[HN Gopher] What is a manifold?
___________________________________________________________________
What is a manifold?
Author : isaacfrond
Score : 311 points
Date : 2025-11-04 09:58 UTC (13 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| hshdhdhehd wrote:
| Funny how a car manifold is also a mathematical manifold but the
| word seems to come from different roots.
| dvt wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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).
| leshokunin wrote:
| 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-wrangler wrote:
| 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 wrote:
| 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.
| robotresearcher wrote:
| Carnegie Libraries, Nobel Prizes, Rhodes scholarships?
| gowld wrote:
| Mathematica?
| robot-wrangler wrote:
| Good game and a hard question, especially if you make
| "comparable" more explicit. I'd add "noncommercial, open-
| access", and "modern" in the sense that it happened under
| the current norms with respect to legacy and the social
| contract.
| drob518 wrote:
| 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 wrote:
| 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.
| aleph_minus_one wrote:
| > Many tech publications write as if they're showing off,
| and you just end up feeling tired after reading them.
|
| I like this honestly because this shows that I learned
| something intelligent. On the other hand, if I don't feel
| exhausted after reading, it is a strong sign that the
| article was below my intellectual capacity, i.e. I would
| have loved it if I could have learned more.
| mulmen wrote:
| Seems superficial. If a simple concept is presented in a
| complex way what did you actually learn?
| aleph_minus_one wrote:
| Often, if the concept is presented in a more complex way
| the reason is that the author wants to emphasize and
| explain how the concept relates in a non-trivial way to
| some other deep concept; thus you learn a lot more than
| when the author explains things in the most simple (and
| shallow) way.
| mejutoco wrote:
| IMO the most common reason why something is presented in
| a more complex way is that it is badly explained.
|
| Of course, most common or not, each case is different.
| jaennaet wrote:
| Also speaks to a lack of understanding on the author's
| part; people who truly understand some subject are
| generally much more adept at explaining it in simpler
| terms - ie without adding complexity beyond the subject's
| essential complexity
| mulmen wrote:
| I don't see how that is beneficial. If a simple concept
| relates to a complex one then explain the complexity,
| don't add it.
| gowld wrote:
| It's OK to keep going deeper into the material if you
| aren't tired yet.
| Workaccount2 wrote:
| 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.
| jordanpg wrote:
| Remarkably, they don't even ask for money anywhere on the
| site. Now _that_ is a rare thing on the modern internet,
| especially for high quality writing.
| pferde wrote:
| _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 wrote:
| Nice find, thank you. Your sleuthing is appreciated.
| leshokunin wrote:
| Oh dope. Added to my feedbin!
| Angostura wrote:
| 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-whale wrote:
| 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 wrote:
| 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,2p[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.
| gowld wrote:
| > Precisely, why more as a manifold than as a square [O,2p[2?
|
| Because, as the article explains, it's a torus (loop crossed
| with a loop), not a square (segment crossed with a segment).
| griffzhowl wrote:
| 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 wrote:
| 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 wrote:
| > 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.
| elashri wrote:
| 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.
| omnicognate wrote:
| 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.
| messe wrote:
| 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)
| omnicognate wrote:
| Yeah, I had a slightly odd introduction to these things as
| I studied joint honours maths and physics. That meant both
| that I had a bit more mathematical maturity than most of
| the physics students and that I was being taught the more
| rigorous underpinnings of the maths while it was being
| (ab)used in all sorts of cavalier ways in physics. I liked
| the subject matter of physics more, but I greatly preferred
| the intellectual rigour of the maths.
|
| Eric Gourgoulhon is a product of the French education
| system, and I often think I would have done better studying
| there than in the UK.
| messe wrote:
| Mine was similar actually, just in Ireland.
|
| I had started in a theoretical physics degree which was
| jointly taught by the maths and physics department. By my
| final year I had changed into an ostensibly pure maths
| degree, although I did it mainly to take more advanced
| theoretical/mathematical physics courses (which were
| taught by the maths department), and avoid having to do
| any lab work--a torsion pendulum experiment was my final
| straw on that one, I don't know what caused it to fuck
| up, but fuck that.
|
| In the end I took on more TP courses than the TP
| students, nearly burnt out by the end of the year, and...
| didn't exactly come out with the best exam results.
| tonyarkles wrote:
| > 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.
