Sunday, October 23, 2005

Your Kitchen is so Babylonian

This post is very geeky. It meanders almost aimlessly between the subjects of math, history, and my personal life. Please bear with me. I promise the math will be accessible to even the most severe mathphobe who has at least a vague recollection of algebra.

ball-and-chain and I and our kids have spent much happy time at the home of Ralphie and Mrs. Ralphie. Even though I've been to their house an Avogadro's number of times, it was only about six months ago that I noticed their kitchen tile floor. It looks something like this.

Well, it looks nicer that that, since I've only reproduced a black and white sketch of it, but it's the geometry I'm trying to convey, not the aesthetics. A single tile looks like this.

To some, this would only be a pretty pattern, but I remembered that it was a graphical representation of a proof of the Pythagorean Theorem. So as soon as I noticed the tile, I dashed home, grabbed pencil and paper and convinced myself by jotting down the proof.

Now the Pythagorean Theorem isn't just any statement in mathematics. It was known by the Babylonians at least 3,600 years ago. It was proven by the ancient Greeks. It forms the bedrock of geometry and analytic trigonometry since it is used to calculate the distance between two points. Hundreds of different proofs of the Theorem have been discovered (one by President Garfield!) and Elisha Loomis, an early twentieth century professor, published a collection of 367 proofs. (If I had a wish list, that book would be on it.) Just as any educated well-rounded person should be able to recognize a couple of Shakespeare's sonnets even if her training is in engineering, so I propose that any well-educated person in the languages, or humanities, or arts should know the Pythagorean Theorem and be able to prove it. (In fact, Ralphie tells me that in one of the

*Planet of the Apes*books, one of the human protagonists convinces his ape captors that he is intelligent and civilized by jotting down a proof of the Theorem. I would certainly campaign vigorously for voting rights for any chimp which could do the same.)

So, with your indulgence, allow me to present a simple proof of the Pythagorean Theorem memorialized by the tile on Ralphie's kitchen floor. It is so simple you may remember it and use it later to impress your captors.

First, let's just remind ourselves what the Theorem says. It says that for any right triangle (that's a triangle in which one of the angles is 90 degrees) the lengths of the sides obey a certain relationship.

If the length of side opposite the right angle is c, and the lengths of the other two sides are a and b, the Pythagorean Theorem states that a²+b²=c².

Here's the proof. Let's first take a second look at a tile and label some of the lengths on it.

You can see that the tile is composed of a large square of side c which is cut into four identical right triangles and a smaller square in the center of side b-a. So the area of the larger square is the sum of the areas of the smaller square and the four triangles. Since the area of a right triangle is half the product of the base and height, we have

c² = (b – a)² + 4(ab)/2

Simplifying (just squaring b-a) we get

c² = b² – 2ab + a² + 2ab

c² = b² + a²

Tah da! Strangely satisfying, no?

You can learn much more about the Pythagorean Theorem and much less about Ralphie's house here.

Comments:

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good lord, you are a geek.

OTOH, I may have enjoyed that post more than anything I've ever read in blogdom, so what does that say about me?

Now you need to get a book on Tangrams.

OTOH, I may have enjoyed that post more than anything I've ever read in blogdom, so what does that say about me?

Now you need to get a book on Tangrams.

Og: It says that you too are a huge nerd. I love tangrams and play with them with my kids, though I've never read a book on them. If you have one you like, post a link.

Bean, your memory of my floor is astounding (I'm looking at it right now, and you're right about its appearance). However, your memory of other things is a little shaky. For example, I have never read a Planet of the Apes book and have no idea what you're talking about. Also, when you first noticed my floor, you didn't go home immediately to do research. You first started telling me about it. Then, when I fell asleep, you ran home.

In fact, the only reason I even read this post after realizing it had something to do with math is that I saw my moniker. Which is to say, I'm a vain simpleton.

In fact, the only reason I even read this post after realizing it had something to do with math is that I saw my moniker. Which is to say, I'm a vain simpleton.

Og: Thanks. I'll try to check it out.

Stacey: We math geeks have to stick together. I googled "math" in your blog and found a treasure trove of puzzles I hope to have a chance to get to. I hope your boyfriend appreciates your mathiness.

Ralphie: The weird thing is I can't figure out who told me that story now (about Planet of the Apes and about the book of hundreds of proofs). I asked of our friends in synagogue this morning and they have no recollection of it. It seems too detailed for me to have made up entirely, but I have no clue where it came from.

Psychotoddler: Sometimes it's good to communicate in ways that allow others to understand you. Was that another Homestar reference?

Stacey: We math geeks have to stick together. I googled "math" in your blog and found a treasure trove of puzzles I hope to have a chance to get to. I hope your boyfriend appreciates your mathiness.

Ralphie: The weird thing is I can't figure out who told me that story now (about Planet of the Apes and about the book of hundreds of proofs). I asked of our friends in synagogue this morning and they have no recollection of it. It seems too detailed for me to have made up entirely, but I have no clue where it came from.

Psychotoddler: Sometimes it's good to communicate in ways that allow others to understand you. Was that another Homestar reference?

Oh my. I am a man of words. I am not afraid of numbers but they certainly do not hold the same interest for me as they do for some of you.

Not that there is anything wrong with that.

Not that there is anything wrong with that.

Jack: We who are not like others appreciate your tollerance. Thanks for not dragging us off to the math sanitoriums.

Shmata Queen,

Go diagram a sentence. ;)

I see that the spammers finally got to this place, the old word verification tool is here.

Go diagram a sentence. ;)

I see that the spammers finally got to this place, the old word verification tool is here.

When I grow up, I wanna be a geek just like you! I can't figure out what I love more - Math or Grammar. Or is it Biology? Or maybe Literature. Perhaps History? No not history...

Thanks for this always entertaining blog! It's a joy (or as my word verification saye: a jehoy).

Thanks for this always entertaining blog! It's a joy (or as my word verification saye: a jehoy).

Jack & Stacey: Play nice.

Psychotoddler: No comprende. It's a riddle.

Ayelet: It takes hard work to be a geek, but if you put your mind to it, you can do it. Step 1: be a nerdy introvert and alienate all your friends.

Psychotoddler: No comprende. It's a riddle.

Ayelet: It takes hard work to be a geek, but if you put your mind to it, you can do it. Step 1: be a nerdy introvert and alienate all your friends.

Doc,

I do play nice which is why I helped the poor little math girl from the slums of cleveland.

She loves this stuff. Ask her. http://jpbrown.i8.com/cubesolver.html

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I do play nice which is why I helped the poor little math girl from the slums of cleveland.

She loves this stuff. Ask her. http://jpbrown.i8.com/cubesolver.html

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