An ancient geometry problem falls to new mathematical techniques(quantamagazine.org)
quantamagazine.org
An ancient geometry problem falls to new mathematical techniques
https://www.quantamagazine.org/an-ancient-geometry-problem-falls-to-new-mathematical-techniques-20220208/
18 comments
> “I’d bet a beer that you can square the circle, provably, with less than 20 pieces,” he said. “But I wouldn’t bet $1,000.”
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.
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.
Here, the pieces are fractal holy messes-- not something you can make a puzzle out of.
(On the other hand, it is trivial to cheat with small gaps, etc, and make such a puzzle).
(On the other hand, it is trivial to cheat with small gaps, etc, and make such a puzzle).
I think the gif at the top is an approximation of what that result would look like -- cutting the square into just 6 "pieces" to make a circle. But it's absolutely not something that can be cut up and given to children.
It also couldn't be farer away from how I imagined the solution would like.
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.
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.
My High School Geometry teacher caught a bunch of us goofing off and talking in class, so he assigned us the squaring the circle problem (with compass and straightedge) as an extra assignment. Said if we solved it, he would give us an A for the semester. We had no idea and worked really hard on that for a few weeks before he told us.
I get a kick out of these kinds of pure math problems where there's such a large gap between what is provable and what might be the best answer.
"We can prove it can be done with 10^200 pieces, but it can probably be done with less than 20".
Close enough.
"We can prove it can be done with 10^200 pieces, but it can probably be done with less than 20".
Close enough.
A great example of such a range of bounds is the problem that Graham's number was an upper bound for :
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!
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!
It's not hard to type in lambda calculus [1]:
(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)))
[1] https://mindsarentmagic.org/2012/11/22/lambda-graham
(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)))
[1] https://mindsarentmagic.org/2012/11/22/lambda-graham
it is when your keyboard doesn't have a lambda key
It's just an ALT + 955 away!
alt 955 gives me ╗
That should be 187: https://theasciicode.com.ar/extended-ascii-code/box-drawing-...
both 955 and 187 give me that character. I wonder if it has to do with the keyboard I'm using because 955 mod 256 = 187 and I'm using an old keyboard with a usb adapter, so it might be truncating to 8 bits for some reason
Of all the discussion of lambda calculus I've seen, this may be the most interesting bit. I've never really understood the purpose, not how to read it. This might actually encourage me to understand.
You might like this:
https://flownet.com/ron/lambda-calculus.html
Or the movie version: https://www.youtube.com/watch?v=8qC1iZN5ozw
https://flownet.com/ron/lambda-calculus.html
Or the movie version: https://www.youtube.com/watch?v=8qC1iZN5ozw
Now I just wanna play Half-Life
The best thing about Graham's Number, apart from it being unrepresentable in the universe(!) because it's just THAT BIG, is that they know the last few digits (7262464195387).
In terms of all the choices, it's a pretty small sliver
Wait a minute. I thought there was a simple non-existence proof something along the lines of: Given the square, you'll need to cut pieces with the outer arc(s) of the circle. When you cut any length of convex arc, you also create a piece with and equal length of concave arc which will then require that much more convex arc to fill in the end. In other words, the circle requires a certain amount of arc length, and every time to create a piece to fit that you add a requirement for an equal amount of arc somewhere else.
I suppose this could be resolved by some kind of fractal, but that's going to have an infinitely long perimeter.
I suppose this could be resolved by some kind of fractal, but that's going to have an infinitely long perimeter.
Yes, as discussed in the article, the shapes are not piecewise smooth curves (i.e., not the sort of shapes you can construct by making a finite number of straight and smoothed cuts). Furthermore, as also mentioned, the areas of the shapes are non-measurable, so the perimeter is probably non-measurable too.
The areas aren't non-measurable, that's part of the point. Doing it with non-measurable pieces was already known.
The Quanta article says that doing it non-constructively with non-measurable pieces was already known, and that later results did it partially and then fully constructively (without leaving a measure-zero set). But it does not say in the Quanta article whether or not the pieces in these later results were individually measurable. (The union of non-measurable sets can be measurable.) Are you saying that you know they are?
