What is a manifold?
28 comments
·November 4, 2025leshokunin
pferde
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/
robot-wrangler
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
hufdr
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.
drob518
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.
getnormality
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.
ChrisGreenHeur
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
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
> you can put a cd shaped object on
You're thinking of open sets.
elashri
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
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
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)
tonyarkles
> 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.
omnicognate
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.
sega_sai
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.
genoveffo
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
This is not really something limited to mathematics.
stelliosk
Lobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"
xanderlewis
боже мой.
neuralkoi
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:
hshdhdhehd
Funny how a car manifold is also a mathematical manifold but the word seems to come from different roots.
dvt
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
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
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.
p1dda
A very tight poker player
nelox
A $1,500 trip to the mechanic
huflungdung
[dead]
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.