I Learned the Pythagorean Theorem

Summary

The younger child and I were talking about maths on the school run this morning, and today’s topic was geometry. I was pleased to discover that he’s already got a reasonable comprehension of the Pythagorean Theorem: I was telling him that I was about his age when I first came across it, but in my case I first had a practical, rather than theoretical, impetus to learn it. It was the 1980s, and I was teaching myself Dr. Logo, Digital Research‘s implementation of the Logo programming language (possibly from this book). One day, I was writing a program to draw an indoor scene, including a window through which a mountain would be visible. My aim was to produce something like this: All of these graphics were made using my own 2019 implementation of Logo, TRRTL.COM: click on any graphic to continue drawing! My window was 300 “steps” tall by 200 steps wide and bisected in both directions when I came to make my first attempt at the mountain. And so, naively, starting from the lower-left, I thought I’d need some code like this: RIGHT 45 FORWARD 100 RIGHT 90 FORWARD 100 But what I ended up with was this: Hypotenuse? More like need-another-try-potenuse. I instantly realised my mistake: of course the sides of the mountain would need to be longer so that the peak would reach the mid-point of the window and the far side would hit its far corner. But how much longer ought it to be. I intuited that the number I’d be looking for must be greater than 100 but less than 250: these were, logically, the bounds I was working within. 100 would be correct if my line were horizontal (a “flat” mountain?), and 250 was long enough to go the “long way” to the centrepoint of the window (100 along, and 150 up). So I took a guess at 150 and… it was pretty close… but still wrong: I remember being confused and frustrated that the result was so close but still wrong. The reason, of course, is that the relationship between the lengths of the sides of a triangle don’t scale in a 1:1 way, but this was the...