Quanta Magazine consistently explains mathematics/physics for an advanced lay audience in ways that don't terribly oversimplify / still expose you to the true ideas. It's really nice! I don't know of any other sources like this.
Unfortunately I have not found this to be true (though it is in this case). There are quite a number of articles that are misleading or flat out false.
The best example is the quantum wormhole article and video[0,1], because it is egregious and doesn't take much nuance or expertise see the issues. I'm glad they made a note and wrote a follow-up[2], but all this illustrates what is wrong with the picture. For one, the article and video were published the same day as it was published in Nature[3]. Sure, they are getting wind of the preprints, but in this case there was none! They're often acting as a PR firm for many of the big universities and companies, unfortunately so is Nature.
The article was published Nov 30th, but the note didn't come till March 29th![4] You might think, oh it took that much time to figure out that there were problems, but no, only a few days after the publication (Dec 2nd) even Ars Technica was posting about the misinformation. They even waited over a month after Kobrin, Schuster, and Yao placed their comment on ArXiv[6]. Scott Aaronson had already written about it[7]. There was so much dissent in that time frame that it is hard to explain it as an accident. A week or two and it wouldn't be an issue.
But I think Peter Woit explains it best[8] (published, yes, Nov 30th).
This work is getting the full-press promotional package: no preprint on the arXiv, embargoed info to journalists, with reveal at a press conference, a cover story in Nature, accompanied by a barrage of press releases. This is the kind of PR effort for a physics result I’ve only seen before for things like the Higgs and LIGO gravitational wave discoveries. It would be appropriate I suppose if someone actually had built a wormhole in a lab and teleported information through it, as advertised.
I hate to say it, but you need to be careful with Quanta and others that __should__ be respectable. And I don't think we should let these things go. They are unhealthy for science and fundamentally create more social distrust for science. Now science skeptics can point to these same things as if there isn't more nuance all because they were more willing to take money from Google and CIT than wait a day and get some comments from other third party sources. (The whole peer review thing is another problem, but that's a different rabbit hole).
I've always wondered if it's possible to harness teen minds to solve significant math problems in high school, if you formulated them well and found the right scope. I think it's possible.
MIT's PRIMES program does exactly this - they give advanced high school students a mentor who picks out a problem, gets them up to speed on what is known, and then they work on the problem for a year and publish their results. It tends to work best with problems which have a computational aspect, so that the students can get some calculations done on the computer to get the ball rolling.
Just this year these girls discovered a proof for the Pythagorean theorem using nothing but trigonometry, a feat considered impossible until they did it: https://youtu.be/VHeWndnHuQs
Unfortunately it seems their proof already had the Pythagorean Theorem embedded within its implicit assumptions - they define measure of an angle through rotation of a circle. They don't explicitly define circle, but from their diagram they hint at the "understood" definition, namely a set of points equidistance from a central point, while using Euclidean distance as the metric.
Intuitively, just imagine picking a starting point on each of the circle and the knot. Now walk at different speeds such that you get back to the starting point at the same time.
In fact, that's what the knot is: a continuous, bijective mapping from the circle to the image of the mapping, i.e., the knot. (As the linked answer says.)
Edit: I see now that the article already has this intuitive explanation but with ants.
If you regard two spaces being homeomorphic as meaning — roughly — that if you lived in either space you’d not notice a difference, it makes sense. To a one-dimensional being (that has no concept of curvature or length, since we’re talking about topology here), they’d all feel like living in a circle.
I agree - I would happily pay for a print subscription and donate a second one to the nearest high school (although teens may no longer visit the library, so you'd probably have to spread the magazines in toilets, corridors and cafes).
You’d probably have a harder time finding a school library or even harder a librarian in the school. Between funding prioritization and challenging public policy requirements many schools have removed their libraries.
> But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large.
I'm struggling to understand what is counterintuitive here. Am I missing something?
Also, it's still (always) going to be in the shape of a cube. And if we are going to argue otherwise, we can do that without invoking infinity—technically it's not a cube after even a single iteration.
I won't say it isn't possible that someone might struggle with this—it's quite subjective, obviously—but I do think it's unlikely that anyone with a general understanding of both volume and surface area would struggle here.
