What is a manifold?

https://www.quantamagazine.org/what-is-a-manifold-20251103/

by isaacfrond • 9 months ago

393 133 9 months ago

hshdhdhehd• 9 months ago

Funny how a car manifold is also a mathematical manifold but the word seems to come from different roots.

dvt• 9 months ago

I just looked it up because I was interested in their etymologies, but it seems that the words actually have the same (Old English/Germanic) root: essentially a portmanteau of "many" + "fold."

andycrellin• 9 months ago

This has always caused me trouble when learning new concepts. A name for something will be given (e.g. manifold) and it sounds very much like something that I've come across before (e.g. a manifold in an engine) - and that then gets cemented in my brain as a relationship which I find extremely difficult to shake - and it makes understanding the new concept very challenging. More often than not the etymology of the term is not provided with the concept - not entirely unreasonable, but also not helpful for me personally.

It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!

Enginerrrd• 9 months ago

On many occasions in my mathematics education I was able to figure out and use a concept based solely on its name. (e.g. Feynman path integral)

Names are important.

namibj• 9 months ago

I thought those are ethymologically about the thin-walled containment of a volumetric interior space where said space is connected to only specific ports/holes, and is often but not necessarily mandatory intertwined with a second such containment for a second space (intake+exhaust).

tetrisgm2• 9 months ago

This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.

It’s actually much more well written than the majority or articles we usually come across.

robot-wrangler2• 9 months ago

I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles.

Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either

getnormality• 9 months ago

It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.

drob518• 9 months ago

I agree. I find their articles very enjoyable. And even though they stay technical, they don’t descend into becoming a technical journal. The content is still accessible to a non-expert like me.

hufdr• 9 months ago

Quanta’s greatest strength is that it doesn’t pretend to be clever. Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.

Workaccount2• 9 months ago

Quanta is amazing because it doesn't have to worry about money. It's a publication run by the Simons Foundation, funded with the proceeds of the wildly successful RenTec hedge fund. So they get pretty much full editorial control.

For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.

pferde• 9 months ago

And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/

lemonberry• 9 months ago

Nice find, thank you. Your sleuthing is appreciated.

Angostura• 9 months ago

Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.

ur-whale2• 9 months ago

When you use the word "engine" on HN, it can be understood as many things that aren't what you think (e.g. game engines).

idoubtit• 9 months ago

Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain.

For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as a square [O,2π[²?

I also expected more talk about atlases. In simple cases, it's easy to think of a surface as a deformation of a flat shape, so a natural idea is to think of having a map from the plan to the surface. But, even for a simple sphere, most surfaces can't map to a single flat part of the plan, and you need several maps. But how do you handle the parts where the maps overlap? What Riemmann did was defining properties on this relationship between manifold points and maps (which can be countless).

BTW, I know just enough about relativity to deny that "space-time [is] a four-dimensional manifold", at least a Riemmannian manifold. IIRC, the usual term is Minkowski-spacetime.

gowld4• 9 months ago

Precisely, why more as a manifold than as a square [O,2π[²?

Because, as the article explains, it's a torus (loop crossed with a loop), not a square (segment crossed with a segment).

griffzhowl2• 9 months ago

Spacetime is a four-dimensional manifold (at least theoretically - who knows what it is in reality). Technically it's a pseudo-Riemannian manifold since the metric is not positive definite: it can be negative or zero for non-zero vectors. A Riemannian manifold proper has a positive definite metric, but in popularizations like this I wouldn't really expect them to get into these kinds of distinctions.

mr_mitm• 9 months ago

Minkowski spacetime is the term in special relativity, i.e. the flat case, or zero curvature. In general relativity, spacetime is a pseudo Riemannian manifold, like the sibling comment says. Unlike Minkowski spacetime, it can be curved.

doctoboggan• 9 months ago

Precisely, why more as a manifold than as a square

In a double pendulum, each arm can freely rotate (there is no stopping point). This means 0 degrees and 360 degrees are the same point, so the edges of the square are actually joined. If you join the left and right edges to each other, then join the top and bottom edges to each other, you end up with a torus.

frutiger• 9 months ago

BTW, I know just enough about relativity

Unfortunately this is one of those things where that knowledge is not enough.

