A video on what proofs in mathematics are for, using Pick's theorem as an example. PBS Infinite Series's video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bYW1zOMCQno.html
Dude, I independently discovered Pick's Theorem while in elementary school. We had to find the area of shapes on a grid and I distinctly remember discovering this trick and being very exited about it. I didn't even know algebra yet, so I didn't write up a formula or think it would be special to anyone else though. What's even more amazing to me though is that I discovered this by randomly stumbling on the "wrong" proof. I didn't think about it with water, but rather realised that, after I had counted up the squares entirely contained within a shape, the squares which had been cut in two typically had another square right next to them which was inside the total shape enough for both squares to count for 1 whole square inside the entire shape. It took a little fiddling to get from there to caring about points. Your video has turned this from a cool memory into one of my fondest. Thank you.
@@Ashiya-Ichiro I am aware of homology classes, and homologies (1to1 onto functions between sets), but have never heard the term "homology trap" hmmmm So were going from 0 dimensional points to 2 dimensional areas. Simplified version, let's put the points in a row and find length. Without assuming a metric, we got nothing so length is simply sum over all distances between consecutive points as defined by the metric in the 1d space. So spread them out in 2d and its just the sum of the areas of "consecutive (i.e touching, non overlapping) triangles with lattice-point vertices" So the space of coordinates of 0d lattice points in {Z}^k, the space of lengths in {Z}, and the space of areas in {Z}^2, probably volumes on {Z}^3 and so on are all isoomorphically equivalent, or of the same homology class. Makes sense, the video is interested in a specific homology for euclidean spaces, but the idea applies to any sparse metric space. I fail to see the "trap"
My favorite proof is even more wrong. It's so wrong that it stuck and helped me to remember the surface area formula of a sphere. "The surface of a sphere is quite easy to derive. (draws a sphere). We are enclosing the sphere with a cube (draws a cube) and project the sphere on the cube's surfaces (draws a circle on one surface). This circles' area is r²π. Because the sum of all projected circles equals the surface of a sphere and because a cube has 4 surfaces (draws circles on the 3 visible surfaces and one on a hidden surface), we get the surface of a sphere equal to 4r²π." "But a cube has 6 sides" "Yes but then the area formula would be wrong."
@@spacematt5418 Congrats for being featured in the final 3b1b video! It's fun to see my favorite Math RU-vidrs encouraging each other, nice graphics btw :D
@@spacematt5418 You are very smart and very good idea. I respect your attitude to prove this theory and mathematics. But I think this proof is not correct and wrong. You explain more details about this proof.It is seems like a correct but you only indexed point and simple example. If I knowledge were not wrong, homology group is singular chain and in free abelian groups , average model Cn(x) and singer n-chain An:Cn→Cn-1 Long story make a short, it is seems like can explain homology . For example, group of singer-n cycles and group of singer-n boundaries and relative homology. I’m not good at homology theory, I only understand basic homology theory(local homology and Con CX and so on.) I think it’s is possible to explain in homology theories or algebraic topology.
Hi there, This proof is not "wrong!" at ALL. It is already published in 2018 ^^, for more details take a look at my comment on this video, and enjoy :)
Don't get me wrong, this is amazing! But I think that his proof is not necessarily unrigorous to the core. It can be formalized with a bit of effort(maybe instead of considering units of water you can consider circles!). "Unrigorous ideas" are beautiful and they are the moving force of mathematics, but they are not "wizardry": if they are indeed correct, they can be formalized. And when you don't manage to formalize your idea, that may be an indicator that it's wrong: intuitive reasoning is risky! That's why rigour is important in mathematics: it stops the mathematician from flying too close to the sun. I wrote this comment because often there is the misuderstanding: -rigour=boring -ideas=amazing Formalizing an intuitive idea is often a beautiful challenge that may help you understand even more about that idea. :)
I agree! I don't want to put down the "right" proof with this video. I think both proofs are valuable and complement each other, and hey, we can have both! While it may be possible to make the "wrong" proof rigorous I think it'd be really hard and haven't tried. Even making rigorous the notion of area is very hard (you can sidestep doing this for the "right" proof).
@@spacematt5418 I think the question underneath is: how different is the rigorous proof from this one? It shouldn't be too different, should it? Rigor doesn't push into a whole different direction, rather it forces us to check upon the validity of our intuitive, explanatory proof. If the formalization of hitherto unformalized concepts leads to new math, then we win twice.
