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What is a manifold?
- hshdhdhehd 1y agoFunny how a car manifold is also a mathematical manifold but the word seems to come from different roots.
- dvt 1y agoI 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 1y agoThis 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 1y agoOn 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 1y agoI 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).
- tetrisgm 1y agoThis 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 1y agoI'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 1y agoIt'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 1y agoCarnegie Libraries, Nobel Prizes, Rhodes scholarships?
- Iwan-Zotow 1y agoClay Mathematics Institute, with its 7 Millennium problems
- elashri 1y agoThis 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 1y agoThis 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 1y agoI 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)
- genoveffo 1y agoI 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 1y agoThis is not really something limited to mathematics.
- BigTTYGothGF 1y agoIn English, not all varieties are manifolds, see forex https://math.stackexchange.com/a/9017/120475 https://math.stackexchange.com/a/9017/120475
- shmageggy 1y agoFTA > The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
- huflungdung 1y agoStand 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 1y agoSee 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 1y agoA 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 1y agoNicely 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 1y ago> you can put a cd shaped object on You're thinking of open sets.
- BigTTYGothGF 1y agoIn 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.
- deleted 1y ago[deleted]
- ChrisGreenHeur 11mo agoThe position of the points on the surface can be changed without worrying about the topology and its the topology we are worried about
- dist-epoch 1y agoIncluding the hole at the center of the CD?
- ChrisGreenHeur 11mo agothere 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 1y agoLobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"
- xanderlewis 1y agoбоже мой.
- ur-whale 1y ago> боже мой. Pokemon?
- xanderlewis 1y agoYou must not know great Lobachevsky.
- bigdict 1y agoThere's antimony, arsenic, aluminum, selenium…
- fsloth 1y agoPlagiarize!
- p1dda 1y agoA very tight poker player
- nelox 1y agoA $1,500 trip to the mechanic
- ericcholis 1y agoThat's why I clicked the title...thought for sure I was getting some engine knowledge
- neuralkoi 1y agoThis 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 https://www.paulinarowinska.com/about-me
- jordanpg 1y agoMan, 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 1y agoIts 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 1y agoI was reading a book on string theory and I remember the Calabi–Yau manifold https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_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 https://www.google.com/search?q=calabi+yau+manifold+images
- tamnd 1y agoI 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 1y 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?
- tamnd 1y ago> 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.
- thaumasiotes 1y 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.
- mmooss 1y agoI 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 1y agoI 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 1y agoIt'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 1y agoThat'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 1y agoWhat's the relation between the different Lee manifolds? Is it a sequence you're supposed to read in order?
- ducttapecrown 1y agoLee 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 1y agoTbh, 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 1y agoI 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 1y agoOne 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 1y agoThanks. 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 1y ago
- mathgradthrow 1y agoWhat 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 1y agook but she was talking about riemann
- kristopolous 1y agoWell 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
- mathgradthrow 1y agoIf this article broke some loop, you could answer my question and tell me one thing you got out of it. What's a manifold?
- kristopolous 1y agoI'm pretty confident I'm not going to give a definition that's suitable for you and I don't wish to engage with diminutive minutiae or arcane jargon in a hostile combative atmosphere. I don't think I can improve this silence. Have a good day.
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- tug2024 1y ago[dead]
- gowld 1y agoWikipedia has a thorough intro article https://en.wikipedia.org/wiki/Manifold https://en.wikipedia.org/wiki/Manifold
- hoseja 1y agoSo what is the "Not a manifold." part? The actually interesting part.
- HarHarVeryFunny 1y agoDoes 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 1y agoIt'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 1y agoPeople 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.
- hansvm 1y agoThose are less problematic than you might imagine. - For language, individual words might be discrete, but concepts being communicated have more nuance and fill in the gaps. - For language, even to the extent that discreteness applies, you can treat the data as being sampled from a coarser manifold and still extract a lot of meaningful structure. - Images of cars are more continuous than you might imagine because of hue differences induced by time of day, camera lens, shadows, etc. - Images of cars are potentially smooth even when considering shape and color discontinuities. Manifolds don't have to be globally connected. Local differentiability is usually the thing people are looking for in practical applications.
- britzkopf 1y agoEvery 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 1y agoManifold: 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.