Intro to Mathology: 1

I'm a geometer - or in the very least, I'm supposed to be one.

Geometry feels like it should be all about drawing shapes like triangles and squares, and modern geometry isn't too different. Pretty much all that I do is draw "curvy" (hyperbolic) triangles and (hyperbolic) "pairs of pants" - and every so often, I might drawn only half of the pants.

One way of thinking about geometry is that it's the study of (patterns to do with) angles and lengths. And a HUGE chunk of what we know about triangles and squares - the constituents of more complicated polygons - comes from the ancient Greeks (alright, fair enough, other civilizations also did stuff, but their stuff seem to be a bit more guessy than mathsy). In particular, Euclid (originally Εὐκλείδης) wrote some stuff on this. But screw that, let's talk about something that (supposedly) the Babylonians did: Pythagoras' Theorem.

Given a right-angled triangle \(\triangle\), so that its three sides are of lengths \(a\leq b<c\), then:
\[a^2+b^2=c^2.\]
This is usually illustrated with the following picture (on the left), which is supposed to suggest that the areas of the two smaller squares add up to the area of the large one.

 
But if you think about it: if \(a^2+b^2=c^2\), then
 \[\frac{1}{2}a^2+\frac{1}{2}b^2=\frac{1}{2}c^2\]
 must be true. This tells us that if we halve the squares, then the areas still add up correctly.

But why take \(\frac{1}{2}\) times the squares?
Why not take a \(\frac{\sqrt{3}}{2}\) (for equilateral triangles)?
Why not take \(\frac{\pi}{2}\) (corresponding to semicircles)?

So, we see that \(a^2+b^2=c^2\) is really just a relationship about the areas of any shape that you'd like to use - so long as you scale them correctly.

I picked whales. Splash.

But let's forget about all that, for now. Here's a simple, practical question: when people build houses, how can they make right-angles? Okay, maybe it's not that simple, because we quickly realise (*) that using set squares probably won't work. I mean, sure, we'll have a right angle for the first \(20\)cm or something, but after that, when we try to extend the the line using rulers and what-not, we'll probably be off by a degree or two. But then, a degree or two makes a pretty big difference over a long distance. And all of this is made even harder if we're building on slightly hilly ground.

So...what to do? Or as Koreans might say in any given k-drama: this.

Let's use Pythagoras' Theorem.
Step 1: take a really long piece of string, maybe \(24\)m long or something. Yea, \(24\) is a nice number, and it's a nice game too (not the one based on the TV show).
Step 2: mark the string \(6\)m and \(14\)m. This effectively breaks the string up into three segments of lengths \(6\), \(8\) and \(10\) metres(*).
Step 3: O LOOK! \(6^2+8^2=10^2\), what a coincidence. I guess these three lengths must form the sides of a right-angled triangle...my my, I wonder how I might use this to my advantage...
Step 4: form this right-angled triangle and use the right-angle in it to measure the right angle you need for building the house.

So, that was a kinda cute use for Pythagoras' theorem (if you don't have more high-tech equipment available). But in all fairness, most walls aren't all that long and larger versions of the set square (called carpenter's squares) suffice.

Good thing maths isn't about application - it's about ideas.

(*): I'm Aussie, we spell it with a "s" here.