Even just comparing two consecutive iterations, I feel confident that any child who has learned the basic concepts would be able to reliably tell you which has more enclosed volume or surface area.
I will happily concede that the part you quoted could be quite unintuitive without the context of the article or the animation included in it. :)
I think Gabriel's Horn is a great explanation of how this is counter-intuitive[1]. This is a shape which you could fill with a finite amount of water, say a gallon. Yet it would take an infinite amount of paint to paint the surface. Of course, part of the reason it's counter-intuitive is that there is no 0-thickness paint that exists.
Think of a 3-dimensional object (unlike a surface, which is 2-dimensional, regardless of the shape), with the volume zero. That's not easy to wrap your head around.
I don’t think the question is an active research area, or the problem would probably already be solved. However, it’s nonetheless extremely impressive. I couldn’t have done this at 21 with a lot more experience under my belt.
Quanta Magazine consistently explains mathematics/physics for an advanced lay audience in ways that don't terribly oversimplify / still expose you to the true ideas. It's really nice! I don't know of any other sources like this.
I'm sure the article author would love to know this!
Unfortunately I have not found this to be true (though it is in this case). There are quite a number of articles that are misleading or flat out false.
The best example is the quantum wormhole article and video[0,1], because it is egregious and doesn't take much nuance or expertise see the issues. I'm glad they made a note and wrote a follow-up[2], but all this illustrates what is wrong with the picture. For one, the article and video were published the same day as it was published in Nature[3]. Sure, they are getting wind of the preprints, but in this case there was none! They're often acting as a PR firm for many of the big universities and companies, unfortunately so is Nature.
The article was published Nov 30th, but the note didn't come till March 29th![4] You might think, oh it took that much time to figure out that there were problems, but no, only a few days after the publication (Dec 2nd) even Ars Technica was posting about the misinformation. They even waited over a month after Kobrin, Schuster, and Yao placed their comment on ArXiv[6]. Scott Aaronson had already written about it[7]. There was so much dissent in that time frame that it is hard to explain it as an accident. A week or two and it wouldn't be an issue.
But I think Peter Woit explains it best[8] (published, yes, Nov 30th).
I hate to say it, but you need to be careful with Quanta and others that __should__ be respectable. And I don't think we should let these things go. They are unhealthy for science and fundamentally create more social distrust for science. Now science skeptics can point to these same things as if there isn't more nuance all because they were more willing to take money from Google and CIT than wait a day and get some comments from other third party sources. (The whole peer review thing is another problem, but that's a different rabbit hole).[0] https://www.quantamagazine.org/physicists-create-a-wormhole-...
[1] https://www.youtube.com/watch?v=uOJCS1W1uzg
[2] https://www.quantamagazine.org/wormhole-experiment-called-in...
[3] https://www.nature.com/articles/s41586-022-05424-3
[4] https://web.archive.org/web/20230329191417/https://www.quant...
[5] https://arstechnica.com/science/2022/12/no-physicists-didnt-...
[6] https://arxiv.org/abs/2302.07897
[7] https://scottaaronson.blog/?p=6871
[8] https://www.math.columbia.edu/~woit/wordpress/?p=13181
A browser puzzle, based on "Knot Theory". Not sure I learned anything from playing this, but that was fun:
https://brainteaser.top/knot/index.html
Kind of reminds me of https://www.jasondavies.com/planarity/
I've always wondered if it's possible to harness teen minds to solve significant math problems in high school, if you formulated them well and found the right scope. I think it's possible.
MIT's PRIMES program does exactly this - they give advanced high school students a mentor who picks out a problem, gets them up to speed on what is known, and then they work on the problem for a year and publish their results. It tends to work best with problems which have a computational aspect, so that the students can get some calculations done on the computer to get the ball rolling.
It is very possible!
Just this year these girls discovered a proof for the Pythagorean theorem using nothing but trigonometry, a feat considered impossible until they did it: https://youtu.be/VHeWndnHuQs
Unfortunately it seems their proof already had the Pythagorean Theorem embedded within its implicit assumptions - they define measure of an angle through rotation of a circle. They don't explicitly define circle, but from their diagram they hint at the "understood" definition, namely a set of points equidistance from a central point, while using Euclidean distance as the metric.
