> 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.
hinkley · 2h ago
So these are all powers of 2 minus 2, and it looks like from the article that the pattern doesn’t exist in 2^8 - 2 or higher. Is there any description a layperson might understand as to why it stops instead of going on forever!
It’s more than 200 pages of pretty technical mathematics, so I’m reasonably confident that there is no description a layperson might understand.
codebje · 2h ago
They’re all double the last dimension plus two, without skipping any in that sequence - but that offers no insight into why it wouldn’t hold for 254.
aleph_minus_one · 30m ago
>
They’re all double the last dimension plus two, without skipping any in that sequence - but that offers no insight into why it wouldn’t hold for 254.
Wikipedia at least gives a literature reference and concise explanation for the reason:
"Hill, Hopkins & Ravenel (2016) showed that the Kervaire invariant is zero for n-dimensional framed manifolds for n = 2^k− 2 with k ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
* The coefficient groups Ω^n(point) have period 2^8 = 256 in n
* The coefficient groups Ω^n(point) have a "gap": they vanish for n = -1, -2, and -3
* The coefficient groups Ω^n(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension n is nonzero then it has a nonzero image in Ω^{−n}(point)"
Paper:
Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016). "On the nonexistence of elements of Kervaire invariant one"
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.
core-explorer · 2h ago
We usually don't talk about "the dimensions", we talk about the general case: n-dimensional spaces (theorems covering all dimensions simultaneously) or infinite-dimensional spaces (individual spaces covering all finite-dimensional spaces).
Of course, when you try to generalize your theorems you are also interested in the cases where generalization fails. In this case, there is something that happens in a 2-dimensional space, in a 6-, 14- or 30-dimensional space. Mathematicians would say "it happens in 2, 6, 14 or 30 dimensions". I never noticed that this is jargon specific to mathematicians.
Problems in geometry tend to get (at least) exponentially harder to solve computationally as the dimensions grow, e.g. the number of vertices of the n-dimensional cube is literally the exponential of base 2. Which is why they discovered something about 126-dimensional space now, when the results for lower dimensions have been known for decades.
Karliss · 1h ago
But that's not how the article says it. It says "in dimensions 2, 6, 14, 30 and 62" instead of "in 2,6,14 or 30 dimensions". The later sounds fine, but "dimensions 8 and 24" to me sounds too much like something is happening in "8th and 24th dimension". It even uses singular "dimension 126" as if you took >=126 dimensional space, ordered the axis and something interesting happened along 126th and only that one.
Sniffnoy · 1h ago
Yeah, that's not what that means. In math "dimension" is used as a statistic. As in, "this manifold has a dimension of 4". So you can say things like "in dimension 4" to mean "when the dimension is equal to 4". We do also say "in 4 dimensions"; it just varies. The two phrases are equivalent. There is no ordering of dimensions or anything like that.
aleph_minus_one · 28m ago
> Is it conventional for mathematicians to talk about “the dimensions” like this
There is an old joke:
How do you imagine a 126-dimensional space? - Simple: imagine an n-dimensional space and set n=126.
enasterosophes · 2h ago
It's a good question. It's easy to assume they're talking about R^126 (where R is the reals) but digging a bit deeper I don't think it's true.
The Kervaire invariant is a property of an "n-dimensional manifold", so the paper is likely about 126-dimensional manifolds. That in turn has a formal definition, and although it's not my specialization, I think means it can be locally represented as an n-dimensional Euclidean space.
A simple example would be a circle, which I guess would be a 1-dimensional manifold, because every point on a circle has a tangent where the circle can be approximated by a line passing through the same point.
So they're saying that there are these surfaces which can be locally approximated by 126-dimensional Euclidean spaces. This in turn probably requires that the surface itself is embedded in some higher-dimensional space such as R^127.
elchananHaas · 1h ago
Manifolds are generally considered objects of themselves, and it may be difficult to embed then in higher dimensional objects. This is especially the case for tricky manifolds like those with a Kervaire invariant of 1.
duskwuff · 1h ago
Not using the language of this article. Referring to e.g. a two-dimensional space as "Dimension 2" is irregular. One might say that the space has dimension 2 (as shorthand for "has a dimension of 2"), but "Dimension 2" is not used as the proper name of such a space.
