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 · 1h ago
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 · 53m ago
> 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.
bee_rider · 6m ago
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.
amelius · 1h ago
> "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.
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 · 58m ago
When you untie a knot, it’s ends are fixed in time.
Humans also unravel language meaning from within a hyper dimensional manifold.
robocat · 19m ago
> 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.
elpocko · 1h ago
The "Mathematical Surgery" illustration is funny. Mathematicians can make a sphere from a torus and two halves of a sphere. Amazing!
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?
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.
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.
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!
Humans also unravel language meaning from within a hyper dimensional manifold.
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.