Well, when I wrote that I was just inferring it from the article, which while it doesn't explicitly say that they are, is written in a way so as to clearly suggest that they are. But you're right that it doesn't eplicitly say that, now that I look again. But if you click on the link to the actual paper, it says right in the abstract that they're Borel, so yes, they are indeed measurable.
Thanks!
Yes, the pieces here are highly nontrivial in shape, as discussed in the article. But more well-behaved than in earlier results.
The intellectual level, complexity of research-level math is so great these days . Your kid has a greater chance of being a multi-millionaire NBA player than being smart enough to understand this stuff or do cutting-edge math research, compared to something like history or literature. As a field, modern mathematics is so far ahead of what laypeople can do but also even much of the field itself. It's like, imagine getting a PhD in math, which is a hard thing to do, and then multiply by a factor 100 in difficulty. Even 18th century math would be a challenge for many math grad students. Just crazy
Sorry, but the breathless way, that maths is often discussed on HN, makes me feel uneasy.
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.
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.
I was good enough in maths to get a PhD from Commutative Algebra, but the really good ones were on another level, where you could barely follow their thoughts (especially real-time; anything can be attacked with enough patience, but it was precisely the speed of their train of thought that humiliated you the worst).
People like Erdös were gods in the mathematical universe.
People like Erdös were gods in the mathematical universe.
This is exactly right. I could also get a PhD degree in math myself (I dropped out after obtaining Master during which I obtained novel results in algebraic geometry), but after meeting and interacting with actually smart people, it became clear to me that I’m just not nearly on the same level. Research level mathematics requires completely another level of sheer brainpower that most people don’t even imagine exists.
I don't think you need that super fast brain to be good at math. I'm sure it helps, and I know some people who have it and I felt I could never be like them. But I've also known some really top mathematicians (one Fields medalist in geometry and one of the biggest cheeses in mathematical logic) who weren't like that, and I felt like I could keep up with them, at least on a conversational basis (I don't claim that I would have been much of a researcher if I had stayed with it, much less at their level). Pure brainpower goes a long way, but personality and commitment count for a lot all by themselves.
Adding my voice to this too. I have a PhD in Differential Geometry and would consider myself to have been a decent student and researcher. The "good" people in my field were more than a head and shoulders above me, and the "great" people were somewhere off in the stratosphere.
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.
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.
Is that because the "good" people in your field were just way more obsessive about the topic?
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.
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.
I 100% agree that some people are innately superior at math, the mental arithmetic abilities (at a very young age) of human calculators like Von Neumann are proof enough of that.
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).
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).
I saw a study from long ago, maybe the 1980s, which researched US and Chinese high school education. As I recall, people in the US high schools mostly thought that success in education was due to natural talent, while people in the high schools in China thought it was overwhelmingly due to hard work. The kids in the Chinese schools did much better on the tests.
> 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.
> 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.
The point here is that while it is perfectly reasonable and probably desirable to encourage people to become proficient in mathematics, there is an enormous chasm from there to most modern research mathematics. The sports analogy is that while it is good idea to encourage people to be physically active, for example by playing recreational basketball, it would be completely ridiculous to expect even a strong amateur basketball player to be able to hold his own in the actual NBA.
I understand. I think math research is much more in reach than the NBA, and thus is a bad analogy that discourages people, etc.
If you looked instead at the number of people who obtain tenure at a research university, it would look much more comparable to getting into the NBA.
I think you're equating "mathematician" with "Fields-medallist-adjacent"? As such, I think you'd need to equate such a mathematician to the list of "greatest of all time" in basketball, who are still alive. I suspect those numbers would still tilt in favor of there being more mathematicians than NBA players.
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.
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.
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I almost feel bad making such a trivial point in response to such a nontrivial article, but they're not solving the ancient problem of doing it with a compass and a straightedge (which was proven impossible in the late 1800s), they're solving another problem that shares the similarity that there's a square and a circle.
That's well explained in the article:
> 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
> 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
The problem is that the title is misleading. It's like an article with the title "New world record in the 100 meters dash using a new technique" and in the middle of the article it explains that it's about bungee jumping a 100m fall.
I think it's fair to say that this is an "ancient problem." The ancients only had a compass and straightedge, but they were asking a general question, "can you square the circle?"