@@knotwilg3596 There is no one single rigorous proof. The usual way to prove Pick's theorem uses very different ideas than the "wrong" proof presented here. This "wrong" proof can be made rigorous using multivariate calculus which is not something many people outside college knows. There is also one proof involving tropical geometry which I've seen in some seminar (don't remember the details, unfortunately) and I bet there are many other proofs as well, each with very different perspectives.
@@spacematt5418 You are very smart and have a good idea watching this movie. But I think this proof is look like using looping theory quickly and easily.
Great video! You articulated an idea I've had which I've been calling "textbook" vs. "classroom" proofs. The dry, often unintuitive proofs aren't really suited for the classroom lecture format; yes, they follow logically from one step to the next but they take time to decipher and they're meant to answer "how is this true" more than "why is this true". The 'wrong' proofs are not only way more helpful, they capture better what it means to do math.
Interesting distinction and good point about how the context of a classroom (or, in my case, a tutoring session) can make some kinds of proof/presentations better or worse suited towards that format. I think I've definitely made the mistake of trying to fit a 'textbook' proof into a tutoring session in the past and didn't understand why it wasn't as engaging for the student as I found it was for me when I learned it. Your insight gave me an "Aha!" moment, so thanks!
Hi there, by the way This proof is not "wrong!" at ALL. It is already published in 2018 ^^, for more details take a look at my comment on this video, and enjoy :)
I thought a similar idea where B/2 represents how one side of the boundary contributes to the inside (and the other, outside) . The "-1" would be necessary to account for how the boundary line loops in on itself, which is like one full rotation when tracing the perimeter.
Brilliantly put. This is the way I think about the theorem, and one which makes it seem obvious (the best kind). This proof was actually in the script, though not so well phrased, and I recorded audio but ran out of time for the graphics so it got cut.
@@spacematt5418 Hey, you can always make another video (or as many as you like!) to flesh out the topic, if you'd like to. No pressure, just saying it'd be fine and welcome if you did. Don't have to get everything 'perfect' (I no longer believe in 'perfection', even though I've been and still (unfortunately) remain a life-long perfectionist) into one single video. Anyway, just a thought! Cheers! 😊
As I'm watching this, the video reportedly has 102 views, yet 225 likes. Congratulations on being one of the winners of SoME! It's quite well deserved.
This technology is apparent in origami. I have been folding paper for about 9 years now. We use a method called " circle packing" to help make crease patterns for complex models. You first figure out how many flaps the base of the origami will have, and then you translate that information into a bunch of circles on square piece of paper. Fascinating art. Thank you for the video!
I used to dislike this formula because it was taught to us as some sort of trick for exam problems without explanation or proof. Now, having an intuition of how it works, I will certainly use it and share my knowlegde of such an amazing proof. Thanks to you :)
Hi Ким, this proof was originally discovered by Russian mathematician G. Morzon and it's not Wrong! at ALL , please take a look : + Video Titel : ПЛОЩАДЬ ФИГУРЫ НА КЛЕТЧАТОЙ БУМАГЕ - ГРИГОРИЙ МЕРЗОН
I was going to vote for the previous video since the technical aspects were better than your video. But after watching your video, I see a very important message that resonates with me and i think this video deserves to reach more people.
In maths, there are two kinds of valid proofs: One is rigorous, self contained, and covers every argument that could support or shut down the idea. The other is something I'm able to explain to my neighbor's kid across the street. Both are fine for their different situations. In all honesty, I'd say trying to find both kind of proofs for the many ideas in maths would probably make the subject more fun while also being serious when the time is right.
Hi @Steven ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
really enjoyed the format of this video. Succinct enough to be never boring yet never confusing, and with enough detail "left as an exercise" to give me something to think about. :D
There is a happy-medium proof that starts by proving that "small" triangles (triangles with no interior points or boundary points other than the vertices) have area 1/2. That's the dry, boring part. Armed with that, take any polygon with vertices on the lattice, triangulate it into small triangles and now count the triangles by summing up the angles and dividing by 180 degrees. EDIT: I just watched the PBS video and it's quite terrible. It doesn't do the dirty part of proving small triangles have area 1/2, and then it counts the number of triangle with a bunch of algebraic manipulations that kill all intuition. You just need to count the total sum of interior angles of the triangles in the triangulation: Each interior point contributes 360 degrees; each point on the boundary contributes 180 degrees, except vertices, which contribute about 180 degrees, but in the aggregate contribute 360 degrees less, as you go around the polygon. Divide that by 180 degrees to get the number of triangles and divide again by 2 to get the area. Much nicer.
Great video and a fascinating proof I've never seen before. Also I really miss PBS Infinite Series. There are some really spectacular maths channels on RU-vid but Infinite Series was one of the best for the short time it lasted. At least we still have Space Time.
