An ancient geometry problem falls to new mathematical techniques
quantamagazine.org
quantamagazine.org
The authors proved [1, Thm. 1.3] that given any two sets in R^d with equal non-zero measure and boundaries that are "not too horrible" (i.e. box / Minkowski dimensions of their boundaries less than d), one can cut one of the sets into finitely many Borel pieces and rearrange them (i.e. apply isometries in R^d) to obtain the other set.
You can also guarantee that the pieces have positive measure under a mild technical assumption.
This is the kind of math that I love because when the results get better they get more appealing to the less-math-savvy masses. The decomposition of a square into those pieces will quickly become a puzzle you give to children and they think it's hard but everyone has seen it before.
(On the other hand, it is trivial to cheat with small gaps, etc, and make such a puzzle).
Not that I had anything non-vague in mind, I'm mostly just an interested layman, but surely not anything like that gif! Truely amazing.
Also I can't even start to get my head behind "Yes there are shapes but they are hard to visualize" so how do they even work with them?
Or "We have a gap left, of zero area". This is not a gap in my book, but you are the experts :) Math at its best.
"We can prove it can be done with 10^200 pieces, but it can probably be done with less than 20".
Close enough.
https://en.wikipedia.org/wiki/Graham%27s_number
"Thus, the best known bounds for N* are 13 ≤ N* ≤ N''." Where N'' is ... really hard to type, it's so large!
(λ (λ 2 1 (λ 1 (λ λ 1 2 (λ 1)) (λ λ 5 (2 1)) 3) 1) (λ λ 2 (3 2 1))) (λ λ 2 (2 (2 1)))
https://flownet.com/ron/lambda-calculus.html
Or the movie version: https://www.youtube.com/watch?v=8qC1iZN5ozw
I suppose this could be resolved by some kind of fractal, but that's going to have an infinitely long perimeter.
On the other hand, I think this is a great way for us science-y types to get a good handle on how hard being an NBA player is: NBA players are the moral equivalent of near-Fields-medallists; that's how good they are compared to the rest of us.
I'm no mathematical slouch — I've done grad work in Math, taught myself differential geometry, etc.; but it'd be fruitless to compare me to Terry Tao. There's really no reference for how good he is at math compared to me. I think, analogously, you wouldn't be able to compare a college-level basketball player to, say, Michael Jordan.
It feels strange to see adults that opine on every subject, from nuclear fusion energy, to virology and financial markets, like they know it all, to suddenly "I was never good at math", like a clichee party conversation.
I mean, I get it: It first feels strange and magical, since even the explanations of some of the vocabulary take more time than we are willing to devote to a single thought. But instead of digging in and looking up what "Borel measurable" might mean, the HN crowd rather watches the x-th numberphile video/emotionalized Quanta blurb.
/rant
More to your points:
> Your kid has a greater chance of being a multi-millionaire NBA player than being smart enough to understand this stuff
There are >5000 math phds each year, so no, getting into the NBA is harder.
> Even 18th century math would be a challenge for many math grad students. Just crazy
Not sure, what this is supposed to mean. Certainly as a math grad you should be able to _understand_ 18th century math. Now, to _come up_ with the stuff is something else entirely. But I'm not sure how many engineers would claim they had discovered the telegraph, were they be born instead of Gauss.
People like Erdös were gods in the mathematical universe.
The nature of Mathematics is that the potential depth of understanding and progress is essentially infinite, which frees truly spectacular minds from the constraints they would experience in other fields.
I feel like there are some topics that I'm obsessed with that I'm so much more informed on than most people in my field that I can run circles around them. They would call me super smart if the things I'm obsessive about mattered. Sometimes they have mattered. But I know better than to talk about them at length because people get bored.
But I also agree with the other poster that it's kind of dangerous/distasteful to imply that mathematical ability is something that is not necessary to cultivate, or at least not worthwhile unless you're the next Galois.