The GR model of spacetime is that it is locally Minkowski but globally a manifold of Minkowski patches.

elashri• 9 months ago

This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers.

Looking at things from abstract view does allow us not to worry about how we visualize the geometry which is actually hard and sometimes counter intuitive.

omnicognate4• 9 months ago

This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations.

In special relativity, for example, a huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.

For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.

messe2• 9 months ago

I agree that focusing on Lorentz transformations is the wrong way to approach thinking about special relativity. But It might be the right way to teach it to physics students.

The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.

The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.

The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".

(As an aside: I really love abstract index notation for dealing with tensors)

NoMoreNicksLeft• 9 months ago

Thanks for the book recommendation.

senderista5• 9 months ago

One of the worst examples is Weinberg’s book on GR, which I found nearly unreadable due to the morass of coordinates/indices. So much more painful to learn from than Wald or other mathematically modern treatments of GR.

senderista5• 9 months ago

I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.

antognini2• 9 months ago

Taylor & Wheeler's Spacetime Physics is similar. They emphasize the importance of frame invariant representations. (I highly recommend the first edition over the second edition, the second edition was a massive downgrade.)

Kip Thorne was also heavily influenced by this geometric approach. Modern Classical Physics by Thorne & Blandford uses a frame invariant, geometric approach throughout, which (imo) makes for much simpler and more intuitive representations. It allows you to separate out the internal physics from the effect of choosing a particular coordinate system.

sega_sai• 9 months ago

I found the physicist definition of a tensor is actually more confusing, because you are faced with these definitions how to transform these objects, but you never are really explained where does it all come from. While the mathematical definition through differential forms, co-vectors, while being longer actually explains these objects better.

lisper7• 9 months ago

You might find it circular reasoning but it is not

Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.

seanhunter3• 9 months ago

To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way <....explanation of the specific characteristics that mean that a tensor transforms in a way that's independent of the chosen coordinate system...>".

When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.

My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones, which is pretty nifty.

https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...

ChadNauseam4• 9 months ago

Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".

KalMann• 9 months ago

I don't get why people act like this definition is so circular. If you were to explain in detail what "transforms as a second rank tensor" means then it wouldn't be circular anymore. This just isn't the full definition.

genoveffo• 9 months ago

I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"

psychoslave• 9 months ago

This is not really something limited to mathematics.

BigTTYGothGF3• 9 months ago

In English, not all varieties are manifolds, see forex https://math.stackexchange.com/a/9017/120475

shmageggy• 9 months ago

FTA

The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”

huflungdung• 9 months ago

Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!

AnimalMuppet• 9 months ago

See you and raise you.

Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/2 way around the equator, turn 90 degrees again. Walk back to the pole. Now the triangle sums 360 degrees!

ChrisGreenHeur3• 9 months ago

A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.

snthpy• 9 months ago

Nicely done!

Initially I recoiled at the thought of the stiffness of the CD, but of course your absolutely right, at least for 2d manifolds.

BigTTYGothGF3• 9 months ago

you can put a cd shaped object on

You're thinking of open sets.

BigTTYGothGF3• 9 months ago

In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.

dist-epoch• 9 months ago

Including the hole at the center of the CD?

ChrisGreenHeur3• 8 months ago

there can be no hole on the cd shaped object, technically disc is the word to use but the point of the sentence is to provide a vivid picture

stelliosk• 9 months ago

Lobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"

xanderlewis2• 9 months ago

боже мой.

ur-whale2• 9 months ago

боже мой.

