I've played that to completion with 2500 dots. I tried to go bigger but on my old machine I don't have the memory (or maybe nobody can with that architecture, i dunno, it got really slow)
you can play in a browser on the web. you can download the web versions and play locally in your browser. you can download the source and build. you can download builds for many different platforms.
This isn't true. Knots are not topological spaces, they are imbeddings, K:S^1 -> S^3 (more generally, S^n-> S^n+2). Therefore there isn't an obvious notion of homeomorphism. What the poster points out is that restriction to the image is a homeomorphism from the circle (because it is an imbedding of the circle)
As maps between topological spaces (and almost always we pick these to be from the smooth or piecewise linear categories, which further restricts them), the closest "natural" interpretation of isomorphism is pairs of homeomorphisms, f:S^1 -> S^1, g:S^3 -> S^3 satisfying Kf=gK. Ie natural transformations or commuting squares.
This gives us almost what we want, except that I can flip the orientation of space, or the orientation of the knot with homeomorphisms, neither of which correspond to the physical phenomenon of knots, so we give our spaces an orientation, which requires us to move to the PL or smooth category, or use homotopy/isotopy.
So, in other words, the notion of the homomorphism is almost useless in this context - which is why its mentioning when talking about knots may create confusion.
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.
Only because "homeomorphic" is a highly technical term that most people don't even know the definition of, let alone have an intuition for, and because "knot" in math is a closed loop, unlikely "knot" in common language.
Once you know the definitions and have learned to tie your shoes, it's quite intuitive. Even a small child can easily constructively prove that knots are homeomorphic.
Homeomorphic is also interesting because it's a very fancy sounding word for a concept that has very few requirements to be met. It's usually a basic requirement for concepts in topology, but it does very little to distinguish topological spaces. It's essentially a highfalutin word for "these things are basically the same thing in a very basic way".
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.
And soon we’ll have an administration with a major goal of deliberately defunding and crippling what remains of our school system to make sure nobody reads enough to realize what’s happening to their country.
Can anyone explain why they bothered with the fractal at all, instead of using a 3 dimensional grid? Doesn't a grid of the appropriate resolution provide the exact same? Or is it to show that they can do everything within even a subset of a 3D grid limited in this way?
The 3D grid result is well known, perhaps it would be fair to call it trivial (you can even throw away all but 2 layers of one of your dimensions). As you say, the Menger Sponge is a subset of the 3D grid, so the students had to find a construction which dodges the holes. To me, the result isn't surprising, but it is pleasing. But the really cool part of the article in my mind is the open problem at the end: can you embed every knot in the Sierpiński tetrahedron?
Like teeray mentions, if we can define any knot as a path through a Menger Sponge then that path could be realized in a Minecraft world (since it is cube/block/voxel based.)
If you placed minecart rails along the knot plot then you could ride it like a roller coaster.
Maybe this was the initial motivation for the teenagers.
Not parent, but I assume the idea is that you could form this fractal by hand (all or in part) in any 3D Minecraft terrain. Then you could route rails according to the theorem.
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.
Fair, but I think the article itself is very clear on the importance of the mentor. It's an fun question that hasn't previously been posed, which itself is quite impressive.
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.
Geometry has made a bit of progress since Euclid's time. Its become a bit more rigorous.
Euclidean geometry is based on five axioms, and some other terms left undefined.
The fifth postulate - the parallel postulate - was considered so irksome that for hundreds of years, many attempted to prove it using the other four, but failed to do so, and almost drove some crazy. In the late 19th century it was shown you can generate perfectly valid geometries if you assume it to be false somehow - either no-parallel (spherical geometry) or infinite parallel (hyperbolic)
Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it. Like I could draw a "circle with a radius of 1" using taxicab distance, and it would look like a diamond shape.
Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
The (non-generalized) Pythagorean theorem is part of Euclidean geometry, so non-Euclidean geometry is irrelevant to this discusion.
> Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it.
You do it using an axiomatic compass, a device that copies length in a circular pattern but does not measure it. Lengths are measured using constructable line segments.
Are you implying that nearly all the hundreds of proofs of Pythagorean theorem, which do not use modern rigorous definitions, are not valid proofs?
> Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
So? The Pythagorean theorem is very easy to prove. There are hundreds of proofs created by amateurs. That doesn't make them "not proofs" simply because other proofs exist.
> 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.
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 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.
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/
Or the "Untangle" puzzle from Simon Tatham's Portable Puzzle Collection https://www.chiark.greenend.org.uk/~sgtatham/puzzles/
Constantly one of the first additions to any new device I acquire (Android, Linux, Windows).
I've played that to completion with 2500 dots. I tried to go bigger but on my old machine I don't have the memory (or maybe nobody can with that architecture, i dunno, it got really slow)
How do you install this?
I am not sure I understand the question. Doesn't the "Download" section of the page provide links to the binaries for different platforms? https://www.chiark.greenend.org.uk/~sgtatham/puzzles/#downlo...
you can play in a browser on the web. you can download the web versions and play locally in your browser. you can download the source and build. you can download builds for many different platforms.
Quanta looks like a magnificent magazine. Thank you for bringing it into my life! This is the first time I've come across it
they have a couple of cool podcasts too, I quite like the Joy of Wh(y)
> Every knot is “homeomorphic” to the circle
Here's an explanation:
https://math.stackexchange.com/questions/3791238/introductio...