A lot of students are already lacking in grit and give up on difficult subjects, not realizing that areas like math require a lot of discipline, struggle, and engagement to cultivate. This hierarchical nonsense about it only being worthwhile for the "chosen few" NBA superstars is not productive, especially with Ameria trailing most developed nations in mathematical and scientific literacy (which has real societal consequences, IMO).
> This hierarchical nonsense about it only being worthwhile for the "chosen few" NBA superstars is not productive
Agreed. It also takes away the dreams of and opportunities from a lot of people.
> Because a previous result had demonstrated that it’s impossible to use a compass and a straightedge to construct a length equal to a transcendental number, it’s also impossible to square a circle that way.
> That might have been the end of the story, but in 1925 Alfred Tarski revived the problem by tweaking the rules. He asked whether one could accomplish the task by chopping a circle into a finite number of pieces that could be moved within a plane and reassembled into a square of equal area — an approach known as equidecomposition
We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
- You can't do it with a compass and a straightedge, like the ancients were trying (1880's?)
- You can't do it with scissors, either (mid-20th century?)
- But with modern mathematics, and really complicated shapes-- you can. (now).
They weren't asking "can you square the circle using only these two tools currently known to us?"
[You wrap the rope around the circle and straight it to get a segment of length 2πR, and take the middle point to get a segment of length πR. Then use the compass to continue it with a segment of length R. And then calculate the square root like in https://www.geogebra.org/m/edtecfcv to get a segment of length sqrt(π)R that is the side of your square. I'm sure this was known in ancient times.]
Using only compass and straightedge is more like a esthetics decision.
The old problem is difficult (impossible) because you have strong restrictions about which points you can draw. You have no rope and no magic rule to get any arbitrary length.
The new problem is difficult because you must cut one figure and rearrange the parts to get the other figure.
They have very different restrictions, in spite both are about a circle and a square with the same area.
Huh. The Banach-Tarski theorem ("you can chop a sphere into a finite number of pieces and, by moving them within 3-space, reassemble them into a sphere of double the radius") strongly suggests this is possible. What's so interesting about the revised question?
I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
As the other responses point out, the Banach-Tarski construction uses properties of 3D space that do not occur in 2D space. But I don't think the "spherical shape" of the original volume or the final volume is relevant; the point of the construction is that the intermediate pieces can't really be said to have a shape.
Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.
The crucial idea that makes Banach-Tarski work in 3D is the insight that the set of rotations around an axis through the origin in 3-space has a free subgroup F on 2 generators (finite strings of A's, B's and their inverses). From this fact the proof is quite easy, but this comment is too small for it.
From what I can tell from a cursory search, there is no surviving fragment concerning squares and circles from Anaxagoras himself, and the mention on Wikipedia goes back to a quote from Plutarch:
> There is no place that can take away the happiness of a man, nor yet his virtue or wisdom. Anaxagoras, indeed, wrote on the squaring of the circle while in prison.
> It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.
Ever since I saw the missing square puzzle https://en.wikipedia.org/wiki/Missing_square_puzzle
I have been very leery of any geometry proof that requires visualization. It is so easy to hide a small difference.
I can kind of see, though, why considerations of what integer ratios are 'good' for such diagrams and questions like angle bisection or intersections between circles and lines become interesting topics. It can really affect how easy or hard it is to draw such a diagram
If you follow a manual of how to construct something with compass and straightedge, the job of the circles is often only to intersect with something else, and these points of intersection are of actual interest (as far as I remember).
As it happens, I sometimes find that the same drawing can be achieved by a simpler construction path that involves (say) only midpoints of squares, which makes life a lot easier.
There is something very "right" about it.
All I really mean is that actually using these tools for an artistic, constructive purpose gives me a feel for why these problems might of been of interest. Of course, without knowing much about the history of mathematics this far back, I cannot be sure.
The fundamental objects in projective geometry are points and lines:
- Given two distinct points, a unique line joins them
- Given two distinct lines, they meet at a unique point
We can't do anything with just a single line/point. With a pair we can find their meet/join, but that's it. With three we can find all the meets and joins (forming a triangle). It's only once we have four objects that things get interesting, and we can start joining points, then meeting those lines, then joining those meets, and so on.
