What is geometry? That's a very good question. First, let us tell you what geometry is not. It's not two-column proofs. It's not that teacher who always took off points for not doing it the way you were told to do it. It's not about remembering postulates that are so obvious you shouldn't have to. It's not a high school class. It's not that stuff that guy with the weird name made up. Alright, alright. What IS it already?!
Geometry is a collection of games. Each has a set of rules. The most popular is Euclidean geometry. Of all the games, it is the easiest to play. I will spare you some of the other names for now. And right now, you could care less about it. But soon enough, you will care by the time we're done.
Euclid is to the timeline of science as Jesus is to the timeline of humans. His book is as popular as His book. What? That WAS confusing, wasn't it. Let us explain. His book is The Elements and His book is The Bible. Still not clear? Ok. Euclid wrote The Elements and Jesus wrote The Bible. Their books vie for most-popular and best-selling of all time. The Elements was a breakthrough in understanding outer space while The Bible was a breakthrough in understanding inner space. Like there was a time before Jesus, there was a time before Euclid. Nah. Really? That don't mean nothin'. What we want you to understand is that The Elements is so important that we decided to divide scientific history into a time before Euclid and a time after him, just as we did with Him. By "Him" you mean Jesus? Yes. Why didn't you just say "Jesus"? It was more fun for us. Shall we continue?
Let's start with the time before Euclid, as we're in the time after Jesus. Why don't they use AJ, instead of AD for that? Let's talk about that later. We could use SBE and SAE to refer to science before Euclid and science after Euclid. Can we agree on that? Great. More than 70,000 years ago, someone in Africa left behind a piece of ochre they were making pretty designs on. In 2323 SAE, we are arguing over how to best interpret (or how not to interpret) this artifact. Are you mad? It's not 2323! Not at all. We told you we were going to mark time according to Euclid. Get with it already! Now, as we were saying, we don't care how to best interpret the Blombos Ochre. We see it as evidence that Middle Stone Age humans were interested in geometric designs. In particular, it suggests an attempt to tile the plane with equilateral triangles. We don't care that others may contend against our interpretation, it works for our purposes. Besides, as long as there is no evidence to the contrary, we are free to speculate within the bounds of reason. Now, let's imagine ourselves playing with this idea of tiling as they did so long ago. What else could we come up with? Starting with a recreation of what they may have been hinting at, we suggest
This brings some questions to mind. What other shapes do the same thing? What a great question! Let's find out. How about a different triangle?
Starting with a unit square
we build a square from its diagonal to observe that it is equal in content to two unit squares. If d denotes the diagonal length, then the figure below shows us that d2 = 12 + 12.
Taking the 2 × 1 rectangle from the prior figure, we build a square from its diagonal to observe that it is equal in content to the unit square with its gnomon and an additional unit square. If d denotes this diagonal length, then the figure below shows us that d2 = 22 + 12.
Taking the 3 × 1 rectangle from the prior figure, we build a square from its diagonal to observe that it is equal in content to the 2 × 2 square with its gnomon and an additional unit square. If d denotes this diagonal length, then the figure below shows us that d2 = 32 + 12.
Taking the 4 × 1 rectangle from the prior figure, we build a square from its diagonal to observe that it is equal in content to the 3 × 3 square with its gnomon and an additional unit square. If d denotes this diagonal length, then the figure below shows us that d2 = 42 + 12.
It may prove interesting to look at the contents of these squares as gnomons in and of themselves.
Now, let's look at the diagonals of 2 × n rectangles. The 2 × 1 case has been demonstrated. The 2 × 2 case is merely a dilation of the 1 × 1 case. If d denotes this diagonal length, then the figure below shows us that d2 = 32 + 22.
What about these square roots? How do we calculate them? Back in 800 BCE, Baudhayana wrote a Sulbasutra presenting his approximation of the side of a square whose area is 2: ⅓ + 1 / (3 * 4) − 1 / (3 * 4 * 34). How on earth did that happen? Well, he may have attempted to cut up a unit square and add the pieces to another unit square to approximate the square having area 2. If we affix each half of a unit square we cut to the top and right sides of another unit square, then we get the gross approximation 1 + ½. You could mention it's so gross that no one touched it. Yes, but that would be vulgar. So, how did he get ⅓ out of it? That is a very good question. What? Oh, you want me to just tell you what he did? Where's the satisfaction in that? What does it profit a man to gain the world if he loses his soul? That's the bible! Can we just stick to math, please? Ok. So, let's instead cut a unit square into 3 pieces and see where that gets us.