4 ms·

Proofs Without Words

111 points by squircle 1y ago

fiforpg 1y ago
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 https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf Caution: proofs of some of the statements in it are difficult.
- JadeNB 1y ago
  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 1y ago
    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 1y ago
I made a game out of creating proofs without words: https://brianberns.github.io/Tactix/ https://brianberns.github.io/Tactix/
vonnik 1y ago
Anyone who enjoys this should read David Bessis’s Mathematica.
- bwfan123 1y ago
  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-Curiosity/dp/0300270887 https://www.amazon.com/Mathematica-Secret-World-Intuition-Cu...
vismit2000 1y ago
There is also this youtube channel called 'Mathematical Visual Proofs' on similar theme: https://www.youtube.com/@MathVisualProofs https://www.youtube.com/@MathVisualProofs
downboots 1y ago
Another great site is https://theoremoftheday.org/ https://theoremoftheday.org/ with a neat one-pager overview of each theorem
jupitr 1y ago
and also how to lie with visual proofs: https://www.youtube.com/watch?v=VYQVlVoWoPY https://www.youtube.com/watch?v=VYQVlVoWoPY
ViscountPenguin 1y ago
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 1y ago
  I'm the opposite. I am not convinced until I "see it". Probably has to do with our innate talents.
  - seanhunter 1y ago
    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 1y ago
  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 1y ago
  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.
  - 1y ago
cuber_messenger 1y ago
There's a book called "Proofs without words". Fun to have a glance. (https://ia801405.us.archive.org/24/items/proofs-without-words-roger-nelsen/Proofs%20without%20Words%20-%20Roger%20Nelsen_text.pdf https://ia801405.us.archive.org/24/items/proofs-without-word...) It also has a sequel.
stared 1y ago
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/ https://www.c82.net/euclid/ (reproduction in CSS by Nicholas Rougeux)
Someone 1y ago
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-in-a-serious-math-journal-john-conway-alexander-soifer https://fermatslibrary.com/s/shortest-paper-ever-published-i...
- seanhunter 1y ago
  Notice that one of the authors is John H. Conway. Serious badass, among many many other things known for: - The game of life - The fractran language (a Turing-complete programming language consisting only of fractions) - Surreal numbers - The “Conway base 13 function” (a brain-scramblingly hideous function that is everywhere discontinuous and yet somehow takes on every real number on every interval - invented as an analysis counterexample to prove that a function can satisfy the intermediate value property and yet not be continuous). - A lot of work on sporadic simple groups. The three groups Co1, Co2 and Co3 are named after John H. Conway, and he was co-author of the “Monstrous Moonshine” paper and conjecture that Richard Borcherds won the Fields medal for proving. … and a bunch of other whacky stuff, such as inventing an algorithm to figure out what day of the week any given date in history was (he used to do this in his head) Sadly he died of complications from Covid. https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/John_Horton_Conway
tel 1y ago
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 1y ago
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.
- downboots 1y ago
  The second one https://en.wikipedia.org/wiki/Viviani%27s_theorem https://en.wikipedia.org/wiki/Viviani%27s_theorem
zem 1y ago
"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) choose 2", the number of ways of selecting 2 elements out of a set of (n + 1). and if you look at the picture, selecting any two balls in the bottom row uniquely identifies a ball in the triangle, and vice versa (selecting a ball in the triangle picks a unique pair of balls in the bottom row). so the sum of all the balls in the first n rows is indeed the number of ways of choosing two balls from the bottom row!

111 points by squircle 1y ago

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 https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf Caution: proofs of some of the statements in it are difficult.

JadeNB 1y ago
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 1y ago
  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 1y ago

I made a game out of creating proofs without words: https://brianberns.github.io/Tactix/ https://brianberns.github.io/Tactix/

vonnik 1y ago

Anyone who enjoys this should read David Bessis’s Mathematica.

bwfan123 1y ago
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-Curiosity/dp/0300270887 https://www.amazon.com/Mathematica-Secret-World-Intuition-Cu...

vismit2000 1y ago

There is also this youtube channel called 'Mathematical Visual Proofs' on similar theme: https://www.youtube.com/@MathVisualProofs https://www.youtube.com/@MathVisualProofs

downboots 1y ago

Another great site is https://theoremoftheday.org/ https://theoremoftheday.org/ with a neat one-pager overview of each theorem

jupitr 1y ago

and also how to lie with visual proofs: https://www.youtube.com/watch?v=VYQVlVoWoPY https://www.youtube.com/watch?v=VYQVlVoWoPY

ViscountPenguin 1y ago

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 1y ago
I'm the opposite. I am not convinced until I "see it". Probably has to do with our innate talents.
- seanhunter 1y ago
  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 1y ago
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 1y ago
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.
- 1y ago

cuber_messenger 1y ago

There's a book called "Proofs without words". Fun to have a glance. (https://ia801405.us.archive.org/24/items/proofs-without-words-roger-nelsen/Proofs%20without%20Words%20-%20Roger%20Nelsen_text.pdf https://ia801405.us.archive.org/24/items/proofs-without-word...) It also has a sequel.

stared 1y ago

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/ https://www.c82.net/euclid/ (reproduction in CSS by Nicholas Rougeux)

Someone 1y ago

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-in-a-serious-math-journal-john-conway-alexander-soifer https://fermatslibrary.com/s/shortest-paper-ever-published-i...

seanhunter 1y ago
Notice that one of the authors is John H. Conway. Serious badass, among many many other things known for: - The game of life - The fractran language (a Turing-complete programming language consisting only of fractions) - Surreal numbers - The “Conway base 13 function” (a brain-scramblingly hideous function that is everywhere discontinuous and yet somehow takes on every real number on every interval - invented as an analysis counterexample to prove that a function can satisfy the intermediate value property and yet not be continuous). - A lot of work on sporadic simple groups. The three groups Co1, Co2 and Co3 are named after John H. Conway, and he was co-author of the “Monstrous Moonshine” paper and conjecture that Richard Borcherds won the Fields medal for proving. … and a bunch of other whacky stuff, such as inventing an algorithm to figure out what day of the week any given date in history was (he used to do this in his head) Sadly he died of complications from Covid. https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/John_Horton_Conway

tel 1y ago

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 1y ago

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.

downboots 1y ago
The second one https://en.wikipedia.org/wiki/Viviani%27s_theorem https://en.wikipedia.org/wiki/Viviani%27s_theorem

zem 1y ago

"The sum of the first $n$ positive integers is ${n+1 \choose 2}$" is beautiful! for anyone lacking the background to get it, the right hand side is "(n + 1) choose 2", the number of ways of selecting 2 elements out of a set of (n + 1). and if you look at the picture, selecting any two balls in the bottom row uniquely identifies a ball in the triangle, and vice versa (selecting a ball in the triangle picks a unique pair of balls in the bottom row). so the sum of all the balls in the first n rows is indeed the number of ways of choosing two balls from the bottom row!