OK, OK, I know. There are lots and lots of proofs of the Pythagorean Theorem. (My favorite appears at the bottom of this entry, in case you want to skip to it.) I’m not trying to overthrow thousands of years of math here. I’m just wondering about something, and I’ve been unable to find any treatment of this particular problem.
Suppose you’re walking along and come to an empty lot. You have to get from the northwest corner of the lot to the southeast corner of the lot. You could just walk all the way down the west side (side A), then walk all the way along the south side (side D). You’d find the total distance equals the length of side A plus the length of side D.
We all know that to shorten your distance, you simply cut across the lot. H (the hypotenuse) does not equal A + D, but is always less than A + D.
But let’s suppose for just a minute that we don’t really know what it means to cut across. We don’t, in other words, understand diagonal. Suppose you’ve got some strange condition where you can only move north-south and east-west, but nothing in between. Now you try to take a shortcut across the lot, but you get this:
Now if you add up your distance, you’ll find that it’s (more or less) 1/2 A + 1/2 D + 1/2 A + 1/2 D = A + D ! You didn’t do any better by cutting across the lot in this way.
OK, maybe if we do a little better. What about this zigzag path?
This path looks a lot more like the hypotenuse. In fact, you can make the zigzags arbitrarily small, maybe a million turns, or a billion, or any other ridiculous number you want. Your total distance traveled is still exactly A + D ! You haven’t done any better by taking the shortcut!
And yet, if you make even a small diagonal movement, you instantly see the benefit. Consider this path, in which you travel most of A, cut across the corner, then travel most of D. Your total distance will be less than A + D. So what’s going on here? Clearly the hypotenuse is the best path, but I can’t approximate that hypotenuse, even a little, by using only north-south and east-west movements. What is this “diagonal” and why is it so powerful? Any thoughts?
OK, here’s the elegant proof of Pythagoras that I like, the one that Albert Einstein found on his own when as a boy he fell in love with what he called “holy geometry.”
The first thing to notice is that there are three three similar triangles in this picture. Why? Because:
1) similar triangles all have the same internal angles.
2) all triangles have the same internal angle measure (we know it’s 180 degrees, but that doesn’t matter – what matters is that all triangles have the same measure, whatever it is).
3) by 1 and 2, if two right triangles share an angle, then they have to be similar.
The three similar triangles are:
ABC, ACX, and CBX (notice that the order of the letters is crucial here, matching the appropriate angles).
In similar triangles, the ratios of the sides are equal. So the triangle CBX compared to the triangle ABC gives
a/x = c/a
Cross-multiplying gives a^2 = cx
The triangle ABC compared to the triangle ACX gives
c/b = b/(c – x)
Cross-multiply again and you get
b^2 = c^2 – cx
But we already know cx = a^2 from before, so now we have
b^2 = c^2 – a^2
Rearranging, we get the familiar
a^2 + b^2 = c^2
The Pythagorean Theorem! THAT is a beautiful bit of math.
It occurs to me that the proof depends on the very existence of triangles, and triangles only exist in a two-dimensional world. Somehow, the Pythagorean theorem emerges from the existence of two dimensions as opposed to one. I still don’t see how, but it’s interesting.
Here’s something else. If you made it through my light clock blog entry, you know that Special Relativity comes right out of the Pythagorean Theorem. The light clock is a two-dimensional argument, so it seems that even if we reduced our universe from three spacial dimensions to only two, Special Relativity would still work. Flatlander twins would still experience the paradox if one rocketed off in a two-dimensional spacecraft, then returned to find her twin aged and wrinkled (would flat creatures wrinkle?) But what isn’t clear to me is whether or not Special Relativity would apply in a one-dimensional universe, one in which everything existed on a line, since in such a universe, there are no shortcuts.
Anyway, if anyone has any thoughts on why the Pythagorean Theorem works, or if you’ve ever seen an analysis of this sort of north-south, east-west approach and why it fails, I’d love to hear about it.