|
| That was one of the most interesting things of my EE/CS
| dual-degree and the exact concept you're describing has
| stuck with me for a very long time... and very much
| influences how I teach things when I'm in that role.
|
| EE taught basic linear algebra in 1st year as a necessity.
| We didn't understand how or why anything worked, we were
| just taught how to turn the crank and get answers out.
| Eigenvectors, determinants, Gauss-Jordan elimination,
| Cramer's rule, etc. weren't taught with any kind of
| theoretical underpinnings. My CS degree required me to take
| an upper years linear algebra course from the math
| department; after taking that, my EE skills improved
| dramatically.
|
| CS taught algorithms early and often. EE didn't really
| touch on them at all, except when a specific one was needed
| to solve a specific problem. I remember sitting in a 4th
| year Digital Communications course where we were learning
| about Viterbi decoders. The professor was having a hard
| time explaining it by drawing a lattice and showing how you
| do the computations, the students were completely lost. My
| friend and I were looking at what was going on and both had
| this lightbulb moment at the same time. "Oh, this is just a
| dynamic programming problem."
|
| EE taught us way more calculus than CS did. In a CS systems
| modelling course we were learning about continuous-time and
| discrete-time state-space models. Most of the students were
| having a super hard time with dx/dt = A*x (x as a real
| vector, A as a matrix)... which makes sense since they'd
| only ever done single-variable calculus. The prof taught
| some specific technique that applied to a specific form of
| the problem and that was enough for students to be able to
| turn the crank, but no one understood why it worked.
| codethief wrote:
| > But It might be the right way to teach it to physics
| students.
|
| Having studied physics, I would disagree rather strongly. I
| only really started understanding Special Relativity once I
| had a clear understanding of the math. (And then it becomes
| almost trivial.) Those of my fellow class mates, however,
| who didn't take the time to take those additional
| (completely optional) math classes, ended up not really
| understanding it at all. They still got confused by what it
| all meant, by the different paradoxes, etc.
|
| I saw the same effect when, later, I was a teaching
| assistant for a General Relativity class.
| NoMoreNicksLeft wrote:
| Thanks for the book recommendation.
| senderista wrote:
| 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.
| omnicognate wrote:
| That's good to know about Wald. I bought a copy to finally
| get my head round General Relativity, but its brief
| explanation of Special Relativity right at the start made
| it clear that I hadn't properly understood that, which led
| to me getting Gourgoulhon's book. I should be better placed
| to tackle it now.
| codethief wrote:
| Weinberg [?] Wald. Wald's book is great! (For GR, of
| course, not SR.)
| omnicognate wrote:
| Indeed! I meant that it's good to know Wald is
| mathematically modern and not encrusted with coordinates.
| Saves me buying another book :-D
|
| (The comment I replied to mentioned both.)
| senderista wrote:
| I think _Spacetime Physics_ takes roughly the same approach
| (they call it "the invariant interval"), but with much less
| mathematical sophistication required.
| antognini wrote:
| 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 wrote:
| 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.
| lisper wrote:
| > 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_.
| seanhunter wrote:
| 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.
| ..
| lisper wrote:
| > a convenient shorthand for the bit that I put in angle
| brackets above.
|
| Yes, but the "convenient shorthand" only makes sense if you
| already know what a tensor is. That renders the
| "definition" useless as an explanation or as pedagogy. It's
| only useful as a social signal to let others know that you
| understand what a tensor is (or at least you think you do).
|
| > My favourite explanation is that "Tensors are the facts
| of the universe"
|
| That's not much better. "The earth revolves around the sun"
| is a fact of the universe, but that doesn't help me
| understand what a tensor is.
|
| What matters about tensors are the properties that
| _distinguish_ them from other mathematical objects, and in
| particular, what distinguishes them from closely related
| mathematical objects like vectors and arrays. Finding a
| cogent description of that on the internet is nearly
| impossible.
|
| > the reality of the tensor ... is independent of the
| coordinate system chosen by the observer
|
| Now you're getting closer, but this still misses the mark.
| What is "the reality of a tensor"? Tensors are mathematical
| objects. They don't have "reality" any more than numbers
| do.
|
| > 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
|
| That is closer still. But I would go with something more
| like: tensors are a way to represent vectors so that the
| representation of a given vector is the same no matter what
| basis (or coordinate system) you choose for your vector
| space.
