[HN Gopher] Dimension 126 Contains Twisted Shapes, Mathematician...
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Dimension 126 Contains Twisted Shapes, Mathematicians Prove
Author : baruchel
Score : 82 points
Date : 2025-05-05 15:34 UTC (7 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| uxhacker wrote:
| 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?
| amelius wrote:
| > "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 wrote:
| 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!
| zmgsabst wrote:
| Yep -- you can always "knot" a sphere of two dimensions
| lower, starting with a circle in 3D and a sphere in 4D.
| nandomrumber wrote:
| When you untie a knot, it's _ends_ are fixed in time.
|
| Humans also unravel language meaning from within a hyper
| dimensional manifold.
| AIPedant wrote:
| 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.)
| Sniffnoy wrote:
| > 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 wrote:
| 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 wrote:
| 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)
| robocat wrote:
| > 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 wrote:
| 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.
| bee_rider wrote:
| 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 wrote:
| It's not just LLMs. Deep learning in general forms these
| multi-d latent spaces
| elpocko wrote:
| The "Mathematical Surgery" illustration is funny. Mathematicians
| can make a sphere from a torus and two halves of a sphere.
| Amazing!
| bee_rider wrote:
| 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.
| codetrotter wrote:
| As someone who is also not a mathematician it sounded perfectly
| normal to me.
| enasterosophes wrote:
| 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 wrote:
| 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.
| core-explorer wrote:
| 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 wrote:
| 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 wrote:
| 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.
| duskwuff wrote:
| 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 wrote:
| 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 wrote:
| 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. :)
| aleph_minus_one wrote:
| > 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.
| zchrykng wrote:
| 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 wrote:
| 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
| baruchel wrote:
| I have always been fond of the following quote by Jacobi:
| "Mathematics exists solely for the honor of the human mind"
| zchrykng wrote:
| 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.
| core-explorer wrote:
| 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 wrote:
| >Seeing as mathematicians proving things in math has minimal
| relation to the real world, I'm not sure how important this is.
|
| Evariste Galois says hi and Satoshi-sensei greets him back.
| duskwuff wrote:
| > 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.
| kiicia wrote:
| > 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 wrote:
| 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!
| codebje wrote:
| 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 wrote:
| > 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:
|
| > https://en.wikipedia.org/w/index.php?title=Kervaire_invar
| ian...
|
| "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
| O with the following properties from which their result
| follows immediately:
|
| * The coefficient groups O^n(point) have period 2^8 = 256
| in n
|
| * The coefficient groups O^n(point) have a "gap": they
| vanish for n = -1, -2, and -3
|
| * The coefficient groups O^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 O^{-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
| robinhouston wrote:
| The proof that it stops instead of going on forever is here:
| https://arxiv.org/abs/0908.3724
|
| It's more than 200 pages of pretty technical mathematics, so
| I'm reasonably confident that there is no description a
| layperson might understand.
| lifefeed wrote:
| Well, shit.
| anthk wrote:
| Network optimizing problems are just better with 4D hypercubes.
| ReptileMan wrote:
| >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 wrote:
| Not wishing to be flippant, but I have lots of bits of string
| that are knotted.
| m3kw9 wrote:
| This is some Dr Strange stuff
| impish9208 wrote:
| This got me thinking -- is there a version of "in mice" for math
| papers?
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