Dimension 126 Contains Twisted Shapes, Mathematicians Prove
quantamagazine.org
quantamagazine.org
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"
It’s more than 200 pages of pretty technical mathematics, so I’m reasonably confident that there is no description a layperson might understand.
I only found a technical article that didn't show any images at all, and 3D looking images in image search
Visually seeing a 2D one might help understand what it is though, and it shouldn't be too bad to render something in 2 dimensions :)
So I guess the 126-dimensional shape actually also is in 127-dimensional space then
But the article says "Over the years, mathematicians found that the twisted shapes exist in dimensions 2, 6, 14, 30 and 62.".
To me "Exists in dimension 2" sounds like a shape in 2D space, not in 3D space, but apparently that's not what they mean and the way I understand this language is wrong
Sometimes you need more dimensions to embed the manifold. For a 2-dimencional object, the most famous example is the Klein bottle https://en.wikipedia.org/wiki/Klein_bottle You can construct one of them in 3-dimmension only if you cheat. Yhey look nice and you can buy a few cheating-versions. But you can embed the Klein bottle in 4-dimensions (without cheating).
For the manifold in the article, I'm not sure how many additional dimensions you need. Perhaps 127 (n+1) is enough or perhaps you need 252 (2n) or perhaps something in between. You can always embed an n-dimensional manifold in the 2n space, but that is the worst case. https://en.wikipedia.org/wiki/Whitney_embedding_theorem
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?
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.
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)
[1] note that a [loop of] rope is actually a 1-dimensional object (it only has length, no width), so the next dimension up should be a 2-dimensional object, which is true of the surface of a ball. a topologist would call these things a 1-sphere and a 2-sphere, respectively
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.
> learning is compression
I don't think I know enough about compression to find that metaphor useful
> occupy considerable space in their heads
I reckon this is a terribly misleading cliche. Our brains don't work like hard drives. From what I see we can keep stuffing more in there (compression?). Much of my past learning is now blurred but sometimes it surfaces in intuitions? Perhaps attention or interest is a better concept to use?
My favorite thing about LLMs is wondering how much of people's (or my own) conversations are just LLMs. I love the idea of playing games with people to see if I can predictably trigger phrases from people, but unfortunately I would feel like a heel doing that (so I don't). And catching myself doing an LLM reply is wonderful.
Some of the other sibling replies are also gorgeously vague-as (and I'm teasing myself with vagueness too). Abstracts are so soft.
The closer one can get to this ideal, the closer one has to a complete description of the distribution.
I think this is the sort of thing they were getting at with the compression comment.
What is true is that you can get good results by projecting lower dimensional data into higher dimensions, applying operations, and then projecting it back down.
The internal space of an LLM would also have things in common with how, say currents flow in a body of water because that too is a vector space. When you study this stuff you get this sort of zen sense of everything getting connected to everything else. Eg in one of my textbooks you look at how pollution spreads through the great lakes and then literally the next example looks at how drugs are absorbed into the bloodstream through the stomach and it’s exactly the same dynamic matrix and set of differential equations. Your stomach works the same as the great lakes on a really fundamental level.
The spaces being described here are a little more general than vector spaces, so some of things which are true about vector spaces wouldn’t necessarily work the same way here.
You probably mean considerably more special than a general vector space. We do have differentiable manifolds here.
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.
> Every smooth n-dimensional manifold can be embedded into R^{2n}.
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.
>> but "dimensions 8 and 24" to me sounds...
Note that the article says
> In dimensions 8 and 24, it’s possible to...
you didn't quote the "In". With the "In" it's usual math jargon that means
> "in dimension 4" to mean "when the dimension is equal to 4"
But the title has no "In" and it sounds very weird, perhaps even incorrect. Anyway, note that most of the times the title is not written by the author.
There is an old joke:
How do you imagine a 126-dimensional space? - Simple: imagine an n-dimensional space and set n=126.
I don't really get the whole surgery concept, if you can just cut the torus (and this could be anywhere, eg along the length of its inner or outer diameters) and glue the edges together aren't you just waving away the problem? By that logic I can slice open a sphere and call it a sheet, or conversely declare all shees to be spheres that haven't been glued together yet.
They have played us for absolute fools
You can do this. If you remove a point (or a line, or really any connected component), you get a space which is the same as the plane. What happens if you remove two distinct points? You end up with with a very thick circle. Three points? It starts to get harder to visualize, but you end up with two circles joined at a point. As you remove more points you will get more circles joined together. From a mathematical perspective, these spaces are very different. If we start to allow gluing arbitrary points in the sphere together it gets even worse, and you can get some pretty wild spaces.
The point of surgery is that by requiring this gluing in of these spheres along the boundary of the space we cut out, the resulting spaces are not as wild - or at least are easier to handle than if we do any operation. To give an example, one might have some space and we want to determine if it has property A. The problem is our space has some property B which makes it difficult to determine property A directly. But by performing surgery in a specific way, we can produce a new space which has property A if and only if the original space did, and importantly, no longer has property B.
For property As that mathematicians care about, surgery often does a good job of preserving the property. In contrast things like just cutting and gluing points together without care will typically change property A, so it does not help as much.
> Likewise I wonder why we need to import a sphere rather than just pinch the ends of the tube shut and say it's now a sphere.
I am not an expect on surgery, but I think from a mathematical perspective, pinching the ends of the tube shut and gluing in a new sphere would be equivalent operations. This pinching operation would be formalized as a "quotient space", and you can formalize the sphere as a "quotient" space equivalent to the pinching.
It's admittedly been a good 15 years since I cracked open a topology textbook, but the high-level, hand-wavey idea behind this sort of topological surgery isn't that you slice up manifolds and glue them together willy-nilly, but that you do so in very precise and controlled ways, which (and this is the very important bit) you've proven ahead of time preserve (or modify in a knowable fashion) some property of the manifold you care about. Rather than waving away the problem, you're decomposing it into a set of simpler ones which are ideally more tractable for some reason or another (perhaps you can compute a given property from first principles for a sphere, but not for a torus, for instance).
The wikipedia page on this stuff (https://en.wikipedia.org/wiki/Surgery_theory) is quite technical, but you might be able to squint at it and get a sense for how it works.
Ok. So, this is a BDSM universe. Interesting.
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
Évariste Galois says hi and Satoshi-sensei greets him back.
You might be surprised; there have proven to be a number of surprising connections between abstract 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.
The most obvious examples are number theory and group theory, which are respectively the study of numbers and how they behave under basic operations like arithmetic, and the study of a type of set with a single operation that satisfies very basic rules[1]. How could this possibly have any relevance or practical application? And yet it turns out they are central to cryptography and information theory. Joseph Fourier trying to solve the equations that govern how heat diffuses through a metal came up with the theory that forms the basis for how we do video and audio compression (and a ton of other things).
Finally mathematicians don’t speculate about how many dimensions the universe has, they study 4- and higher- dimensional objects and spaces to understand them. This theory is used all over the place. You can’t have a function like a temperature map without 4 dimensions (3 for the spatial coordinates of your input and one for the output).
[1] this turns out (non-obviously) to be the study of symmetry.