If an argument is well known despite it being wrong, then it's surely a really interesting one. I'd love to see a compilation of wrong proofs. Also only 3k views yet? I assume that'll change soon.
I think the 'wrong' is (in the video) and should be (in general) in 'quotation marks', because while it is not as formally rigorous as some other proof(s), it's not actually wrong logically speaking. It might take a bit of work to make it formally logical (like 'premises, deductions, conclusion' style), but I'm sure it could be done. Informally, it is a logically valid and sound argument (though requires human spatial/visual intuition to confirm some premises), IMHO.
Hi @Pedro ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
@@robharwood3538 Hi there ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Hi there,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
There's a version of the 'wrong' proof where instead of water, you use miniature squares which expand to unit squares. The difference in my mind is that you don't need to think about the physics of water, and it appears a bit easier to formalize, though maybe it takes away some of the amazement.
Hi there,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
Hi @Flobbled ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Damn, I know this is unrelated to your video but Infinite Series was so good. I remember I would set aside a few hours to work on the problems from the videos every time they came out :(
Hi @Lotschi ,by the way this is not wrong! at ALL, this proof was published | 2018 and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
I'm not sure I understood. The formula for the sum of the interior angles of a polygon only takes in consideration angles referred to vertexes. What about points on the sides of the polygon?
130 views and 293 likes... Well I'm glad to be one of the 130 to actually appreciate the great explanation of the so called "wrong" but amazing proof :D. And also, how did you manage to draw the lines in the beginning so perfectly in one go xD
Hi @Maxim ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Hi @Franklin ,you are right but also this Water analogy proof exist and by the way it is not wrong! at ALL, this proof was published | 2018 and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
6 месяцев назад
i studied surveying, i don't recall not once having this theorem in class
okay i started smiling when i used pick's theorem with the example and got the area, out of the beauty and sheer awe of it, but i heard at 2:46 "1 unit of water" and my mind went boom, i got it, and i loved it
Hi,may be it's a good work but unfortunately there are lot of issues to point out here : - The water or melting - ice analogy proof is not wrong at ALL!, why? Simply continue reading ^^ and enjoy. - The original work proving this formula using physical analogy is done by the german mathematician Christian Blatter - 1997 - [ Mathematics Magazine 70(3),200 ] using Thermal-diffusion analogy proof. - In 2018, the article - in Russian - : "Pick's Formula and Melting - Ice" by the russian mathematician Grigory. A. Merzon in the "Kvant" magazine (9), 36-37, 2018. Where he proved Pick's formula using this Melting - Ice analogy which is a variant of the Blatter's former proof, the so-called " wrong proof!! " in this video. - In addition to the published article, G. A. Merzon had done a beautiful video - in Russian - explaining graphically his charming! proof, without forgetting to point out the originality of the german C. Blatter's proof. + Video Title : ПЛОЩАДЬ ФИГУРЫ НА КЛЕТЧАТОЙ БУМАГЕ - ГРИГОРИЙ МЕРЗОН * Link : ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-z7tZf8NpSuQ.html N.B : If someone would like to translate this video into English using subtitles, it would be so great.
Hi @Aviv ,by the way this is not wrong! at ALL, this proof was published | 2018 and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Hi @Dan ,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
Does the 'wrong' proof give a good guideline about what arrangements of points work, that is, any arrangements of points where a line between two of them results in the remaining points being rotationally symmetric about the midpoint will have Pick's formula also work?
Hi @sizur ,by the way this is not wrong! at ALL, this proof was published | 2018 and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Way to make it rigorous: consider that quantity, of the amount of the water that's inside the polygon early on, which you showed equals I+B/2−1. That water idea can be easily used to show that it's _additive,_ meaning if you break a polygon into two pieces its value is the sum of the values of the pieces. Once you know it's additive, it falls into place like lego bricks. The unit square, by computation, has value 1, so its value is its area. Adding the square to itself tells us rectangles have the right area. Slicing the rectangles in half gives us that the value for right triangles is their area. Slightly more trickily, any non-right triangle is a rectangle minus some right triangles, so we get triangles as well. And any polygon can be broken into triangles.
Hi there ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Beautiful, but what if the symmetrical is also inside the polygon (in a non convex case) then the flows doesn t cancel out (I guess dividing the polygon into small convex peace works and you can sum the area)
There is actually a third, also "wrong" proof, described in V. I. Arnold's book "Chain fractions". It is based on an infinite cover of the plane with parallelograms. Unfortunately, i don't know whether or not the book was translated in English. If you are interested, I can send you the detailed proof in English or a link to the book in Russian.