They weren't asking "can you square the circle using only these two tools currently known to us?"
They weren't asking "can you square the circle using only these two tools currently known to us?"
The old problem is solvable with a compass, a straightedge and a rope.
[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.
[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.
The squaring a circle with compass and straightedge has been known to be impossible for 140 years.
We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
We're obviously talking about the version of the problem everyone's been working on for roughly the last 100.
Then you must agree that the problem that was solved is not in the literal sense of the word "ancient."
This is just a further result:
- 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).
- 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).
It might be obvious to people who are familiar with the problem. In fact, the title confused (and baited) me because I was familiar with the problem, but not the new formulation.
> 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
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?
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?
Actually, a Banach-Tarski-like result is impossible in 2D space, since there is a Banach measure (= volume definition to all subsets of the plane) that extends the usual volume definition (e.g. for circles).
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.
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.
Banach-Tarski doesn't work in 2-dimensional space; there isn't a finite collection of subsets of the plane which can be assembled to make both one disc of radius one and two discs of radius one.
I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
I believe that Banach-Tarski would make it much easier to disect a sphere and make a cube.
Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle. Presumably, if it were easy, Tarski himself would have shown it, given that he's the one who posed the question.
> Chopping and rearranging something that's the same spherical shape (but different size) is different from chopping a 2D square and rearranging into a circle.
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.
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.
As other people point out, 2-space is not 3-space. Only weakened version of B-T work in 2-space (e.g. infinite number of pieces).
Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.
Tarski, of course, was familiar with his work of the year before formulating the B-T paradox when he posed this question.
maybe misleading but still interesting article . Maybe "new mathematics offers a solution to an Ancient problem"
I think this is overly narrow: Anaxagoras considered the problem of turning a circle into a square of the same area, and these mathematicians shed new light on that problem. He was probably thinking of compass and straightedge because it was the only language he had for attacking the problem, but it's not he published a paper with the precise definition of the terms and the theorem he tried to prove.
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.
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.
https://en.wikipedia.org/wiki/Squaring_the_circle#History
> It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
> It is believed that Oenopides was the first Greek who required a plane solution (that is, using only a compass and straightedge).
I took an abstract algebra course in college. I don't think we talked about squaring the circle, but we did visit the problem of trisecting the angle. What we learned is that compass-straightedge construction can execute a finite number of primitive operations, notably the arithmetic operators +, -, *, and /, along with square root. And that's it. Then we showed that trisecting the angle requires a cube root button, which is what makes it impossible.
The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.
The cool thing was finding a correspondence between geometry and algebra, followed by completing the proof as an almost trivial algebra problem.
I agree with you completely. That it was explained in the article, does not change the article's title/premise.
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> The authors show how a circle can be squared by cutting it into pieces that can be visualized and possibly drawn. It’s a result that builds on a rich history.
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.
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.
To be clear, the proof is not dependent on visualization.
(But is more easily visualized than previous proofs).
What's interesting to me is how many of the ancient problems involve using compass and straightedge. Recently I have been trying to draw Islamic geometric patterns (or other tilings, like quasitilings) using compass and ruler, and it can be really difficult!
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
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
Compasses and straightedges were their attempt to distill the nature of plane geometry down to its essential and simplest form (lines and circles).
A note to the circles: It is not so much about circles, but rather "Some points which share the same distance to another point X"
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).
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).
Exactly this. There are many patterns that can be constructed by drawing a regular array of circles, then connecting various intersection points with lines, then erasing the original circles.
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.
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.
How do you find the midpoint of a square without drawing circles?
Draw the diagonals.
That will work, but you need to have the entire square already marked. (As opposed to e.g. one or two of the sides marked and representing a hypothetical square.)
It can even be reduced to Clifford algebra very cleanly: https://en.wikipedia.org/wiki/Conformal_geometric_algebra
There is something very "right" about it.
There is something very "right" about it.
I wonder, though, is it not a coincidence that they're practical tools to do rather precise multi-step constructions? (E.g. Durer wrote a whole book about type design by those methods.) And with all the geometric algebra in Euclid, did they ever use them for calculations that aren't originally geometrical? Would we know?