Note that lines don't always meet in Euclidean geometry, since parallel lines never meet. Projective geometry avoids this by including "points at infinity". Modelling that with normal 2D diagrams is quite mysterious, especially since opposite directions approach the same point at infinity. Yet it becomes very simple if we switch to 3D:
- We can draw our points on a hemisphere instead of a plane, with great-circles for lines: distinct great-circles will always meet, and the "points at infinity" are simply those on the equator.
- Instead, we can choose an "origin" sitting above our plane, and connect it to our points (forming lines) and lines (forming planes). In that case, the "points at infinity" are just the lines through the origin which are parallel to the plane (thus never meeting it).
There's lots of fascinating results in this framework, from the Greeks to modern times:
- Here's a nice video with physical 3D models https://www.youtube.com/watch?v=dBH-Id8VC3U
- Here's a lecture on its history https://www.youtube.com/watch?v=NYK0GBQVngs
In fact the latter channel has loads of videos on the subject https://www.youtube.com/results?search_query=insights+into+m...
That is the definition of "constructible". In order to perform (with straightedge and compass) the squaring of the circle, you need to construct a line of length sqrt(pi) in a finite number of steps. However, since sqrt(pi) is a transcendental number, that's impossible.
Draw a circle of diameter 1
Draw a square touching it on all sides, perimeter 4
Cutting at right angles to the existing edges, cut smaller squares out of all the corners so they touch the circle
Perimeter remains 4
Repeat this corner cutting infinity times
Perimeter of the cut square (4) matches the circumference of the circle (pi)
pi = 4
Unlike traditional geometry, it's just abstract symbol manipulation with no relevance to real shapes.
>Perimeter of the cut square (4) matches the circumference of the circle (pi)
Calculus will show that the area of the fractal approaches the area of the circle. But it will not show that the perimeter of the fractal approaches the circumference of the circle. It remains 4 at every step in the iteration, so the limit is still 4.
It also is about a 3D sphere, and the strong form (cutting a sphere in finitely many parts and reassembling those into two equal-sized spheres) doesn’t work in 2D or 1D (in contrast, in 3D, five pieces suffice. I don’t know whether that is a tight bound)
It changes the volume by a discretionary amount; you can create two spheres of the same size as the original sphere, or 500 spheres of the same size as the original sphere, or you can create one sphere of double the radius [= four times the size] of the original sphere.
I see no reason to believe that you couldn't also make one cube of equal volume to the original sphere?
Perhaps they should try making it into two identical circles or a square of twice the area instead?
This is true for almost all curves you can think of and draw and is the basis of calculus. Calculus is the study of functions whose graph locally looks like a straight line.
Inscribe a 30 sided polygon inside a circle of radius 10 cm. Visually you’ll find it hard to see the difference between the polygon and the circle. Using the formula for the area of a triangle you can calculate the area of the inscribed polygon very easily. This provides an approximation to the area of the circle.
Now do this for a 40 sided polygon. Then a 50 sided polygon. A pattern will emerge and one then sees that the limit, which is what happens as the number of sides gets larger and larger without bound, is the familiar formula for the area of a circle. This is how you can prove what the formula for the area of a circle is. You can think of a circle as an infinite sided regular polygon.
And your approach is close enough to calculus that "totally naive..." seems a tad too modest.
The problem with our plan, I guess, is that you'd always be close at every pie slice size, never exactly there. These guys have figured out how to do it exactly, with a finite number of slices, instead of approaching it in the limit of infinity slices.
Imagine that you have this construction drawn as a svg file in the computer, a really big screen, and you can use the zoom. The size of the pieces will match the size of the circle.
For example, consider the number 2 and 2.000000000001. To engineers and scientists, these two numbers are basically the same, and there's no practical difference. To a mathematician, the former is an integer, and the latter isn't - you could add as many zeros to the second number as you want, and the difference won't go away, unless you add an infinite number of zeros, at which point it becomes identical to 2.
Ah. You couldn’t. Hence it’s then 2.