Hi there,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
I'd guess it could be done but the result would be so complicated as to be unrecognisable. One of the difficulties in making the "wrong" proof rigorous is that you have to define what area is, which is really hard (in the "right" proof, you can sidestep it and use a kind of special definition of area that gets the job done). Let alone making rigorous the step where the water flows...
@@spacematt5418 Do you think that the existence of the short, elegant, "wrong" proofs are a sign that the foundational definitions and propositions used in the "right" proof could be modified to simplify the "right" proof?
It feels a bit like something one should be able to do with measure theory? Don't put all the mass (water) at a point mass on the edge, spread it out some small circle. Create a continuous flow over time that takes this measure to the uniform measure and preserves the total integral. Check the flow (some sort of integral) across the boundary (symmetry still applies), check the fraction of the initial circles that started inside the shape. I guess it's no longer accessible, like this proof. But that's just a difference in what tools are taught when.
Actually, I wonder if some version of the following would work: There's actually a very important idea--classically related to solving some differential equations--where you evolve a measure over time and let it "diffuse". (If you don't know what a measure is, just think of it as a distribution of the density of your water). At time zero, the measure is all concentrated at one point, but then it spreads out. In the most classic example (associated with the heat equation), the measure goes from concentrated at one point (which we can think of as an infinitely steep bell curve) to a bell curve whose width keeps widening and widening as time increases, but it always has total mass 1. It also happens that the amount of mass in a given area at any given time is precisely the probability that a Brownian motion started from the starting point lands in that area at that time (if you've heard of Brownian motion---if you haven't, it's a sort of the canonical "random path" in a plane starting at a given point, like an infinitesimal version of a random walk; but also that doesn't necessarily matter for what I'm saying, it's just a cool connection). Anyway, I like the idea of using this diffusion because it does capture this idea of starting in one place and then spreading out to become uniform, and it also has a nice radial symmetry that I think lends itself well to the proof ideas put forth in the video. I also suspect that in this framework, you could probably actually without too much difficulty make rigorous the notion of flux of the mass through the boundary and the argument that symmetry makes fluxes from different points cancel out. There are a few difficulties: one, here, the mass is very explicitly spreading out infinitely far, and you can see from this perspective that you're dealing with mass coming from everywhere flowing in and flowing out. I think it might not be too bad to make sure that the symmetries mean that all the fluxes cancel, but one also has to be a little careful since we're adding up contributions from all the infinitely many particles on the plane. Secondly, you do have to do a bit of work to see that this diffusion process limits to the uniform measure (again, from the random walk perspective, random walks "equidistribute" which is cool and intuitive! But requires work, or appealing to others' work). Lastly, it's a bit tricky trying to make the (key) part of the argument where you look at how much mass is "inside" at time zero rigorous. In fact, I don't think you could really look at time 0 (in most ways you could try to formalize this, I think that the measure of the polygon does in fact instantaneously change at 0). However! I think that you can actually get around this by looking *very close* to time 0. So again, you're doing some (unfortunately rather involved) analysis, but you should be able to argue that, for times *very close* to the beginning, the area enclosed in the polygon is *very close* to just being what you argued in the video: it's well approximated by the tiny circles around the dots inside or on the boundary of the polygon, and then we make the argument based on the angles you made in the video. Of course, the other issue with this is that it definitely goes far out of the "self-contained" area, so if you're teaching this to high schoolers or even undergraduates, the proof with this level of detail is not really acceptable. However!! I do think that an argument along these lines might be totally rigorous, and I would say that if it works, it's exactly in the same spirit as the "wrong" proof presented. I might actually try to think about whether this could be made totally rigorous, because I think it would be pretty cool if it works. I do actually really love (in perhaps a perverse way) this sort of argument; the quantity we want is the limit as the time goes to infinity; we calculate this quantity by computing the limit as time goes to 0, and then noting that the quantity in question doesn't change over time!
