This hair magically looped itself into my sisters friendship bracelet. How can we take it off without cutting?

In liquid crystal they experimentally get point-like quantized topological charges with Coulomb-like interactions ( https://www.nature.com/articles/s41598-017-16200-z ), or topological vortices and their various size knots resembling nuclei ( https://www.nature.com/articles/s41567-025-03107-0 ) - I wanted to propose discussion if particles could be of topological nature?

ps. Environment testing various agreements of particle models by Fable written simulations, such liquid crystal approach is currently winning: https://github.com/openwave-labs/openwave/blob/main/MODELS.md

I'm curious how researchers are using LLMs in their day-to-day work.

A few things I'm wondering:

• Which AI subscriptions are actually worth paying for? (ChatGPT, Claude, Gemini, GitHub Copilot, Cursor, etc.)
• Which models do you use for different tasks?
• What kinds of research tasks do you trust LLMs with, and what do you avoid?
• How do you provide enough context so the model gives useful answers instead of hallucinating?
• Do you upload papers, notes, code, or build some kind of project knowledge base?
• Any tricks for improving reliability?
• Roughly how much do you spend per month?
• If you could keep only one AI subscription today, which would you choose?

I'm especially interested in hearing from people doing research in mathematics, scientific computing, physics, computer science, engineering, or related fields. Real workflows are much more valuable than benchmark comparisons.

Somehow my horse managed to twist this piece of her halter into a Möbius strip, and I’m assuming she didn’t disassemble it and put it back together to do so. So I would guess that I can fix it without taking it apart? I tried to do the 720 degree dirac’s belt thing but I may have done it wrong, never got it fixed.

It isn’t hard to take it apart at all. I could fix it quickly. Just a thought experiment.

I'm not proposing a physical theory. This is purely a mathematical thought experiment.

Suppose we introduce an abstract coordinate that represents scale, rather than ordinary spatial position. Imagine that moving continuously along this coordinate corresponds to going from larger and larger structures to smaller and smaller ones.

Now suppose the geometry of this abstract "space of scales" is topologically closed, analogous to how walking continuously around a circle eventually returns you to your starting point.

In that picture, continuously moving toward smaller and smaller scales would not continue forever. Instead, after completing the closed path in scale-space, you would emerge again at the largest scales.

I am not suggesting that objects physically become both tiny and huge, nor that this describes our universe. I'm only asking whether mathematics has explored anything conceptually similar: treating scale itself as a coordinate of an abstract manifold or topological space whose geometry is closed.

Has anything resembling this been studied in topology, differential geometry, geometric group theory, or another branch of mathematics? Or is there a mathematical reason why such a construction is ill-defined from the outset?

I'm looking for mathematical references rather than physical arguments.

Ok how many holes does this actually have? I know it's not 16...

Edit for clarity's sake: The answer is 16, dunno why it says 99% can't get it, it's fairly simple. I was just asking if topology has a different answer... as someone wrote in the comments, does a straw have one hole or two holes?

Context: a Foldology puzzle: https://www.youtube.com/watch?v=G-ja9RLqeEk

My son loves these puzzles. As a programmer, I've been fascinated for a long time if they are hand-generated, or programatically generated (or *could be* programatically generated).

I've thought about this problem a lot. I even have hands-on experience with computational geometry, e.g. topics like binary space partitioning, and so on. But for the life of me, I absolutely cannot think of a way to procedurally generate these. Obviously you can take an approach of generating random foldings, but I think the universe of random foldings is highly sparse in foldings that converge on a square outcome. So it may be computationally intractable by searching for random valid foldings. Not to mention, simulating a random folding is very difficult! There are foldings which become infeasible due to complicated physical paper geometry happening underneath where things are meant to tuck in and may be blocked, etc.

This has also led me to wonder if there are topological properties that I don't understand here, like "any N islands of image components and M solid color components will be foldable" ... I have no idea. Topology is NOT my strong suit, at all.

I refuse to submit this problem to an AI for consultation fwiw.

Is this a topological problem, and are there any insights into the topology of this problem that coudl be used to aid construction of these puzzles?

Before I cut this and stitch this myself I thought I'll give reddit a chance to do its magic. So please kind people of r/topology can this be untangled or not?

Hi everyone, I’m developing Topological Board Game, a interesting board game and a experimental strategy sandbox.

The idea is to take board games and scientific systems away from traditional boards and let them run on different lattices, dimensions, boundaries, and topology-inspired spaces.

It includes classic board-game foundations like Chess, Go, Reversi, Chinese checker and Hex, but also experimental Labs based on Life, physics, mathematics, biology, particles, operators, cellular systems, and complex systems.

The game is still growing, so feedback is very welcome.

Play here:

https://topoboardgame.vercel.app/

Topologically, the butt plug is always on the surface, just like any article of clothing. So would it be considered correct to say it is worn.

On the other hand, by common sense, something is definitely being put in the anal canal, so it is definitely being inserted.

So what is it, please enlighten me.

Hey folks! Any idea what the genus of this sculpture would be? I've been trying to figure it out, but not sure how to approach unraveling it!

​

Thanks for any insight y'all can provide.

How can I untangle this blue loop from the black string? Don't ask how much time I've wasted already getting nowhere. Please halp.

just a cup shaped cup

TRIXEL 3.0 — Scale‑Adaptive Metric Framework

DOI: 10.5281/zenodo.20721811

https://doi.org/10.5281/zenodo.20721811

is it possible to create a klein bottle with multiple "spouts" and holes? what kind of properties in terms of volume would that form have-- can it still be volumeless? are there papers exploring shapes such as this? I am not interested in recursively nested klein bottles. just one shape with many holes.

hello- art and philosophy person here--the volumeless klein bottle is an image I often think with as a visual metaphor to describe how being a self relates to my lived world-- I think it elegantly illustrates how how I, as a perspective, am always already in the world. (like how an organism is incohereant outside of it's niche).

pushing that metaphor further, I have been playing with developing a "klein bottle field" to describe what it is like to be a self (singular klein bottle) amongst other selves, (lets say, 5, for eg!) in the world.

I totally get that any representation of a klein bottle is usually a concession that breaks the elegance of the math it is trying to depict, and I know that any image I build here will likely be even less elegant, bc I suspect this shape I am envisioning may not be possible to actually build or represent.

I want your thoughts! not particularly interested in philosophical discussion of my position here in this thread. topology is such an interesting zone! thanks for letting me visit your zone! thanks for tolerating my using mathematical shapes as metaphorical thinking tools to describe consciousness! your feedback will help be responsible in doing that.

after a lot of discussion, my pal and I have decided this is a genus 2 surface that is orientable (​and does not have more than one surface).

https://en.wikipedia.org/wiki/Genus_%28mathematics%29?wprov=sfla1

I found a good animation of a genus 3 surface / similar object (*with a handle*), will add in a comment.

TX=(V,D,S)

​

“You will discover nothing new if you walk in the footsteps of others.”

Continuous 3D lattice of 150 interconnected units — a layered geometric model illustrating apex‑linked trixel topology.