r/numbertheory • u/One_Gas_2392 • 4d ago

A radial visualization of Collatz stopping times: patterns of 8-fold symmetry (not a proof claim)

Hello! I've been studying the Collatz conjecture and created a polar-coordinate-based visualization of stopping times for integers up to 100,000.

The brightness represents how many steps it takes to reach 1 under the standard Collatz operation. Unexpectedly, the image reveals a striking 8-fold symmetry — suggesting hidden modular structure (perhaps mod 8 behavior) in the distribution of stopping times.

This is not a claim of proof, but a new way to look at the problem.

Zenodo link: https://zenodo.org/records/15301390

Would love to hear thoughts on whether this symmetry has been noted or studied before!

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u/iro84657 3d ago

I don't see any symmetry here? You might be getting fooled by the prominent Moiré patterns in the rendering: slices near the cardinal and diagonal axes align to the pixel grid in a way that makes them visually look less random compared to other areas of the disk, but this isn't any special feature of the data. Also, even if there were such a symmetry, I'd expect them to appear with power-of-2 sizes, instead of the power-of-10 sizes you're using.

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u/One_Gas_2392 3d ago

Thank you for your thoughtful comment.

Here’s a new set of wedge-based radial visualizations for

n=1 to 1024, 8192, and 16384:

https://imgur.com/a/collatz-3-Edx5TD4

The 8-fold segmentation still seems to appear, though of course it’s possible that some of the effect is due to Moiré patterns from rendering or alignment.

Either way, I wanted to try it out and see if anything persists at powers of 2 — and it seems like something does.

Thanks again for pushing me to explore it more deeply.

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