Pokemon?

bigdict• 9 months ago

There's antimony, arsenic, aluminum, selenium…

fsloth• 9 months ago

Plagiarize!

p1dda• 9 months ago

A very tight poker player

nelox• 9 months ago

A $1,500 trip to the mechanic

ericcholis• 9 months ago

That's why I clicked the title...thought for sure I was getting some engine knowledge

neuralkoi• 9 months ago

This is such a well written article and the author is such a good communicator. Looks like they've written a book as well called Mapmatics:

[0] https://www.paulinarowinska.com/about-me

jordanpg2• 9 months ago

Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.

pm90• 9 months ago

Its unfortunately all too common for Physics/Math to be taught in that way (extremely technical, memorizing or knowing equations and derivations). The best teachers would always give a ton of context as to why and how these came about.

lizknope• 9 months ago

I was reading a book on string theory and I remember the Calabi–Yau manifold

https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold

I'm not going to pretend to understand it all but they do make pretty pictures!

https://www.google.com/search?q=calabi+yau+manifold+images

tamnd5• 9 months ago

I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing.

What makes Calabi Yau manifolds special is that their curvature balances out perfectly so the space does not stretch or shrink overall.

In physics especially in string theory Calabi Yau manifolds are used to describe extra hidden dimensions of the universe beyond the three we can see. The shape of a Calabi Yau manifold affects how particles and forces behave which is why both mathematicians and physicists study them.

moralestapia2• 9 months ago

their curvature balances out perfectly so the space does not stretch or shrink overall

Could you elaborate a bit on this? I find it fascinating. Thanks.

The shape of a Calabi Yau manifold affects how particles and forces behave [...]

Do you know if there's any experimental evidence of this?

thaumasiotes• 9 months ago

They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi (opens a new tab), a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”

Something's gone badly wrong here. "Without learning Cyrillic" is the normal way to learn Russian. Pick a slightly less prominent language and 100% of learners will do it without learning anything about the writing system.

mmooss5• 9 months ago

I thought the same - many languages don't have a writing system and children learn without being able to write. But that's beside the point; the point is just as valid even if the analogy is poor.

tamnd5• 9 months ago

I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.

WhyOhWhyQ• 9 months ago

It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.

tamnd5• 9 months ago

That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation.

I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.

perihelions• 9 months ago

What's the relation between the different Lee manifolds? Is it a sequence you're supposed to read in order?

codethief3• 9 months ago

Tbh, I never quite understood the appeal of John M. Lee's book. It's not bad but I didn't find it great, either, especially (IIRC) in terms of rigor. Meanwhile, the much less well-known "Manifolds and Differential Geometry" by Jeffrey M. Lee (yeah, almost the same name) was much better.

mmooss5• 9 months ago

I rarely see manifolds applied directly to cartographic map projections, which I've read about a bit, though the latter seem like just one instance of the former. Does anyone know why cartographers don't use manifolds, or mathematicians don't apply them to cartography? (Have I just overlooked it?)

brosco• 9 months ago

One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already.

Another is that when working with manifolds, you usually don't get a set of global coordinates. Manifolds are defined by various local coordinate charts. A smooth manifold just means that you can change coordinates in a smooth (differentiable) way, but that doesn't mean two people on opposite sides of the manifold will agree on their coordinate system. On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.

I'm not very well versed in the history, but the study of cartography certainly predates the modern idea of an abstract manifold. In fact, the modern view was born in an effort to unify a lot of classical ideas from the study of calculus on spheres etc.

mmooss5• 9 months ago

Thanks. I've thought about those possibilites, but I really don't know the reasons.

On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.

Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with distance from the poles (i.e., the distance is much greater at the equator than the poles), but there is a unique coordinate for every point on the sphere.

mathgradthrow2• 9 months ago

What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?

The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.

_as_text• 9 months ago

ok but she was talking about riemann

kristopolous2• 9 months ago

Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings.

This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon

mathgradthrow2• 9 months ago

If this article broke some loop, you could answer my question and tell me one thing you got out of it. What's a manifold?

yatopifo• 9 months ago

Can anyone who is not a mathematician tell me one thing they learned from this?

I can share my two take-aways.

- in the geometric sense, manifolds are spaces analogous to curved 2d surfaces in 3d that extend to an arbitrary number of dimensions

- manifolds are locally Euclidean

If I were to extrapolate from the above, i'd say that:

- we can map a Euclidean space to every point on a manifold and figure out the general transformation rules that can take us from one point's Euclidean space to another point's.

