Dimension 126 Contains Twisted Shapes, Mathematicians Prove
18 comments
·May 5, 2025bee_rider
Is it conventional for mathematicians to talk about “the dimensions” like this? I think they are talking about a 126 dimensional space here, but I am a lowly computerer, so maybe this went over my head.
uxhacker
I’m not a mathematician (just a programmer), but reading this made me wonder—doesn’t this kind of dimensional weirdness feel a bit like how LLMs organize their internal space? Like how similar ideas or meanings seem to get pulled close together in a way that’s hard to visualize, but clearly works?
That bit in the article about knots only existing in 3D really caught my attention. "And dimension 3 is the only one that can contain knots — in any higher dimension, you can untangle a knot even while holding its ends fast."
That’s so unintuitive… and I can't help thinking of how LLMs seem to "untangle" language meaning in some weird embedding space that’s way beyond anything we can picture.
Is there a real connection here? Or am I just seeing patterns where there aren’t any?
Sniffnoy
> That’s so unintuitive…
It's pretty simple, actually. Imagine you have a knot you want to untie. Lay it out in a knot diagram, so that there are just finitely many crossings. If you could pass the string through itself at any crossing, flipping which strand is over and which is under, it would be easy, wouldn't it? It's only knotted because those over/unders are in an unfavorable configuration. Well, with a 4th spatial dimension available, you can't pass the string through itself, but you can still invert any crossing by using the extra dimension to move one strand around the other, in a way that wouldn't be possible in just 3 dimensions.
> Or am I just seeing patterns where there aren’t any?
Pretty sure it's the latter.
amelius
> "And dimension 3 is the only one that can contain knots — in any higher dimension, you can untangle a knot even while holding its ends fast."
Maybe you could create "hyperknots", e.g. in 4D a knot made of a surface instead of a string? Not sure what "holding one end" would mean though.
Sniffnoy
Yes, circles don't knot in 4D, but the 2-sphere does: https://en.wikipedia.org/wiki/Knot_theory#Higher_dimensions
Warning: If you get too deep into this, you're going to find yourself dealing with a lot of technicalities like "are we talking about smooth knots, tame knots, topological knots, or PL knots?" But the above statement I think is true regardless!
nandomrumber
When you untie a knot, it’s ends are fixed in time.
Humans also unravel language meaning from within a hyper dimensional manifold.
bee_rider
I think LLM layers are basically big matrices, which are one of the most popular many-dimensional objects that us non-mathematician mortals get to play with.
lamename
It's not just LLMs. Deep learning in general forms these multi-d latent spaces
robocat
> Or am I just seeing patterns where there aren’t any?
Meta: there are patterns to seeing patterns, and it's good to understand where your doubt springs from.
1: hallucinating connections/metaphors can be a sign you're spending too much time within a topic. The classic is binging on a game for days, and then resurfacing back into a warped reality where everything you see related back to the game. Hallucinations is the wrong word sorry: because sometimes the metaphors are deeply insightful and valuable: e.g. new inventions or unintuitive cross-discipline solutions to unsolved maths problems. Watch when others see connections to their pet topics: eventually you'll learn to internally dicern your valuable insights from your more fanciful ones. One can always consider whether a temporary change to another topic would be healthy? However sometimes diving deeper helps. How to choose??
2: there's a narrow path between valuable insight and debilitating overmatching. Mania and conspirational paranioa find amazing patterns, however they tend to be rather unhelpful overall. Seek a good balance.
3: cultivate the joy within yourself and others; arts and poetry is fun. Finding crazy connections is worthwhile and often a basis for humour. Engineering is inventive and being a judgy killjoy is unhealthy for everyone.
Hmmm, I usually avoid philosophical stuff like that. Abstract stuff is too difficult to write down well.
anthk
Network optimizing problems are just better with 4D hypercubes.
zchrykng
Seeing as mathematicians proving things in math has minimal relation to the real world, I'm not sure how important this is.
Mathematicians and physicists have been speculating about the universe having more than 4 dimensions, and/or our 4 dimensional space existing as some kind of film on a higher dimensional space for ages, but I've yet to see compelling proof that any of that is the case.
Edit: To be clear, I'm not attempting to minimize the accomplishment of these specific people. More observing that advanced mathematics seems only tangentially related to reality.
brian_cloutier
You might consider reading Hardy's A Mathematician's Apology. It gives an argument for studying math for the sake of math. Personally, reading a beautiful proof can be as compelling as reading a beautiful poem and needs no further justification.
However, there is another reason to read this essay. Hardy gives a few examples of fields of math which are entirely useless. Number theory, he claims, has absolutely no applications. The study of non-euclidean geometry, he claims, has absolutely no applications. History has proven him dramatically wrong, “pure” math has a way of becoming indispensable
zchrykng
I have no problem studying Math just to study Math. I read the title and jumped to some conclusions, I'm afraid. Was talking to a friend about String Theory and their 11+ dimensions the other day and that is immediately where my brain went to with this one. The article is interesting even though I have zero desire to personally study math just for math's sake.
baruchel
I have always been fond of the following quote by Jacobi: “Mathematics exists solely for the honor of the human mind”
elpocko
The "Mathematical Surgery" illustration is funny. Mathematicians can make a sphere from a torus and two halves of a sphere. Amazing!
lifefeed
Well, shit.
null
> Mathematicians Weinan Lin, Guozhen Wang, and Zhouli Xu have proven that 126-dimensional space can contain exotic, twisted shapes known as manifolds with a Kervaire invariant of 1—solving a 65-year-old problem in topology. These manifolds, previously known to exist only in dimensions 2, 6, 14, 30, and 62, cannot be smoothed into spheres and were the last possible case under what’s called the “doomsday hypothesis.” Their existence in dimension 126 was confirmed using both theoretical insights and complex computer calculations, marking a major milestone in the study of high-dimensional geometric structures.