| seanhunter wrote:
| > But I would go with something more like: tensors are a
| way to represent vectors so that the representation of a
| given vector is the same no matter what basis (or
| coordinate system) you choose for your vector space.
|
| That's just incorrect though for a couple of reasons.
| Firstly, a vector in the sense in which it is used in
| physics _is_ a rank 1 tensor so it has this
| transformation behaviour just like other higher order
| tensors. Secondly the representation is the thing that
| changes, but the meaning of that representation in the
| old basis and the new basis is the same. For example, if
| I take the displacement from me to the top of the Eiffel
| tower, I can represent that in xyz Cartesian coordinates
| or in spherical or cylindrical coordinates, or I can
| measure it relative to an origin that starts with me or
| at sea level at 0 latlong. The representation will be
| very different in each case, but the actual displacement
| from me to the top of the Eiffel tower doesn 't change.
| What has happened is the basis vectors transform in
| exactly such a way as to make that happen. It's a rank 1
| tensor in 3 dimensions because there is a magnitude and
| one direction (one set of 3 basis vectors) in whatever
| case.
|
| Now if I want an example of a rank 2 tensor think about a
| stress tensor. I have a steel beam which is clamped at
| both ends and a weight is on top of it. This is a tensor
| field. For every point in the beam there are different
| forces acting in each direction. So you could imagine the
| beam as made up of a grid of little rubik's cubes. On
| each face of each cube you have different net forces. (eg
| at the middle of the beam the forces are mainly downwards
| due to gravity, at the ends of the beam the fact that the
| middle of the beam is bowing downards will lead to the
| "faces" that point to the middle of the beam to be being
| pulled towards the middle (transverse to the beam and
| slightly downwards) whereas the opposite face is pulled
| in the opposite direction because the ends of the beam
| are clamped. So I need two sets of basis vectors. One set
| indicates the "face" experiencing the force, one set
| indicates the direction of the force. Now just like the
| vector/rank one tensor case I can represent those in
| whatever coordinate system I want, and my representation
| will be different in each case, but will mean the same
| sets of forces in the same directions and applied to the
| same directions because both sets of basis vectors will
| transform to make that true. I would call that a rank 2
| tensor field because I would express it as a function
| from a set of spatial coordinates to a thing which has a
| magnitude and 2 directions (that's what I think of as the
| tensor). However I understand physicists and civil
| engineers and stuff just call the whole thing the stress
| tensor (not the stress tensor field). I could be wrong.
|
| So what I mean when I talk about the reality of the
| tensor I mean whatever it is the tensor is expressing in
| the physical universe (eg the displacement from me to the
| tower or the stress in the beam). From a mathematical
| point of view I agree of course, mathematical objects
| themselves are purely arbitrary and abstract. But if you
| have a bridge and you want to make sure it doesn't buckle
| and fall down, the stress tensor in the bridge is a real
| and important fact of the universe that you need to have
| a decent understanding of.
| lisper wrote:
| > That's just incorrect though
|
| Quite possible. But that's in no small measure because I
| have yet to find an actual cogent definition of "tensor"
| that distinguishes a tensor from an array. (I have a
| similar problem with monads.)
|
| > So what I mean when I talk about the reality of the
| tensor I mean whatever it is the tensor is expressing in
| the physical universe
|
| OK, but then "the reality of a tensor" not depending on
| the coordinate system has nothing to do with tensors, and
| becomes a vacuous observation. It is simply a fact that
| actual physical quantities don't depend on how you write
| them down, and hence don't change when you write them
| down in different ways.
| denotational wrote:
| Right, but if you fill in the shorthand there's no reason
| to think it's circular; it's just a normal definition at
| that point, albeit one without much motivation.
| lisper wrote:
| But it's not possible to fill in the shorthand unless you
| already know what it stands for. Hence: the shorthand is
| not useful for communicating information, only for social
| signaling.
| KalMann wrote:
| 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 wrote:
| 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
| "varieta"
| psychoslave wrote:
| This is not really something limited to mathematics.
| BigTTYGothGF wrote:
| In English, not all varieties are manifolds, see forex
| https://math.stackexchange.com/a/9017/120475
| shmageggy wrote:
| FTA
|
| > _The term "manifold" comes from Riemann's_ Mannigfaltigkeit,
| _which is German for "variety" or "multiplicity."_
| huflungdung wrote:
| 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 wrote:
| 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!
