4 ms·

Proof Without Words: Gregory’s Theorem

zazen 8y ago
I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective.
- laurentl 8y ago
  I didn’t get this feeling from the proof in the article. I thought there was a kind of poetry to it. The diagram and colors make it reasonably easy to understand and they emphasize the geometrical nature of the proof, which uses only elementary math: the formula for the area of a triangle and Thales’ theorem (If my memory is correct, I haven’t used that name since high school!). On the other hand, a “proof with words” would likely have derived the explicit formulas for I_n and C_n in order to prove the relationship and would have been much more clanky. If I had to make a comparison, this felt like reading good functional code. Concise, a bit mysterious at first, but once you understand what it does you feel enlightened. (Vs eg. wading through for loops and index arithmetic, which also does the job but is more tedious to read)
  - whatshisface 8y ago
    I think it wasn't so much a proof as it was an image that made you realize the proof with some thought. Still, that's about what all published proofs are in the end, isn't it?
- agumonkey 8y ago
  Same for education purposes, I deeply believe that most terms cloud the learning process. Most of the time a pattern can be witnessed and learned without giving too many names if at all. Maybe then, after the structure/topology[1] has set in the mind, you can label things to accelerate communication. But words first feels like cart before the wheel. [1] in the fuzzy sense, not the mathematical field
- coldtea 8y ago
  >I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. There are lots of geometrical proofs for e.g. the pythagorean theorem that need no words. It's not like code golf any more than a text-based proof is. Words still need to get interpreted just as images do. If anything, a succinct geometric proof can be much easier to grasp and verify than a text-based proof.
- jonahx 8y ago
  > a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. This is truly, deeply the wrong lesson to take away. I'd say that, in a spirit similar to the Feynman technique, you don't really understand proofs whose central idea you can't explain simply -- and a wordless proof is good evidence that you have in fact grasped the proof's essence. Of course, this doesn't mean you can do away with formal rigor. Mathematics needs that too. But I'd argue that is "the clever game for insiders to play" (albeit an important game), while intuitive understanding is what you should really be after, and is what generates new ideas and new proofs. More beautiful proofs without words: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain#734790 https://math.stackexchange.com/questions/733754/visually-stu... https://math.stackexchange.com/questions/1337750/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthusiastic-ab?noredirect=1&lq=1 https://math.stackexchange.com/questions/1337750/easy-math-p... https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=proofs+without+words https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3...
heinrichhartman 8y ago
Here is another proof without words for the Pythagorean Theorem: https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/File:Pythagoras-proof-anim.svg https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/Fil...
- JBiserkov 8y ago
  Nice. I also like the proof which uses the only the first image a calculates the area of the square twice: c^2 + 4*a*b/2 = c^2 + 2*a*b (a+b)^2 = a^2 + b^2 + 2*a*b
espeed 8y ago
Evidently Oliver Selfridge [1] said a puzzle based on deriving the radius like this once tripped Feynman up... "Puzzles from last week" http://web.media.mit.edu/~walter/MAS-A12/week11.html http://web.media.mit.edu/~walter/MAS-A12/week11.html [1] https://en.wikipedia.org/wiki/Oliver_Selfridge https://en.wikipedia.org/wiki/Oliver_Selfridge
- abecedarius 8y ago
  What's the expected mistaken answer?
gcb0 8y ago
great. now draw a few hundred more n. and don't forget the min-max values.
herodotus 8y ago
My favourite proof without words is the proof that the sum of the interior angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in chalk on the ground. Stand on one of the lines at one of the corners and note the direction you are facing. Now back up until you are at the corner of the triangle behind you. Rotate your self by shuffling inside the angle to the next line. Walk to the next corner, repeat. Walk (backwards this time) to the last corner, repeat. Now shuffle back to your starting position. You are facing exactly 180 degrees in the other direction! Of course it is way better to just do this than write it down! But obviously I can't do it on HN.
- whatshisface 8y ago
  Does that prove that every triangle has angles adding up to 180, or just that one?
  - Corvus 8y ago
    “draw an arbitrary triangle” means this works with any triangle.
    - whatshisface 8y ago
      But you'd have to try an infinite number of triangles before you were convinced that it worked for all of them.
      - justinpombrio 8y ago
        This is an interesting confusion. OP said "draw an arbitrary triangle in chalk on the ground", and from there gave a set of instructions that showed you how to check that the interior angles of that triangle added up to 180 degrees. You're right that that concrete set of instructions only proves it for that one triangle. But that's not the whole proof. If I followed these instructions for a right triangle with angles 45,45,90, I'd only have proved that that triangle's angles added up to 180 degrees. The full proof is that if you imagine doing this, for any triangle, it's clear that it will always work. When I picture this in my head, it's clear that I'm going to end up having turned 180 degrees, regardless of the measurements of the triangle. This leap: starting with a set of concrete instructions that you can do on a particular object, but then verifying that they would work no matter what object you started with, is common in proofs. EDIT: Nevermind, totally unrelated.
louis_ 8y ago
Hi there. Just a precision: the fact that the sum of the interior angles of a triangle is 180 degrees is not really provable, it is a postulate of the Euclidean geometry. It is an intuitive and accepted property, it reflects the fact that the geometric space usually considered is flat. Proving this statement would be possible provided that you change basic postulates. At that moment, it would become a proposition of these new mathematics and would certainly be provable starting from the new postulates. Changing these postulates is possible, that's how we got the parabolic and hyperbolic geometries, see wiki!

