[HN Gopher] Proofs Without Words
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Proofs Without Words
Author : squircle
Score : 95 points
Date : 2025-06-14 11:31 UTC (4 days ago)
(HTM) web link (artofproblemsolving.com)
(TXT) w3m dump (artofproblemsolving.com)
| fiforpg wrote:
| Nice. There's an entire book like this for geometric statements.
| Every picture is a fact, proofs are supplied by the reader:
|
| https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf
|
| Caution: proofs of some of the statements in it are difficult.
| JadeNB wrote:
| I think that is rather different. The traditional meaning of
| "proofs without words" is that the picture _is_ the proof, or
| at least, if you believe that a proof can only be in words,
| that the picture should convey the idea so transparently that
| anyone with reasonable mathematical skill can routinely
| translate it into words.
| fiforpg wrote:
| You are correct, after posting I realized the difference. The
| book is rather "theorems [formulated] without words".
|
| Which is why I added that the proofs are left to the reader
| :P
| brianberns wrote:
| I made a game out of creating proofs without words:
| https://brianberns.github.io/Tactix/
| vonnik wrote:
| Anyone who enjoys this should read David Bessis's Mathematica.
| bwfan123 wrote:
| Thanks for the mention. I loved the book [1], and it started me
| off on a journey to spark intuition, and sensory (visual)
| connection.
|
| On another note, I was shocked to find that some members of my
| family have aphantasia which is a complete inability to
| visually imagine geometric figures or pictures, and yet, they
| were good at math. So, there are faculties beyond visual
| imagination which are invoked, and even within visual
| imagination, there is a spectrum among people as to its
| strength, and quality.
|
| [1] https://www.amazon.com/Mathematica-Secret-World-Intuition-
| Cu...
| vismit2000 wrote:
| There is also this youtube channel called 'Mathematical Visual
| Proofs' on similar theme:
| https://www.youtube.com/@MathVisualProofs
| downboots wrote:
| Another great site is https://theoremoftheday.org/ with a neat
| one-pager overview of each theorem
| jupitr wrote:
| and also how to lie with visual proofs:
| https://www.youtube.com/watch?v=VYQVlVoWoPY
| ViscountPenguin wrote:
| I've never really been a fan of proofs without words; they've
| always felt way too slippery to me, for lack of a better term. A
| well worded proof with nice explanatory diagrams hits the spot
| for me instead.
| paufernandez wrote:
| I'm the opposite. I am not convinced until I "see it". Probably
| has to do with our innate talents.
| seanhunter wrote:
| The problem is not whether you (or anybody) can be convinced
| by seeing something that is true. Mathematics study involves
| a lot of drawing curves etc so you can develop
| geometric/visual intuition about things, and of course that
| is a good idea.
|
| The problem is that it is far too easy to convince someone of
| something which is not true via visual means.
| perlgeek wrote:
| For me, the visual proofs of simple sums (like The sum of the
| first n odd natural numbers is n2) works pretty well for me.
|
| For the more geometry-based ones where you have move triangles
| around and so, it's often not obvious to me that two angles
| that look the same really always are the same, and that things
| that add up to rectangle do so reliably, independently of the
| actual angles used in the examples.
|
| I guess in these cases, a more parameterized, interactive
| version would work better, where you can use sliders to adjust
| some of the angles and lengths used. That should make it much
| more obvious that it's not just an artifact of particular
| angles used in an example.
| seanhunter wrote:
| Feel the same way. It's way too close to the infamous proof by
| "just look at it". Our visual intuition is way too easy to
| trick especially in three dimensions, and our intuition for any
| dimension higher than that is basically zero.
| LegionMammal978 wrote:
| I've found Dudley's _A Budget of Trisections_ pretty fun in
| this regard: the book is filled with depictions of various
| people 's attempts to trisect an angle with straightedge and
| compass. It turns out that a few steps can get you within
| arcseconds of the correct result, easily enough to fool the
| eye.
| wat10000 wrote:
| Yeah, I don't get how you distinguish between a correct visual
| proof and a visual proof that looks right but doesn't actually
| prove what it's trying to prove. You could probably make a
| pretty convincing-looking visual proof that the limit of the
| sum of the harmonic series is below some finite number, that
| 0.9 repeating is less than 1, that there are more rationals
| than integers, that there are the same number of reals and
| rationals, and that sort of thing.
|
| On the linked page, a lot of the proofs are essentially proofs
| by induction that stop at some (pretty small) n. Maybe there's
| a way to make it rigorous by visually showing the induction
| step that proves n+1 given n, but if there is, it's not shown.
|
| This can be great for building intuition for a statement known
| to be true by other means, but I wouldn't consider them to be
| proofs.
| monktastic1 wrote:
| > Yeah, I don't get how you distinguish between a correct
| visual proof and a visual proof that looks right but doesn't
| actually prove what it's trying to prove.
|
| This problem exists not only for visual proofs, but for
| standard written ones too.
| wat10000 wrote:
| Not in the same way. For a written proof it can be _hard_
| but with effort and sufficient background knowledge you can
| figure out if it actually proves the statement or not. If
| the proof doesn't prove the statement there will be a step
| that doesn't follow from the rest of it. You may not be
| able to spot it but it can at least theoretically be
| spotted.
| cuber_messenger wrote:
| There's a book called "Proofs without words". Fun to have a
| glance. (https://ia801405.us.archive.org/24/items/proofs-without-
| word...) It also has a sequel.
| stared wrote:
| See also O. Byrne, "The First Six Books of the Elements of
| Euclid, in Which Coloured Diagrams and Symbols are Used Instead
| of Letters for the Greater Ease of Learners",
| https://www.c82.net/euclid/ (reproduction in CSS by Nicholas
| Rougeux)
| Someone wrote:
| Here's a proof with just a few words that got published in a
| serious math journal: https://fermatslibrary.com/s/shortest-
| paper-ever-published-i...
| tel wrote:
| I'm not a huge fan of these, but this time I noticed that the
| best ones feel a lot like naturality arguments. As in, moving
| structural bits in a way that makes it clear that we're not
| touching anything that ought to be universally quantifiable.
|
| I still don't love this sort of thing being presented as "proof",
| but I thought that idea is interesting. Is there a way to
| formalize naturality into technical diagrams? Probably!
| js8 wrote:
| I know a nice proof of volume of tetrahedron being 1/3 of the
| corresponding paralellepiped. You split it into smaller
| tetrahedra by midpoints and count them.
|
| Also there is a nice visual proof that in an equilateral
| triangle, for every point in it, the sum of distances from all
| the sides is constant.
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