finite set. That then feeds back into complexity explosion. 

There is one more example already in checked data: 

https://github.com/bradleyklee/Artifacts/tree/main/16-more-disks

A pair of 24-a-gons that are a bi-infinite pair by time-reversal symmetry. I'll have 

a video for those and a data extraction soon. 

Again, this is not a final draft code repository, but it is being checked to the 

maximum of what I can afford to the end of this month, and maybe next. 

All the best, 

--Brad

[brad klee]

PS. If there are any math phys. students out there wanting to study chaos, 

I'll just throw out one project suggestion quickly: 

Take the L = 4W central collision dodecagonal pair model and instead 

of assuming the initial collision exactly overlaps facets, allow for a non-zero 

offset while still fixing the collision centroid to the container center.

Is it possible to tune the offset to small enough exact values that we 

can test the "butterfly effect" definition of chaos? Small changes in the 

initial condition must eventually lead to large divergent outcomes. 

What are the longest persistent agreement times for similar ternary

chaos sequences? When they start to disagree, how do they disagree? 

Is it an immediate bifurcation or a gradual onset of differences?

This would be okay for a physics student, and actually the more facets

the better. Rotational impulses scale with edge length, so the limit 

approaching a circle begins to negate complaints about angular 

momentum from experimentalists and over-competing theorists who 

probably wouldn't like this idea relative to a hockey game.  

A hockey game with octagonal pucks isn't a bad idea, but I doubt that 

it would make much use of algebraic number fields.

If anyone takes this suggestion and comes up with comparison sequences 

please let me know and I'll verify them if I'm allowed to. 

Thanks for your time, 

--Brad

 

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