I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective.

laurentl 8y ago
I didn’t get this feeling from the proof in the article. I thought there was a kind of poetry to it. The diagram and colors make it reasonably easy to understand and they emphasize the geometrical nature of the proof, which uses only elementary math: the formula for the area of a triangle and Thales’ theorem (If my memory is correct, I haven’t used that name since high school!). On the other hand, a “proof with words” would likely have derived the explicit formulas for I_n and C_n in order to prove the relationship and would have been much more clanky. If I had to make a comparison, this felt like reading good functional code. Concise, a bit mysterious at first, but once you understand what it does you feel enlightened. (Vs eg. wading through for loops and index arithmetic, which also does the job but is more tedious to read)
- whatshisface 8y ago
  I think it wasn't so much a proof as it was an image that made you realize the proof with some thought. Still, that's about what all published proofs are in the end, isn't it?
agumonkey 8y ago
Same for education purposes, I deeply believe that most terms cloud the learning process. Most of the time a pattern can be witnessed and learned without giving too many names if at all. Maybe then, after the structure/topology[1] has set in the mind, you can label things to accelerate communication. But words first feels like cart before the wheel. [1] in the fuzzy sense, not the mathematical field
coldtea 8y ago
>I haven't previously come across the idea of trying to make wordless proofs. I guess this is the mathematical equivalent of code golf: a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. There are lots of geometrical proofs for e.g. the pythagorean theorem that need no words. It's not like code golf any more than a text-based proof is. Words still need to get interpreted just as images do. If anything, a succinct geometric proof can be much easier to grasp and verify than a text-based proof.
jonahx 8y ago
> a clever game for insiders to play, but basically the opposite of good practice from a readability perspective. This is truly, deeply the wrong lesson to take away. I'd say that, in a spirit similar to the Feynman technique, you don't really understand proofs whose central idea you can't explain simply -- and a wordless proof is good evidence that you have in fact grasped the proof's essence. Of course, this doesn't mean you can do away with formal rigor. Mathematics needs that too. But I'd argue that is "the clever game for insiders to play" (albeit an important game), while intuitive understanding is what you should really be after, and is what generates new ideas and new proofs. More beautiful proofs without words: https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain#734790 https://math.stackexchange.com/questions/733754/visually-stu... https://math.stackexchange.com/questions/1337750/easy-math-proofs-or-visual-examples-to-make-high-school-students-enthusiastic-ab?noredirect=1&lq=1 https://math.stackexchange.com/questions/1337750/easy-math-p... https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3Daps&field-keywords=proofs+without+words https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3...

heinrichhartman 8y ago

Here is another proof without words for the Pythagorean Theorem: https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/File:Pythagoras-proof-anim.svg https://en.wikipedia.org/wiki/Pythagorean_theorem#/media/Fil...

JBiserkov 8y ago
Nice. I also like the proof which uses the only the first image a calculates the area of the square twice: c^2 + 4*a*b/2 = c^2 + 2*a*b (a+b)^2 = a^2 + b^2 + 2*a*b

espeed 8y ago

Evidently Oliver Selfridge [1] said a puzzle based on deriving the radius like this once tripped Feynman up... "Puzzles from last week" http://web.media.mit.edu/~walter/MAS-A12/week11.html http://web.media.mit.edu/~walter/MAS-A12/week11.html [1] https://en.wikipedia.org/wiki/Oliver_Selfridge https://en.wikipedia.org/wiki/Oliver_Selfridge

abecedarius 8y ago
What's the expected mistaken answer?

gcb0 8y ago

great. now draw a few hundred more n. and don't forget the min-max values.

herodotus 8y ago

My favourite proof without words is the proof that the sum of the interior angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in chalk on the ground. Stand on one of the lines at one of the corners and note the direction you are facing. Now back up until you are at the corner of the triangle behind you. Rotate your self by shuffling inside the angle to the next line. Walk to the next corner, repeat. Walk (backwards this time) to the last corner, repeat. Now shuffle back to your starting position. You are facing exactly 180 degrees in the other direction! Of course it is way better to just do this than write it down! But obviously I can't do it on HN.

whatshisface 8y ago
Does that prove that every triangle has angles adding up to 180, or just that one?
- Corvus 8y ago
  “draw an arbitrary triangle” means this works with any triangle.
  - whatshisface 8y ago
    But you'd have to try an infinite number of triangles before you were convinced that it worked for all of them.
    - justinpombrio 8y ago
      This is an interesting confusion. OP said "draw an arbitrary triangle in chalk on the ground", and from there gave a set of instructions that showed you how to check that the interior angles of that triangle added up to 180 degrees. You're right that that concrete set of instructions only proves it for that one triangle. But that's not the whole proof. If I followed these instructions for a right triangle with angles 45,45,90, I'd only have proved that that triangle's angles added up to 180 degrees. The full proof is that if you imagine doing this, for any triangle, it's clear that it will always work. When I picture this in my head, it's clear that I'm going to end up having turned 180 degrees, regardless of the measurements of the triangle. This leap: starting with a set of concrete instructions that you can do on a particular object, but then verifying that they would work no matter what object you started with, is common in proofs. EDIT: Nevermind, totally unrelated.

louis_ 8y ago

Hi there. Just a precision: the fact that the sum of the interior angles of a triangle is 180 degrees is not really provable, it is a postulate of the Euclidean geometry. It is an intuitive and accepted property, it reflects the fact that the geometric space usually considered is flat. Proving this statement would be possible provided that you change basic postulates. At that moment, it would become a proposition of these new mathematics and would certainly be provable starting from the new postulates. Changing these postulates is possible, that's how we got the parabolic and hyperbolic geometries, see wiki!