Visual calculus

(en.wikipedia.org)

231 points | by larion1 12 days ago

14 comments

  • zamadatix 11 days ago
    The caution with visual proofs being just because it looks right doesn't mean it is right. A classic example being the "missing square puzzle". More on point examples would be a curve which ever so slightly changes to concave up for a short portion of a shallow concave down region in an overall wavy function, a curve that looks like it converges at a limit but actually doesn't, or something undefined at only 1 point that otherwise looks continuous.

    What this kind of thing is really good at is giving good intuition for understanding a proved concept or thinking about a potential solution to an unproven one. It doesn't actually replace having to then do the math behind it to see if it really makes sense. Even a bog standard classroom calculus textbook will show a visual representation of e.g. Simpson's rule before dumping the actual equations and derivations on you.

    • pavel_lishin 11 days ago
      Some great examples of "lying" with visual proofs here, from 3Blue1Brown: https://www.youtube.com/watch?v=VYQVlVoWoPY
    • samatman 11 days ago
      Note that this doesn't apply to proofs by construction, as in Euclidean geometry.

      Those are visual in nature, but the rules must be rigorously followed, if they are the resulting proofs are reliable. The steps of the proof may also be written out textually, but that's merely a translation, the construction itself is a proof. It's how we can tell the difference between a diagram which looks like a trisected angle, and a construction of a trisected angle, which has been proved impossible in the general case.

      • zamadatix 11 days ago
        It kind of doesn't and kinda does at the same time. You can follow these visual mappings of rules to get something that is visually very convincing but subtly not actually a correct result (e.g. something very closely approximating a construction of a trisected angle). If you want to be sure the construction works you've got to have already mathematically proven how the construction method used is valid, applies in the way you used it, and really did come out with the right answer instead of something very close to the right answer which really isn't any less pure math than just doing it symbolically altogether. Something that reasonably makes sense and seems to come out to the right answer is often a great way to shortcut to finding such a validation though, it's just also sometimes a disappointment when it turns out to be arbitrarily close instead of exact.
        • samatman 11 days ago
          Correctly following the rules of construction results in a valid proof. That proof is in the steps taken, not the final diagram, because there's no way to tell by looking at a diagram if the rules were in fact followed.

          Part of the point I was making is that proof by construction isn't really a "visual proof" at all, it's a proof system where the steps in the proof are added to a diagram, according to exact rules which must be followed at every step.

          • lupire 11 days ago
            How do you know the proof is valid for the theorem you are trying to proof? You have to check the logic of the construction, to know things like which measurements are exactly equal. So the visual part of the construction doesn't suffice. It's the same issue as the infinite chocolate triangles glitch.
    • photochemsyn 11 days ago
      Another example is the use of the parallelogram method to calculate tangents, which works sometimes and fails in other cases.

      "Birth of Calculus" (1986)

      https://youtu.be/ObPg3ki9GOI?t=334

      • jacobolus 9 days ago
        Is there a more complete explanation? They state that this method fails for the quadratrix, but the chosen vectors drawn in the picture in the video seem clearly nonsensical, so it's not clear to me precisely what procedure was being followed.

        Edit: I don't think the video leaves itself enough time to do a good job covering this point. But there's a clearer description of the history at Wolfson (2001) "The Crooked Made Straight: Roberval and Newton on Tangents" AMM 108(3): 206–216 https://jstor.org/stable/2695381

  • JKolios 11 days ago
    "Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation."

    This is the furthest from encyclopedic language you can possibly get. Vague, unsourced, flowery and subjective.

    • mofunnyman 11 days ago
      That's what they don't get paid for.
  • gogurt2 11 days ago
    Fans of this should absolutely check out "Visual Complex Analysis" by Tristan Needham. No other book out there like it.
    • auggierose 11 days ago
      There is now also "Visual Differential Geometry and Forms", also by Needham.
    • jacobolus 11 days ago
      These are both neat, but entirely unrelated.
      • bainsfather 11 days ago
        I've been considering reading this book - can you tell me what I might get from it in addition to what I learned in undergraduate maths/physics courses?

        I found these courses allowed me to do calculations (e.g. contour integrals, conformal transformations) that were useful but always felt like black magic. I don't have an intuitive feel for the subject. I am wondering if this book will help me with that?

