Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What is a manifold?

quantamagazine.org

Post reply on HN
Print view View original

11–20 of 138 posts

Re: What is a manifold?

#11

Post by messe »

omnicognate wrote: post #8
elashri wrote: post #6

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 visual…

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…

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)

Re: What is a manifold?

#12

Post by robot-wrangler »

tetrisgm wrote: post #5

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.

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

Re: What is a manifold?

#13

Post by omnicognate »

messe wrote: post #11
omnicognate wrote: post #8

Earlier quoted context omitted.

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…

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…

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.

Re: What is a manifold?

#14

Post by psychoslave »

genoveffo wrote: post #7

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"

This is not really something limited to mathematics.

Re: What is a manifold?

#15

Post by snthpy »

ChrisGreenHeur wrote: post #10

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.

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.

Re: What is a manifold?

#16

Post by Enginerrrd »

andycrellin wrote: post #4
hshdhdhehd wrote: post #2

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

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 te…

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.

Post reply on HN