- manifolds enable us to discuss curved spaces without looking at their higher-dimension parent spaces (e.g. in the case of a sphere surface we can be content with just two dimensions without working in 3d).

Naturally, I may be totally wrong about all this since I have no knowledge on the subject...

gowld4• 9 months ago

Wikipedia has a thorough intro article https://en.wikipedia.org/wiki/Manifold

hoseja• 9 months ago

So what is the "Not a manifold." part? The actually interesting part.

HarHarVeryFunny• 9 months ago

Does the way "manifold" is used when describing subsets of the representational space of neural networks (e.g. "data lies on a low-dimensional manifold within the high-dimensional representation space") actually correspond to this formal definition, or is it just co-opting the name to mean something simpler (just an embedded sub-space)?

If it is the formal definition being used, then why? Do people actually reason about data manifolds using "atlases" and "charts" of locally euclidean parts of the manifold?

antognini2• 9 months ago

It's hard to prove rigorously which is why people usually refer to it as the "manifold hypothesis." But it is reasonable to suppose that (most) data does live on a manifold in the strict sense of the term. If you imagine the pixels associated with a handwritten "6", you can smoothly deform the 6 into a variety of appearances where all the intermediate stages are recognizable as a 6.

However the embedding space of a typical neural network that is representing the data is not a manifold. If you use ReLU activations the kinks that the ReLU function creates break the smoothness. (Though if you exclusively used a smooth activation function like the swish function you could maintain a manifold structure.)

macleginn• 9 months ago

People also apply the notion of data manifold to language data (which is fundamentally discrete), and even for images the smoothness is hard to come buy (e.g., "images of cars" is not smooth because of shape and colour discontinuities). I guess the best we can do is to hope that there is an underlying virtual "data manifold" from which our datapoints have been "sampled", and knowing its structure may be useful.

youoy• 9 months ago

The closest thing that you may get is a manifold + noise. Maybe some people thing about it in that way. Think for example of the graph of y=sin(x)+noise, you can say that this is a 1 dimensional data manifold. And you can say that locally a data manifold is something that looks like a graph or embedding (with more dimensions) plus noise.

But i am skeptical whether this definition can be useful in the real world of algorithms. For example you can define things like topological data analysis, but the applications are limited, mainly due to the curse of dimensionality.

qbit42• 9 months ago

Sometimes statistical rates for empirical risk minimization can be related to the intrinsic dimension of the data manifold (and noise level if present). In such cases, you are running the same algorithm but getting a performance guarantee that depends on the structure of the data, stronger when it is low dimensional.

griffzhowl2• 9 months ago

There's a field known as information geometry. I don't know much about it myself as I'm more into physics, but here's a recent example of applying geometrical analysis to neural networks. Looks interesting as they find a phenomenon analogous to phase transitions during training

Information Geometry of Evolution of Neural Network Parameters While Training

https://arxiv.org/abs/2406.05295

britzkopf• 9 months ago

Every time I try to get some handle on the essence of this topic I fail. No different here. In the second paragraph it defines manifolds as "... shapes that look flat to an ant living on them, even though they might have a more complicated global structure"

So manifolds are complicated shapes that are at large enough a scale that an ant (which species?) will think they're flat....ok

d4rkn0d3z• 9 months ago

Manifold: Any m dimensional hyperplane embedded in an n dimensional Euclidean space, where m is less than or equal to n. More simply put, a manifold is any set that can be continuously parameterized, with the number of parameters being the dimension of the manifold.

A continuous manifold will have a line element that allows you to compute distances between its points using its parameters. The simplest line element was first written down by Pythagorus I think, it allows you to compute the distance between two points in a flat manifold. In physics we do away with gravitational forces by realizing that masses move along geodesics (shortest paths) of a manifold, hence the saying,"matter tells spacetime how to curve and spacetime tells matter how to move". We stich together large curvy manifolds like a patch quilt from the locally Euclidean tangent spaces that we erect at any point.

What is a manifold?

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