Sniffnoy · 1h ago
It's common in math to say things like "in dimension 2" to mean "when the dimension is 2". It doesn't necessarily refer to a specific space (although it could based on context). It's just setting a contextual variable. Many problems occur in varying dimension and oftentimes you want to restrict discussion to a specific dimension.
duskwuff · 1h ago
Right - what I meant specifically is the use of names like "Dimension 2" (with the capital D) as if to refer to a specific location with that name. Among other things, it has too many associations with pulp science fiction. :)
codetrotter · 2h ago
As someone who is also not a mathematician it sounded perfectly normal to me.
impish9208 · 23m ago
This got me thinking — is there a version of “in mice” for math papers?
uxhacker · 5h 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 · 4h 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.
stouset · 2h ago
That makes sense for a 2D rope in 4D space, but I’m not convinced the same approach holds for a 3D ”hyperrope” in 4D space.
Sniffnoy · 1h ago
I'm not sure what you mean here. This is discussing a 1-dimensional structure embeded in 4-dimensional space. If you're not sure it works for something else, well, that isn't what's under discussion.
If you just mean you're just unclear on the first step, of laying the knot out in 2D with crossings marked over/under, that's always possible after just some ordinary 3D adjustments. Although, yeah, if you asked me to prove it, I dunno that I could give one, I'm not a topologist... (and I guess now that I think about it the "finitely many" crossings part is actually wrong if we're allowing wild knots, but that's not really the issue)
amelius · 5h 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!
zmgsabst · 21m ago
Yep — you can always “knot” a sphere of two dimensions lower, starting with a circle in 3D and a sphere in 4D.
lamename · 3h ago
It's not just LLMs. Deep learning in general forms these multi-d latent spaces
nandomrumber · 4h ago
When you untie a knot, it’s ends are fixed in time.
Humans also unravel language meaning from within a hyper dimensional manifold.
AIPedant · 1h ago
I don't think this is true, I believe humans unravel language meaning in the plain old 3+1 dimensional Galilean manifold of events in nonrelativistic spacetime, just as animals do with vocalizations and body language, and LLM confabulations / reasoning errors are fundamentally due to their inability to access this level of meaning. (Likewise with video generators not understanding object permanence.)
bee_rider · 3h 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.
robocat · 3h 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.
hinkley · 2h ago
A lot of innovation is stealing ideas from two domains that often don’t talk to each other and combining them. That’s how we get simultaneous invention. Two talented individuals both realize that a new fact, when combined with existing facts, implies the existence of more facts.
Someone once asserted that all learning is compression, and I’m pretty sure that’s how polymaths work. Maybe the first couple of domains they learn occupy considerable space in their heads, but then patterns emerge, and this school has elements from these other three, with important differences. X is like Y except for Z. Shortcut is too strong a word, but recycling perhaps.
elpocko · 5h ago
The "Mathematical Surgery" illustration is funny. Mathematicians can make a sphere from a torus and two halves of a sphere. Amazing!
m3kw9 · 1h ago
This is some Dr Strange stuff
anthk · 3h ago
Network optimizing problems are just better with 4D hypercubes.
lifefeed · 3h ago
Well, shit.
ReptileMan · 2h 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.
Do we have anything in the universe that is knotted? Both on large and small scales. Or it is just coincidence?
mike-the-mikado · 2h ago
Not wishing to be flippant, but I have lots of bits of string that are knotted.
zchrykng · 3h ago
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 · 3h ago
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 · 3h ago
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 · 3h ago
I have always been fond of the following quote by Jacobi: “Mathematics exists solely for the honor of the human mind”
duskwuff · 1h ago
> More observing that advanced mathematics seems only tangentially related to reality.