| ChrisGreenHeur wrote:
| 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 wrote:
| 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.
| BigTTYGothGF wrote:
| > you can put a cd shaped object on
|
| You're thinking of open sets.
| BigTTYGothGF wrote:
| 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 wrote:
| Including the hole at the center of the CD?
| stelliosk wrote:
| Lobachevsky... "the analytic and algebraic topology of locally
| Euclidean metrizations of infinitely differentiable Riemannian
| manifolds"
| xanderlewis wrote:
| bozhe moi.
| ur-whale wrote:
| > bozhe moi.
|
| Pokemon?
| bigdict wrote:
| There's antimony, arsenic, aluminum, selenium...
| fsloth wrote:
| Plagiarize!
| p1dda wrote:
| A very tight poker player
| nelox wrote:
| A $1,500 trip to the mechanic
| ericcholis wrote:
| That's why I clicked the title...thought for sure I was getting
| some engine knowledge
| neuralkoi wrote:
| 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
| jordanpg wrote:
| 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 wrote:
| 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 wrote:
| 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
| tamnd wrote:
| 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.
| moralestapia wrote:
| >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?
| tamnd wrote:
| > Do you know if there's any experimental evidence of this?
|
| As my knowledge, there is no direct evidence that Calabi
| Yau manifolds describe real extra dimensions. In string
| theory, these shapes are used because they fit the math and
| preserve symmetries like supersymmetry. Experiments have
| not found signs of extra dimensions or supersymmetric
| particles, so Calabi Yau manifolds remain a beautiful
| theoretical idea, not something confirmed by observation.
| tamnd wrote:
| > Could you elaborate a bit on this?
|
| Please correct me if I am wrong, I have not touched this
| subject in a long time and only have some intuition. Here
| is how I understand it:
|
| A manifold is a kind of space that looks flat when you zoom
| in close enough. The surface of a sphere or a doughnut is a
| 2D manifold, and the space we live in is a 3D manifold. A
| Calabi Yau is one of these spaces but with more dimensions
| and extra symmetry that makes it very special.
|
| In geometry there are several ways to describe curvature.
| The most complete one is the Riemann curvature tensor,
| which contains all the information about how space bends.
| If you take a specific kind of average of that, you get the
| Ricci curvature tensor. Ricci curvature tells you how the
| size of small regions in space changes compared to what
| would happen in flat space.
|
| Imagine a tiny ball floating in this curved space. If the
| Ricci curvature is positive, nearby paths tend to come
| together and the ball's volume becomes smaller than it
| would in flat space. If the Ricci curvature is negative,
| nearby paths move apart and the ball's volume grows larger.
| If the Ricci curvature is zero, the ball keeps the same
| volume overall. So when I said "the space does not stretch
| or shrink overall" I was describing this situation: the
| Ricci curvature is zero, which means the space does not
| expand or contract on average compared to flat space.
|
| The space can still have complicated twists and bends.
| Ricci curvature only measures a certain type of curvature
| related to volume change. Even if the Ricci tensor is zero,
| there can still be other kinds of curvature present. The
| curvature balances out is just an intuitive way to express
| that the volume effects cancel when you take the average
| that defines Ricci curvature. It does not mean the space
| has matching regions of positive and negative curvature in
| a literal sense, but rather that the mathematical
| combination producing Ricci curvature sums to zero.
|
| Noe back to definition: A Calabi-Yau manifold is defined as
| compact (finite in size), complex), and Kahler (it has a
| compatible geometric and complex structure), with a first
| Chern class equal to zero. Yau's theorem proves that such a
| space always has a way to measure distances so that its
| Ricci curvature is exactly zero. So when I said "the
| curvature balances out perfectly so the space does not
| stretch or shrink overall" I meant it as an intuitive
| description of this Ricci flat property. The space is not
| flat like a sheet of paper, but its internal geometry is
| perfectly balanced in the sense that there is no net
| expansion or contraction of space.
| moralestapia wrote:
| Thanks a lot!
| thaumasiotes wrote:
| > 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.
| mmooss wrote:
| 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.
| tamnd wrote:
| 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 wrote:
| 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.
| tamnd wrote:
| 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 wrote:
| What's the relation between the different Lee manifolds? Is
| it a sequence you're supposed to read in order?