This isn't true. Knots are not topological spaces, they are imbeddings, K:S^1 -> S^3 (more generally, S^n-> S^n+2). Therefore there isn't an obvious notion of homeomorphism. What the poster points out is that restriction to the image is a homeomorphism from the circle (because it is an imbedding of the circle)
As maps between topological spaces (and almost always we pick these to be from the smooth or piecewise linear categories, which further restricts them), the closest "natural" interpretation of isomorphism is pairs of homeomorphisms, f:S^1 -> S^1, g:S^3 -> S^3 satisfying Kf=gK. Ie natural transformations or commuting squares.
This gives us almost what we want, except that I can flip the orientation of space, or the orientation of the knot with homeomorphisms, neither of which correspond to the physical phenomenon of knots, so we give our spaces an orientation, which requires us to move to the PL or smooth category, or use homotopy/isotopy.
So, in other words, the notion of the homomorphism is almost useless in this context - which is why its mentioning when talking about knots may create confusion.
https://en.wikipedia.org/wiki/Embedding#General_topology
"In general topology, an embedding is a homeomorphism onto its image."
a homeomorphism to the image is not the same as a homeomorphism.
onto is not the same as to
everything is onto its image.
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.
Only because "homeomorphic" is a highly technical term that most people don't even know the definition of, let alone have an intuition for, and because "knot" in math is a closed loop, unlikely "knot" in common language.
Once you know the definitions and have learned to tie your shoes, it's quite intuitive. Even a small child can easily constructively prove that knots are homeomorphic.
Homeomorphic is also interesting because it's a very fancy sounding word for a concept that has very few requirements to be met. It's usually a basic requirement for concepts in topology, but it does very little to distinguish topological spaces. It's essentially a highfalutin word for "these things are basically the same thing in a very basic way".
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.
[flagged]
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.
And soon we’ll have an administration with a major goal of deliberately defunding and crippling what remains of our school system to make sure nobody reads enough to realize what’s happening to their country.
Reminds me of one of my sophomore crushes: Leonardo @ MIT: https://direct.mit.edu/leon
I, on the other hand, prefer the modern media for the ability to include animations etc.
> a tetrahedral version of the Menger sponge
Better known as a Sierpiński tetrahedron, AKA the 3d version of a Sierpiński triangle.
Can anyone explain why they bothered with the fractal at all, instead of using a 3 dimensional grid? Doesn't a grid of the appropriate resolution provide the exact same? Or is it to show that they can do everything within even a subset of a 3D grid limited in this way?
The 3D grid result is well known, perhaps it would be fair to call it trivial (you can even throw away all but 2 layers of one of your dimensions). As you say, the Menger Sponge is a subset of the 3D grid, so the students had to find a construction which dodges the holes. To me, the result isn't surprising, but it is pleasing. But the really cool part of the article in my mind is the open problem at the end: can you embed every knot in the Sierpiński tetrahedron?
Thanks, I think that helps me understand the 'why' a lot more! That's kind of cool, and I do hope they can keep making progress in that effort!
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. :(
Don't feel bad: it's mostly a Menger Sponge joke
I love that the proof is so elementary and understandable ( almost reminiscent of the Pythagorean theorem proofs) yet it might have some significance
Super cool. I would have liked to have seen a similar visualisation for how they solved it on the Sierpinski gasket.
Interesting, I'm tempted to apply this towards routing minecart rails in Minecraft.
Can you explain?
Like teeray mentions, if we can define any knot as a path through a Menger Sponge then that path could be realized in a Minecraft world (since it is cube/block/voxel based.)
If you placed minecart rails along the knot plot then you could ride it like a roller coaster.
Maybe this was the initial motivation for the teenagers.
Not parent, but I assume the idea is that you could form this fractal by hand (all or in part) in any 3D Minecraft terrain. Then you could route rails according to the theorem.
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.
You could if you had a good mentor.
Fair, but I think the article itself is very clear on the importance of the mentor. It's an fun question that hasn't previously been posed, which itself is quite impressive.
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.
It’s a project funded by the recently passed Jim Simons:
https://en.wikipedia.org/wiki/Quanta_Magazine
https://en.wikipedia.org/wiki/Simons_Foundation
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
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.
That's not true at all.
To understand why, read Euclid.
Geometry has made a bit of progress since Euclid's time. Its become a bit more rigorous.
Euclidean geometry is based on five axioms, and some other terms left undefined.
The fifth postulate - the parallel postulate - was considered so irksome that for hundreds of years, many attempted to prove it using the other four, but failed to do so, and almost drove some crazy. In the late 19th century it was shown you can generate perfectly valid geometries if you assume it to be false somehow - either no-parallel (spherical geometry) or infinite parallel (hyperbolic)
Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it. Like I could draw a "circle with a radius of 1" using taxicab distance, and it would look like a diamond shape.
Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
The (non-generalized) Pythagorean theorem is part of Euclidean geometry, so non-Euclidean geometry is irrelevant to this discusion.
> Euclid's third postulate - "a circle can be drawn with any center and radius - doesn't define how to do it.
You do it using an axiomatic compass, a device that copies length in a circular pattern but does not measure it. Lengths are measured using constructable line segments.
Are you implying that nearly all the hundreds of proofs of Pythagorean theorem, which do not use modern rigorous definitions, are not valid proofs?
> Conversely, if you take the "conventional" definition, than the Pythagorean theorem falls out almost immediately.
So? The Pythagorean theorem is very easy to prove. There are hundreds of proofs created by amateurs. That doesn't make them "not proofs" simply because other proofs exist.
> 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.
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.
Formulating a problem well is half the solution already.
Monkeys with typewriters, or teenagers with MacBooks?
> 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.
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 see why it would be hard. Surprising, maybe, if one never thought about a limit of iterative processes.
> 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.
> 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