You might be surprised; there have proven to be a number of surprising connections between mathematical structures and more concrete sciences. For instance, group theory - long thought to be an highly abstract area of mathematics with no practical application - turned out to have some very direct applications in chemistry, particularly in spectroscopy.
core-explorer · 2h ago
When you try to solve one problem involving two objects in three-dimensional space, you have a six-dimensional problem space. If you have two moving objects, you have a twelve-dimensional problem space. Higher dimensional spaces show up everywhere when dealing with real-life problems.
Muromec · 1h ago
>Seeing as mathematicians proving things in math has minimal relation to the real world, I'm not sure how important this is.
Évariste Galois says hi and Satoshi-sensei greets him back.
It’s more than 200 pages of pretty technical mathematics, so I’m reasonably confident that there is no description a layperson might understand.
Wikipedia at least gives a literature reference and concise explanation for the reason:
> https://en.wikipedia.org/w/index.php?title=Kervaire_invarian...
"Hill, Hopkins & Ravenel (2016) showed that the Kervaire invariant is zero for n-dimensional framed manifolds for n = 2^k− 2 with k ≥ 8. They constructed a cohomology theory Ω with the following properties from which their result follows immediately:
* The coefficient groups Ω^n(point) have period 2^8 = 256 in n
* The coefficient groups Ω^n(point) have a "gap": they vanish for n = -1, -2, and -3
* The coefficient groups Ω^n(point) can detect non-vanishing Kervaire invariants: more precisely if the Kervaire invariant for manifolds of dimension n is nonzero then it has a nonzero image in Ω^{−n}(point)"
Paper:
Hill, Michael A.; Hopkins, Michael J.; Ravenel, Douglas C. (2016). "On the nonexistence of elements of Kervaire invariant one"
* https://arxiv.org/abs/0908.3724
* https://annals.math.princeton.edu/2016/184-1/p01
Of course, when you try to generalize your theorems you are also interested in the cases where generalization fails. In this case, there is something that happens in a 2-dimensional space, in a 6-, 14- or 30-dimensional space. Mathematicians would say "it happens in 2, 6, 14 or 30 dimensions". I never noticed that this is jargon specific to mathematicians.
Problems in geometry tend to get (at least) exponentially harder to solve computationally as the dimensions grow, e.g. the number of vertices of the n-dimensional cube is literally the exponential of base 2. Which is why they discovered something about 126-dimensional space now, when the results for lower dimensions have been known for decades.
There is an old joke:
How do you imagine a 126-dimensional space? - Simple: imagine an n-dimensional space and set n=126.
The Kervaire invariant is a property of an "n-dimensional manifold", so the paper is likely about 126-dimensional manifolds. That in turn has a formal definition, and although it's not my specialization, I think means it can be locally represented as an n-dimensional Euclidean space.
A simple example would be a circle, which I guess would be a 1-dimensional manifold, because every point on a circle has a tangent where the circle can be approximated by a line passing through the same point.
So they're saying that there are these surfaces which can be locally approximated by 126-dimensional Euclidean spaces. This in turn probably requires that the surface itself is embedded in some higher-dimensional space such as R^127.
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.
If you just mean you're just unclear on the first step, of laying the knot out in 2D with crossings marked over/under, that's always possible after just some ordinary 3D adjustments. Although, yeah, if you asked me to prove it, I dunno that I could give one, I'm not a topologist... (and I guess now that I think about it the "finitely many" crossings part is actually wrong if we're allowing wild knots, but that's not really the issue)
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.
Someone once asserted that all learning is compression, and I’m pretty sure that’s how polymaths work. Maybe the first couple of domains they learn occupy considerable space in their heads, but then patterns emerge, and this school has elements from these other three, with important differences. X is like Y except for Z. Shortcut is too strong a word, but recycling perhaps.
Do we have anything in the universe that is knotted? Both on large and small scales. Or it is just coincidence?
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.
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
You might be surprised; there have proven to be a number of surprising connections between mathematical structures and more concrete sciences. For instance, group theory - long thought to be an highly abstract area of mathematics with no practical application - turned out to have some very direct applications in chemistry, particularly in spectroscopy.
Évariste Galois says hi and Satoshi-sensei greets him back.