There is a form of formal logic called 'natural deduction' (I'll post a link in a second reply in case RU-vid blocks links), in which you can state logical proofs in a, well, more 'natural' way than in some other formal (typically heavily axiomatic) systems. They tend to read more like natural language, and they are based largely on propositional logic (less so than on predicate logic), and so you can probably make a 'wrong' proof such as this one into a more rigorous one by simply bundling up some of the trickier details into a small set of 'premises', such that: If the person will accept the premises as true (for example, that "Water can fill the grid space evenly in all directions" or something to that effect), then the following deductions (again, using the 'natural deduction' system) will lead to the desired conclusion. By using a propositional logic system in this way, and by judicious choice of required premises, you can turn even a typically mathematical argument into a valid propositional one. It won't be as general as the full mathematical argument, but we're not looking for full generality here, just a 'more rigorous' presentation of a 'wrong' proof. And propositional logic is certainly more rigorous than 'argument by pretty pictures and narration'. Link on Natural Deduction to follow in reply. [If it doesn't show up, YT blocked it. Just in case, here's a de-linkified version of it that you'll have to re-linkify yourself or use to do a web search: en wikipedia org / wiki / Natural_deduction]
Do you think this might be a difference between pure and applied maths? In pure, we need rigour to derive other pure results. In applied maths, we need intuition before we can apply it to real-world problems. Nice video
I like the idea and I also wonder whether that proof could be made formal and therefore correct. Obviously you haven't made it formal already but the movement of the water is just a function which you could specify more rigorously. I think this could be made formal. Of course that would probably ruin its simplicity, but the idea that these proofs can be improved makes me feel that they're not really "wrong".
Hi @Ishmael ,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
Hi @Amaar ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
Hi @Cisco ,by the way this is not wrong! at ALL, this proof was published and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
@@ciscoortega9789 Well, indirectly ^^ since the title is ".. wrong and amazing proof.", sort of deduction :), anyway when you find my longer comment you will be hopefully more amazed by this piece of wizardry proof and discovering the real Wizards behind it ^^. Enjoy
Hi @Isak ,Yes Elegant and Correct. by the way this is not wrong! at ALL, this proof was published | 2018 and explained beautifully in a Russian video, for more details take a look at my longer comment on this video, and enjoy :)
You looked as if you were speaking in a lower voice around the corner (by the side of some stairs) trying not to be detected by or found by someone who is searching for you.
Hey Matt, great video - I actually did the same thing, so go check it out. It's not my SoME entry, but you'll find it. And it uses the same "wrong" proof you used, though I don't use the water analogy, but rather start with the premise that the angle divided by 360 is a valid way to measure area of a polygon. Also check out my ACTUAL SoME entry - "Cauchy's Ingenious Solution to the Basel problem." It's a proof, so I think you'll like it! PS - you got my vote!
@@spacematt5418 thx and congrats on top 5. I am kicking myself for not doing Pick Formula - I literally had the same video but decided on Cauchy at the last minute. Go check out my Pick Formula video and lmk what you think! It's got polygon in the title.
Hi there,by the way This proof is not "wrong!" at ALL. It was published in 2018 ^^ in a math magazine , for more details take a look at my comment on this video, and enjoy :)
I like informal proofs, but I didn't like that there was no pause at the step at 2:48-2:58, no comment about it. The rest of the informal proof is essentially the same as many of the formal proofs, for example the proof via integrating Weierstrass P-function, or the many proofs that use subdivisions. All the technicality got hidden in the mysterious "spreading of the unit of water" that took 10 seconds. It is in this step that resides probably the most important facet of the topic, and that is how area connects to sums along arcs (with Green's or Stoke's theorem lurking in there), how it connects to certain sums that are just zero except at encircling certain special points which ultimately allowed to turn the computation into a sum of angles. Note how all the various proofs of Pick's theorem differ precisely in that step. Subdivisions or integrals, are essentially making sense of that spreading of water (or rather the reverse collecting the water at the lattice points). Funnily, at 3:10 it comes the "Now comes the critical step". Well that the flow of water on each side of a segment cancels is ultimately a consequence that integrals change sign when you change the orientation of the manifold that one integrates over. It is true that this is important, every step of a proof is, but compared to the Stoke's theorem phenomenon it is not what is doing the heavy lifting in Pick's theorem. The critical step just happened seconds before and it was not highlighted.
A fairly close analogy, in \sum_{k=n}^{m} 1/(k(k+1)) = 1/n - 1/(m+1) it is more crucial that 1/(k(k+1))=1/k - 1/(k+1) and then the sum telescopes (the water flows to the end points) than the fact that |-a|=|a|.
Sorry mate, just a technicality...at 1:22, why do you count it as 11/2 ? It still becomes 5.5, but that would be unacceptable here in Europe. We first multiple and divide and then do the addition.
The trick is that in this case, a polygon with an "180 degree angle vertex" is still valid for the formula, even though they are typically not considered vertices. So any point on the sides of the polygon could be arbitrarily decided to be a 180 degree vertex, and the equation will still work. Much the same as the non-vertex boundary points
