

Proofs without words - cruise02
http://mathoverflow.net/questions/8846/proofs-without-words

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char
"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.

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

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

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omaranto
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.

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greenlblue
What was your question? Post a link.

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pkrumins
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/>

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lpolovets
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.

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

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pmiller2
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.

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roundsquare
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.

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pmiller2
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.

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roundsquare
How old were the students? What kind of class?

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

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Tichy
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.

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roundsquare
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).

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

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

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anatoly
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.

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stralep
Thanks.

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

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est
tl;dr use geometry.