I believe that ancient geometry was used for governance and engineering. However the compass and straightedge had an element of abstractness or deliberate simplified impracticality even back then: they had rulers and strings, and could have practically used them as well.
Agreed, I am sure that the drive for axiomisation of geometry drove a lot of this interest.
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.
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.
I've enjoyed playing around with projective geometry recently: that's even more basic, since it doesn't involve a compass, only a straightedge!
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...
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...
Is the mentioned proof about not being able to create a transcendental number length segment with a compass and straight edge, I think I'm missing the boundaries of how constructions are permitted within the proof. Is this effectively because you need more than 2 linkages to translate curved motion to linear motion? (As I'd assume any device that converts circular motion to linear motion would produce linear motion in ratios of pi.)
https://en.wikipedia.org/wiki/Constructible_number
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.
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.
This is similar to the claim that an orange, say, can be cut into pieces that can then be put together to make two oranges. It turns out some of these pieces would be infinitesimal, and hence smaller than the atoms making up the orange (or whatever). While such a result may be satisfying to a theoretical mathematician, the engineer in me recoils.
Mathematically, it’s quite different. The Banach-Tarski paradox (https://en.wikipedia.org/wiki/Banach–Tarski_paradox) changes the _volume_ of the objects. That’s requires some of the prices to be immeasurable.
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 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)
> The Banach-Tarski paradox (https://en.wikipedia.org/wiki/Banach–Tarski_paradox) changes the _volume_ of the objects. That’s requires some of the prices to be immeasurable.
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?
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?
The pieces in this particular example seem to be quite a bit better behaved. In fact they're measurable.
But here the pieces aren't infinitesimal. They're fractals, but still measurable.
Fractals are still cheating IMO, because fractals have infinitely small features. Whenever you use infinity you can get all kinds of crazy results. It's like the geometric proof that pi=4:
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.
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.
That proof is just plain incorrect, though. It will break down when trying to prove this statement:
>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.
>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.
I wonder if a square of the same area is too constraining for the axiom of choice?
Perhaps they should try making it into two identical circles or a square of twice the area instead?
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Perhaps they should try making it into two identical circles or a square of twice the area instead?
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Would you be able to elaborate a bit what you mean here? There is no version of Banach-Tarski in two dimensions - you can prove that there exists a finitely additive set function which is invariant under isometries.
[totally naive -no maths background] What happens when you make a circle so large that the boundary in small enough arcs is practically straight?
The radius of the circle doesn’t come into play. This is because instead of enlarging the circle, as you say, what one does is narrow in on smaller and smaller sections of the circle. Given any circle one can narrow in on a small enough piece that, essentially, when looking at it, it will be almost straight.
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.
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.
I think the point of the comment was precisely that I cannot simply let n be infinite. In fact, there's an old joke where that's the punchline.
The construction uses the radius of the circle as a parameter. If the radius is bigger, then each piece is bigger.
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.
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 this sort of math (theoretical), "practically straight" is never straight enough.
And your approach is close enough to calculus that "totally naive..." seems a tad too modest.
And your approach is close enough to calculus that "totally naive..." seems a tad too modest.
I think you could get close by making a lot of pie slices, lining them up in a row, and then flipping every other one upside-down. Then you'd be able to put them back together into a rectangle.
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.
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.
Math isn't concerned with "practically straight", so it won't make a difference.
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.
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.
But if you add an infinite amount of zeros to it how could you have the one on the end?
Ah. You couldn’t. Hence it’s then 2.
Ah. You couldn’t. Hence it’s then 2.
Relevant Numberphile:
https://youtu.be/CMP9a2J4Bqw
https://youtu.be/CMP9a2J4Bqw
Frustrating article. Suggests that somebody has come up with a way to cut jigsaw pieces differently to arrange a rectangular puzzle into a perfect circle. The solution is so simple that the shape of the pieces can actually be described and visualized. And then, after all this tantalizing buildup, the big reveal is... hidden behind a paywall. :(
It doesn't say anything about the new mathematical techniques.
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So we can't claim it's impossible to square a circle now?
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.
[1] https://arxiv.org/pdf/2202.01412.pdf