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Proofs without words

mathoverflow.net

50 pointscruise0225 comments

"Can you give examples of proofs without words?"

I enjoy that by providing an example of a proof without words, it is proving (without words) that this is in fact possible.

Wow, I didn't know about MathOverflow. Good to know!

MathOverflow is serious business, full of seriously accomplished mathematicians. There are at least two Fields medallists there (Gowers and Tao).

Not really. Those guys at MathOverflow are ruthless. Try posting a question and see what happens.

Sorry if we're still appearing hostile. We have a very focused target audience -- people doing mathematical research -- and in order to serve that community best we turn away questions at inappropriate levels (i.e. what should be 'standard material' of a mathematics undergraduate education at a good university). We try to do that politely, and offer suggestions of other places to try, but I know we don't always succeed.

In my experience it's fantastic and you get good answers very quickly. I got an answer from precisely the mathematician I was planning on emailing!

It's not meant for homework type questions, though, and they do shut those down right away.

What was your question? Post a link.

I also didn't realize MathOverflow existed. On a whim, I typed in physicsoverflow.net, and it turns out that exists as well (though not well populated at all).

I wonder what other overflows exist.

Cool, I wasn't aware of the physics one either. There's a list of them maintained here: http://meta.stackexchange.com/questions/4/list-of-stackexcha...

They're going there. But you know how it is... You can know where the physicists are, or where they're going, but not both at the same time.

I was thinking about creating a ProcrastinationOverflow but I haven't gotten around to it yet.

I just started a new article series on my blog about "Visual Proofs". I also explain why they work:

http://www.catonmat.net/blog/visual-math-friday/

Wow, these are beautiful.

Things like this make me really appreciate the power of crowdsourcing. Hard to imagine how one could compile a list like this 25 years ago.

It's a problem that worthwhile material like this is not valued as highly as new discoveries.

I don't think this is completely true. There may not be tons of "proofs without words" in it, but the book Proofs from the Book is a great example of a nice collection of elegant proofs of elementary propositions that convince at a level that's much more effective than a pure, logical demonstration. Sure, you won't find these things in math journals -- you have to look in things like Mathematics Magazine, College Mathematics Journal, and various recreational mathematics and mathematics education journals. The fact that it's out there in print shows that somebody values this stuff.

Wow, the real number line to the open interval one is spectacular. I've always been interested in infinity and came up with some proofs of stuff like this during college, but this is such a nice proof.

I showed nearly that same picture to a class just two weeks ago. (I didn't draw the whole circle or the line segment representing the interval above it -- I just drew a line and a semicircle and said "this (pointing at the semicircle) is the unit interval bent into a semicircle.)" It's a very powerful demonstration, IMO.

How old were the students? What kind of class?

I'm asking because I'm curious how approachable the concept/proof is.

Many of them don't seem to be complete proofs at all. It's not enough to draw pictures and claim that two areas are identical. You have to argue why the areas are identical.

I think you're missing the point. Its not that these are complete proofs or easy to understand proofs. The idea of a proof without words is that by staring at it and thinking about it, you can work out why the statement/theorem is true.

That being said, you're right that you need to be careful, as pointed out by Russel O'Conner's proof that 32.5 = 31.5 (with the colored triangles). However, someone who uses that as a "proof" is doing it wrong. When you find a proof without words, you need to actually write out the formulas that the picture indicates and make sure everything adds up (at least, for proofs of type you mentioned such as proving that areas are equal).

Hm - Zen math! 'A wordless transmission, outside the scriptures.'

i don't get the (n choose 2) identity proof. can someone explain?

Sure. The total number of yellow balls (the ones not in bottom row), is 1+2+3+...+(n-1). But each yellow ball determines exactly two blue balls (the ones in the bottom row): they're the ones you get by travelling left and right from the yellow ball in a straight line until the bottom. You can visually confirm it to be a 1-1 correspondence: different yellow balls will determine different pairs, and each pair is determined by some yellow ball. Therefore the total number of yellow balls is also the number of ways to choose 2 blue balls out of the total number of n blue balls.

Thanks.

It is interesting that for proofs without words, comments are needed :)

aaah, any pair chooses a unique yellow ball - wow! thanks. if the image were animated, or showed a sequence of "selections" i think i would have gotten it.

tl;dr use geometry.

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