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A "houses" proof of the Pythagorean theorem - Lindsey Kuper [entries|archive|friends|userinfo]
Lindsey Kuper

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A "houses" proof of the Pythagorean theorem [Oct. 12th, 2010|01:20 am]
Lindsey Kuper
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For the geometry cognitive science cognitive-science-via-geometry course I'm taking this semester, the assignment for tomorrow is to write a "non-algebraic" and "intuitive" proof of the Pythagorean theorem. As far as I can tell, that means something like: no equations or plus signs, but we're allowed to use the words "equals" and "sum"; no ratios if we can help it; lots of pictures and colors and similar triangles and appeals to various geometric results; bonus points for recursive behavior; and extra bonus points if it's the kind of proof you could explain to someone at a dinner party in a couple of minutes.1

Figuring out how to do this kind of thing has been difficult and uncomfortable. First of all, I don't care for the implication that intuitiveness can be measured objectively; I think that intuitive equals familiar. Second, I've spent the last year being trained to express proofs in ways that computers can understand, which is to say, not very much like this. Third, dammit, I like algebra.

But hey, I'll give it my best shot. What do you think of my proof -- is it convincing? (I didn't come up with the "houses" idea myself -- I'm just expanding on something that was done in class.)


  1. "A dinner party without whiteboards, I mean," I clarified to Chris chrisamaphone. "Sounds like a pretty dull dinner party," she said.
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Comments:
From: markluffel
2010-10-12 06:07 am (UTC)
I like it!
One idea that a young reader (I'm thinking about my 5th grade brother) might be unfamiliar with is that knowing two of the angles of a triangle tells you the third angle. Maybe showing a transformation that flips and then shrinks one of the roofs into another would explain it?
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[User Picture]From: lindseykuper
2010-10-12 12:38 pm (UTC)
I don't know if I have time to make another figure, but I could definitely throw a "Since knowing the two angles tells you the third angle..." in there. Thanks for the feedback!
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[User Picture]From: jcreed
2010-10-12 12:06 pm (UTC)
A pretty dull dinner party indeed!

I hadn't seen this proof before; I rather like it, although I think I share your skepticism about "intuitive" geometry proofs. As a reasonably mature mathematician, I still like knowing a lot of them as a form of memory compression. I know to really prove it, I'd do something else.

The reason I like this proof is it makes some sense if you don't even draw the canonical squares-on-sides-of-triangles, and just focus on the right triangle and how it can be subdivided into two similar triangles by dropping that perpendicular to the hypotenuse. For you have to learn at some point that fundamental fact about n-dimensionalness that when you scale up an n-dimensional thingy by a factor of k, its volume goes up as k^n. And so just to see that you can find widgets "proportional to" the two legs of the right triangle that in (two-dimensional) volume add up to the hypotenuse one means that you are really talking about squared things adding up to a third squared thing.


(And anyhow I actually believe that it's the central limit theorem that's the reason we live in a space where the pythagorean theorem (approximately, locally) holds in the first place. No matter whether you start with a square lattice or a hex lattice or whatever, the purely combinatorial number-of-ways you can get from point A to point B goes like e^(-x^2) * e^(-y^2) * e^(-z^2) = e^(-(x^2 + y^2 + z^2)) and so similar physics obtains for the same sum-of-squares, because QFT is all about weighted counts of paths mumble mumble rant...)
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[User Picture]From: lindseykuper
2010-10-12 02:08 pm (UTC)
For this proof, you needn't know that the k^n thing is true in general -- you just need to know that the area of a square is defined as the length of its side, squared. But there's no reason why they have to be squares -- they could be any old similar polygons that have a side of the triangle as one of their sides. Actually, I don't think they even have to be polygons, but then you have to worry about defining similarity for arbitrary shapes, I guess.

I don't know if it really makes sense to say that this or that proof of a particular fact is the real reason that the fact is true. However, I do know that if I put e^(-x^2) * e^(-y^2) * e^(-z^2) = e^(-(x^2 + y^2 + z^2)) on my homework, Doug would fire me. Or whatever it is that they do to grad students.
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[User Picture]From: jcreed
2010-10-12 02:12 pm (UTC)
I don't know if it really makes sense to say that this or that proof of a particular fact is the real reason that the fact is true. However, I do know that if I put e^(-x^2) * e^(-y^2) * e^(-z^2) = e^(-(x^2 + y^2 + z^2)) on my homework, Doug would fire me. Or whatever it is that they do to grad students.

Hee hee yes.

I mean, I use phrases like "the real reason" very fuzzily and intuitively without any claim that they really mean anything. But I think there is a deep connection between the squarings that show up in the e^-x^2 of the central limit theorem and the squarings that show up in the Euclidean norm (of any dimension, of course, not just 2!). It's not the reason that the Pythagorean Theorem is true in Euclidean space; proofs such as the one you gave are explanations of that fact. But I suspect it has something to with why our universe is locally Euclidean in the first place.
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[User Picture]From: cos
2010-10-12 02:24 pm (UTC)
That seems convincing, though I don't know how much of that is based on knowing things already. I mean, is the idea of "similar triangles" and proportional sizes "intuitive"? To whom?
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[User Picture]From: lindseykuper
2010-10-12 05:50 pm (UTC)
That's a good question. It's irritating to me that we talk about intuitiveness a lot in this course without ever defining what it means. For example, today the claim was made that because human bodies are vertically symmetric, humans "relate better" to drawings that are vertically symmetric -- which is why it's "easier" to bisect an angle that's pointed straight up or down than it is to bisect some other angle. That kind of made my blood boil. My body is mostly bilaterally symmetric, sure, but what if it weren't? What if, for instance, I was missing one leg? Would I then be less able to "relate" to a vertically symmetric diagram? Moreover, isn't part of the point of cognitive science to examine things that are "intuitively" "obvious" to a human and figure out why they are so? Sometimes the best way to do that is to try to take off our human-colored glasses. (Of course, we humans can't help looking at things from human points of view. But it's sure interesting to try not to, and this course seems to take the opposite approach.)
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[User Picture]From: pmb
2010-10-13 04:28 pm (UTC)

You made me do my own!

Nice! My favorite proof is the one I use in Calc class as a "fun fact".

Just rearrange the four equal triangles:

Pythagorean Thm

But I actually like yours quite a bit as well - I have seen that drawing of the triangle with the squares on its sides, but nobody ever was able (until now) to explain to me exactly how and why that managed to prove the theorem. Thank you!
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[User Picture]From: lindseykuper
2010-10-13 04:42 pm (UTC)

Re: You made me do my own!

Have you seen the ones that use tiling?
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[User Picture]From: pmb
2010-10-13 04:47 pm (UTC)

Re: You made me do my own!

Neat!
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