| ducttapecrown wrote:
| Lee taught Intro to Topological Manifolds for one quarter,
| and then the next two quarters where Intro to Smooth
| Manifolds. Then Riemannian, then vector bundles, and then
| complex manifolds.
| codethief wrote:
| 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.
| mmooss wrote:
| 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 wrote:
| 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.
| mmooss wrote:
| 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.
| senderista wrote:
| The problem is that this global coordinate system isn't a
| continuous mapping (see the discontinuity of both angular
| coordinates between 2*pi and 0). Manifolds are required to
| have an "atlas"[0]: a collection of coordinate systems
| ("charts") that cover the space and are continuous mappings
| from open subsets of the underlying topological space to
| open subsets of Euclidean space, with the overlaps between
| charts inducing smooth (i.e., infinitely differentiable)
| mappings in Euclidean space.
|
| Colloquially, this means a manifold is just "a bunch of
| patches of n-dimensional Euclidean space, smoothly sewn
| together."
|
| A sphere requires at least two charts for an admissible
| atlas (say two hemispheres overlapping slightly at the
| equator, or six hemispheres with no overlaps), otherwise
| you get discontinuities.
|
| [0] https://en.wikipedia.org/wiki/Atlas_(topology)
| mmooss wrote:
| This part I don't grasp:
|
| > this global coordinate system isn't a continuous
| mapping (see the discontinuity of both angular
| coordinates between 2*pi and 0).
|
| I'm guessing that the issue is that I don't know your
| definition of 'continuous'.
|
| I believe every point on the planet (sphere, for
| simplification) has unique corresponding coordinates on
| the map projection (chart). The only exceptions I can see
| are, A) surfaces perpendicular to the aspect
| (perspective) of the projection, which is usually
| straight down and causes points on exactly vertical
| surfaces to share coordinates; B) if somehow coordinates
| are limited in precision or to rational numbers; C) some
| unusual projection that does it.
|
| > A sphere requires at least two charts for an admissible
| atlas (say two hemispheres overlapping slightly at the
| equator, or six hemispheres with no overlaps), otherwise
| you get discontinuities.
|
| There are cartographic projections that use two charts.
| Regarding those with one, where is the discontinuity in a
| Mercator projection? I think when I understand your
| meaning, it will be clear ...
| senderista wrote:
| Continuity is fundamentally a topological property of a
| mapping. It just means that for a mapping F and a point
| p, for any neighborhood del of F(p), we can find a
| neighborhood eps of p such that F(eps) is contained
| entirely in del. In simpler terms, if you draw a little
| ball around F(p), I can find a little ball around p whose
| image under F is contained in the little ball you drew
| around F(p). If I have coordinates on the sphere that
| suddenly jump between 0 and 2*pi, I can't satisfy this
| property, because points that are arbitrarily close on
| the sphere will be mapped to opposite sides of the
| "coordinate square" with sides [0,2*pi).
|
| The Mercator projection is obtained by removing two
| points from the sphere (both poles) and stretching the
| hole at each pole until the punctured sphere forms a
| cylinder, then cutting the cylinder along a line of
| longitude. So you can see that the 3 discontinuities in
| the Mercator projection correspond to the top and bottom
| edges (where we poked a hole at each pole) and the
| left/right edges (where we cut the cylinder). (Note that
| stretching the sphere at the poles changes the curvature,
| but cutting the cylinder does not. The projection would
| have the same properties on a cylinder.)
|
| It is possible to continuously map the sphere to the
| entire (infinite) plane if you just remove a single point
| (the north pole): place the sphere so the south pole is
| touching the origin of the plane and for any point on the
| sphere, draw a line from the north pole through that
| point. Where that line intersects the plane is that
| point's image under this mapping (called the Riemann
| sphere).
| mathgradthrow wrote:
| 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 wrote:
| ok but she was talking about riemann
| kristopolous wrote:
| 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
| yatopifo wrote:
| > 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...
| gowld wrote:
| Wikipedia has a thorough intro article
| https://en.wikipedia.org/wiki/Manifold
| hoseja wrote:
| So what is the "Not a manifold." part? The actually interesting
| part.
| HarHarVeryFunny wrote:
| 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?
| antognini wrote:
| 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.)
| youoy wrote:
| 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.
| griffzhowl wrote:
| 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
___________________________________________________________________
(page generated 2025-11-04 23:00 UTC)