Proof Without Words: Gregory’s Theorem

https://divisbyzero.com/2018/09/28/proof-without-word-gregorys-theorem/

97 26 8 years ago

zazen• 8 years 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• 8 years 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)

whatshisface8• 8 years 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• 8 years 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• 8 years 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• 8 years 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-stu...

https://math.stackexchange.com/questions/1337750/easy-math-p...

https://www.amazon.com/s/ref=nb_sb_noss_2?url=search-alias%3...

heinrichhartman• 8 years ago

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

JBiserkov• 8 years 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• 8 years 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

[1] https://en.wikipedia.org/wiki/Oliver_Selfridge

abecedarius• 8 years ago

What's the expected mistaken answer?

gcb0• 8 years ago

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

herodotus• 8 years 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.

whatshisface8• 8 years ago

Does that prove that every triangle has angles adding up to 180, or just that one?

Corvus• 8 years ago

“draw an arbitrary triangle” means this works with any triangle.

louis_• 8 years 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!

Proof Without Words: Gregory’s Theorem

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