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An ancient geometry problem falls to new mathematical techniques
- lscharen 5y agoI 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 5y agoIn terms of all the choices, it's a pretty small sliver
- gilleain 5y agoA 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 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 5y agoIt'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 https://mindsarentmagic.org/2012/11/22/lambda-graham
- dmonitor 5y agoit is when your keyboard doesn't have a lambda key
- richardfey 5y agoIt's just an ALT + 955 away!
- voldacar 5y agoalt 955 gives me ╗
- richardfey 5y agoThat should be 187: https://theasciicode.com.ar/extended-ascii-code/box-drawing-character-double-line-upper-right-corner-ascii-code-187.html https://theasciicode.com.ar/extended-ascii-code/box-drawing-...
- voldacar 5y agoboth 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
- alanbernstein 5y agoOf 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.
- lisper 5y agoYou might like this: https://flownet.com/ron/lambda-calculus.html https://flownet.com/ron/lambda-calculus.html Or the movie version: https://www.youtube.com/watch?v=8qC1iZN5ozw https://www.youtube.com/watch?v=8qC1iZN5ozw
- speed_spread 5y agoNow I just wanna play Half-Life
- zimpenfish 5y agoThe 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).
- mabbo 5y ago> “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.
- mlyle 5y agoHere, 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).
- SamBam 5y agoI 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 5y agoIt 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.
- mc32 5y ago[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 5y agoI think EVERY circle fulfills this condition ;)
- thaumasiotes 5y agoA circle of radius 0 may not.
- syki 5y agoThe 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 5y agoI 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.
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- hw-guy 5y agoThis 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.
- nmilo 5y agoBut here the pieces aren't infinitesimal. They're fractals, but still measurable.
- mrob 5y agoFractals 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 5y agoThat 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 5y agoThe pieces in this particular example seem to be quite a bit better behaved. In fact they're measurable.
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- gilleain 5y agoWhat'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 5y agoCompasses and straightedges were their attempt to distill the nature of plane geometry down to its essential and simplest form (lines and circles).
- abecedarius 5y agoI 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 5y agoI 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.
- riidom 5y agoA 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).
- whatshisface 5y agoI 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.
- mlyle 5y agoThat'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 5y agoThe 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.
- mlyle 5y agoThe 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 5y agoThen you must agree that the problem that was solved is not in the literal sense of the word "ancient."
- EGreg 5y agoSo we can't claim it's impossible to square a circle now?
- RcouF1uZ4gsC 5y ago> 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 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 5y agoTo be clear, the proof is not dependent on visualization.
- mlyle 5y ago(But is more easily visualized than previous proofs).
- sandebert 5y agoRelevant Numberphile: https://youtu.be/CMP9a2J4Bqw https://youtu.be/CMP9a2J4Bqw
- paulpauper 5y agoThe 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
- thechao 5y agoI 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.
- m00n 5y agoSorry, 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.
- Glyptodon 5y agoIs 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 5y agohttps://en.wikipedia.org/wiki/Constructible_number 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.
- kurthr 5y agoI 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 https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
- arutar 5y agoWould 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.
- phkahler 5y agoWait 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 5y agoYes, the pieces here are highly nontrivial in shape, as discussed in the article. But more well-behaved than in earlier results.
- jessriedel 5y agoYes, 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.
- Sniffnoy 5y agoThe areas aren't non-measurable, that's part of the point. Doing it with non-measurable pieces was already known.
- jessriedel 5y agoThe 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?
- xoxxala 5y agoMy 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.
- kmote00 5y agoFrustrating 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. :(
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- arutar 5y agoI'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 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 https://arxiv.org/pdf/2202.01412.pdf
- throwaway81523 5y agoIt doesn't say anything about the new mathematical techniques.