> a feat considered impossible until they did it
Hm? https://www.cut-the-knot.org/pythagoras/TrigProof.shtml
> J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009)
And Zimba's proof terminates in a finite number of steps.
Monkeys with typewriters, or teenagers with MacBooks?
Yes that's the education system. But I suspect you mean in some automated turk fashion.
Sorry, I meant while they were still in high school; I've edited my original comment to make that clear.
> Every knot is “homeomorphic” to the circle
Here's an explanation:
https://math.stackexchange.com/questions/3791238/introductio...
Well, except for in Florida - where homeomorphism is banned in public schools.
Intuitively, just imagine picking a starting point on each of the circle and the knot. Now walk at different speeds such that you get back to the starting point at the same time.
In fact, that's what the knot is: a continuous, bijective mapping from the circle to the image of the mapping, i.e., the knot. (As the linked answer says.)
Edit: I see now that the article already has this intuitive explanation but with ants.
Somewhat counterintuitively, all knots are homeomorphic to each other.
If you regard two spaces being homeomorphic as meaning — roughly — that if you lived in either space you’d not notice a difference, it makes sense. To a one-dimensional being (that has no concept of curvature or length, since we’re talking about topology here), they’d all feel like living in a circle.
Quanta looks like a magnificent magazine. Thank you for bringing it into my life! This is the first time I've come across it
I love quanta so much. I wish there were a print version.
I agree - I would happily pay for a print subscription and donate a second one to the nearest high school (although teens may no longer visit the library, so you'd probably have to spread the magazines in toilets, corridors and cafes).
You’d probably have a harder time finding a school library or even harder a librarian in the school. Between funding prioritization and challenging public policy requirements many schools have removed their libraries.
I, on the other hand, prefer the modern media for the ability to include animations etc.
> But most important, the fractal possesses various counterintuitive mathematical properties. Continue to pluck out ever smaller pieces, and what started off as a cube becomes something else entirely. After infinitely many iterations, the shape’s volume dwindles to zero, while its surface area grows infinitely large.
I'm struggling to understand what is counterintuitive here. Am I missing something?
Also, it's still (always) going to be in the shape of a cube. And if we are going to argue otherwise, we can do that without invoking infinity—technically it's not a cube after even a single iteration.
This feels incredibly sloppy to me.
> shape’s volume dwindles to zero, while its surface area grows infinitely large
I think it's easy to grok when you get it, but that's certainly counter-intuitive on the surface, no?
I won't say it isn't possible that someone might struggle with this—it's quite subjective, obviously—but I do think it's unlikely that anyone with a general understanding of both volume and surface area would struggle here.
Even just comparing two consecutive iterations, I feel confident that any child who has learned the basic concepts would be able to reliably tell you which has more enclosed volume or surface area.
I will happily concede that the part you quoted could be quite unintuitive without the context of the article or the animation included in it. :)
I think Gabriel's Horn is a great explanation of how this is counter-intuitive[1]. This is a shape which you could fill with a finite amount of water, say a gallon. Yet it would take an infinite amount of paint to paint the surface. Of course, part of the reason it's counter-intuitive is that there is no 0-thickness paint that exists.
[1] https://en.wikipedia.org/wiki/Gabriel%27s_horn
Think of a 3-dimensional object (unlike a surface, which is 2-dimensional, regardless of the shape), with the volume zero. That's not easy to wrap your head around.
> That's not easy to wrap your head around.
I am trying to figure out the formal version of this topological conjecture. Even that isn't easy.
Super cool. I would have liked to have seen a similar visualisation for how they solved it on the Sierpinski gasket.
This is relevant to my interests
At my age, I really have to restrain myself of these interests to spare my time for some other stuffs. :(
Sorry to ask this, but is the result itself significant enough to the community, if it's not discovered by teens?
I don’t think the question is an active research area, or the problem would probably already be solved. However, it’s nonetheless extremely impressive. I couldn’t have done this at 21 with a lot more experience under my belt.