    • billfruit 11 days ago
      There was something in that book on the relationship between Taylor series and Fourier series which I haven't seen anywhere else, however I'm unable to recollect the details. Do you know what is the relation between these two structures as described in the book?
  • WillAdams 11 days ago
    A series of books which builds (literally --- the books describe the use of 3D printing and Lego bricks) up to this might be:

    - https://www.goodreads.com/book/show/58059196-make (geometry)

    - https://www.goodreads.com/book/show/123127774-make

    - https://www.goodreads.com/book/show/61739368-calculus

    • lupire 11 days ago
      Wow! Does anyone sell the 3D printed objects as a collection?
      • WillAdams 11 days ago
        Not that I know of, but I'm sure if you asked on some active 3D printing group that someone would do it for you, or you could send the STL files to a site such as Shapeways.
  • JKCalhoun 11 days ago
    Reminds me of the simpler geometric example where you show that the area of a parallelogram is base X height, the same as a rectangle, by making a cut in the parallelogram perpendicular to the base and sliding/joining the severed piece to the other edge to show that you can create a rectangle with the same area.
    • n_plus_1_acc 11 days ago
      That's what de did in fifth grade. I think it's very intuitive because using paper, you can see the area doesn't change.
    • Sharlin 11 days ago
      Also the similar one with a bit more calculus, where you slice a circle (well, a disk to be precise) into an even number n of identical sectors and reorder them to form a parallelogram-ish shape (with "bumpy" bases) that approaches a rectangle with side lengths r and πr as n → ∞. This is an easy way to visualize why the enclosed area of a circle is πr². (You do have to take as a given that the circumference equals 2πr though.)
  • knightoffaith 11 days ago
    I'm reminded of this professor explaining the gradient geometrically and how it leads to really elegant solutions to some problems: https://www.youtube.com/watch?v=uP8V-O8hncI
  • throwoutway 12 days ago
    Unfortunately, all the links/images in a primary source that the wiki uses are broken: http://www.cco.caltech.edu/~mamikon/calculus.html
  • fnordsensei 11 days ago
    A skill in the fantastic game Disco Elysium: https://discoelysium.fandom.com/wiki/Visual_Calculus
  • auggierose 11 days ago
    What I like about the book referenced is that each chapter starts like this:

    > This problem can be easily solved by the methods developed in this chapter. The reader may wish to try solving it before reading the chapter.

    That's a great way to motivate the methods and generate appreciation for them.

  • photon_lines 11 days ago
    I made a little write-up on getting an visual intuition behind calculus / the derivative here, although I'm not sure that it's really that well done and I should have maybe provided a few more examples: https://photonlines.substack.com/p/visualizing-the-derivativ...
  • peter_d_sherman 10 days ago
    >"Mamikon's theorem:

    The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve."

    Intuition tells me that there's something definitely there...

  • Sniffnoy 12 days ago
    Does Mamikon's theorem have any good generalizations, say to higher dimensions?
    • mindcrime 11 days ago
      I have no idea, but your comment does remind me of a joke I heard once:

      "If you want to look smart at a maths conference, just wait for the presenter to say something that everyone reacts to, then raise your hand ask 'Yes, but does it generalize?'"

      And no, there is no intent on my part to suggest that you are doing the equivalent of that here!

      • eru 11 days ago
        That might be especially fun at a conference on 'generalised abstract nonsense', ie category theory.
        • Sharlin 11 days ago
          Shame that nobody can come up with a meta-category theory because category theory is already its own metatheory.
      • bordercases 11 days ago
        Does it scale though?
        • mindcrime 11 days ago
          MongoCalculus™ is Web Scale!
    • jameshart 11 days ago
      This Mathologer video talks about applying a similar approach to some 3D calculus problems - although the tangent swept shapes are arcs not triangles. https://www.youtube.com/watch?v=5q_sfXY-va8&t=0s

      The model of taking the same swept shapes and rearranging them into another shape whose area or volume is easy to calculate is the same approach though.

      • lupire 11 days ago
        I came to post this. Great channel. More in depth than 3B1B but less fancy animations. But he makes his videos in PowerPoint! Not a bullet point in sight, so Edward Tufte would allow it
    • Someone 11 days ago
      There’s “Volumes of Solids Swept Tangentially Around General Surfaces” (https://forumgeom.fau.edu/FG2015volume15/FG201504.pdf) by Tom M. Apostol and Mamikon A. Mnatsakanian (the “Mamikon” of this theorem)
  • ogogmad 11 days ago
    Looks like a special case of an integral substitution, but done visually.
    • lupire 11 days ago
      Yes. Nearly every integration technique is some kind of substitutions.
  • analognoise 12 days ago
    This is too cool thank you