[HN Gopher] An Ancient Geometry Problem Falls to New Mathematica... ___________________________________________________________________ An Ancient Geometry Problem Falls to New Mathematical Techniques Author : theafh Score : 134 points Date : 2022-02-08 15:02 UTC (7 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | whatshisface wrote: | 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. | jnsie wrote: | I agree with you completely. That it was explained in the | article, does not change the article's title/premise. | HotHotLava wrote: | 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. | thaumasiotes wrote: | 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). | mlyle wrote: | 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 | gus_massa wrote: | 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. | dfabulich wrote: | 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?" | gus_massa wrote: | 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 2pR, and take the middle point to | get a segment of length pR. 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(p)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. | mlyle wrote: | 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. | whatshisface wrote: | Then you must agree that the problem that was solved is | not in the literal sense of the word "ancient." | mlyle wrote: | 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). | ruggeri wrote: | 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. | paulpauper wrote: | maybe misleading but still interesting article . Maybe "new | mathematics offers a solution to an Ancient problem" | thaumasiotes wrote: | > 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? | mlyle wrote: | 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. | m00n wrote: | 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. | OscarCunningham wrote: | 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. | ummonk wrote: | 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. | [deleted] | sandebert wrote: | Relevant Numberphile: | | https://youtu.be/CMP9a2J4Bqw | paulpauper wrote: | 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 | m00n wrote: | 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. | jordan_curve wrote: | 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. | inglor_cz wrote: | 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 Erdos were gods in the mathematical universe. | xyzzyz wrote: | 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. | laingc wrote: | 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. | octopoc wrote: | 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. | sapsucker wrote: | 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). | wolverine876 wrote: | 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. | thechao wrote: | 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. | [deleted] | mabbo wrote: | > "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. | SamBam wrote: | 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. | riidom wrote: | 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. | mlyle wrote: | 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). | kurthr wrote: | 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 | arutar wrote: | 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. | RcouF1uZ4gsC wrote: | > 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. | mannykannot wrote: | To be clear, the proof is not dependent on visualization. | mlyle wrote: | (But is more easily visualized than previous proofs). | mc32 wrote: | [totally naive -no maths background] What happens when you make a | circle so large that the boundary in small enough arcs is | practically straight? | toppy wrote: | I think EVERY circle fulfills this condition ;) | thaumasiotes wrote: | A circle of radius 0 may not. | bell-cot wrote: | 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. | yongjik wrote: | 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. | gus_massa wrote: | 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. | syki wrote: | 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. | posterboy wrote: | 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. | whatshisface wrote: | 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. | lscharen wrote: | 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. | cyberbanjo wrote: | In terms of all the choices, it's a pretty small sliver | gilleain wrote: | 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! | tromp wrote: | It's not hard to type in lambda calculus [1]: | | (l (l 2 1 (l 1 (l l 1 2 (l 1)) (l l 5 (2 1)) 3) 1) (l l 2 (3 | 2 1))) (l l 2 (2 (2 1))) | | [1] https://mindsarentmagic.org/2012/11/22/lambda-graham | dmonitor wrote: | it is when your keyboard doesn't have a lambda key | richardfey wrote: | It's just an ALT + 955 away! | phkahler wrote: | 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. | Sharlin wrote: | Yes, the pieces here are highly nontrivial in shape, as | discussed in the article. But more well-behaved than in earlier | results. | jessriedel wrote: | 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. | [deleted] | hw-guy wrote: | 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. | Someone wrote: | 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) | thaumasiotes wrote: | > 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? | OscarCunningham wrote: | It is a tight bound. | | http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm34125.pdf | https://www.irregularwebcomic.net/2339.html | nmilo wrote: | But here the pieces aren't infinitesimal. They're fractals, but | still measurable. | mrob wrote: | 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. | mb7733 wrote: | 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. | contravariant wrote: | The pieces in this particular example seem to be quite a bit | better behaved. In fact they're measurable. | xoxxala wrote: | 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. | arutar wrote: | I'm not sure why this is not mentioned in the article, but there | is nothing special about circles and squares (or 2 dimensions, | for that matter). If anything, phrasing it like this gives the | (misleading) impression that somehow features of squares and | circles are important! | | 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 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 | EGreg wrote: | So we can't claim it's impossible to square a circle now? | Glyptodon wrote: | 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.) | Jtsummers wrote: | 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. | kmote00 wrote: | 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. :( | gilleain wrote: | 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 | whatshisface wrote: | Compasses and straightedges were their attempt to distill the | nature of plane geometry down to its essential and simplest | form (lines and circles). | ogogmad wrote: | 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. | riidom wrote: | 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). | gilleain wrote: | 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. | thaumasiotes wrote: | How do you find the midpoint of a square without drawing | circles? | whatshisface wrote: | Draw the diagonals. | abecedarius wrote: | 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? | whatshisface wrote: | 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. | gilleain wrote: | 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. ___________________________________________________________________ (page generated 2022-02-08 23:01 UTC)