"One of the more interesting features I hadn’t appreciated before is that when you get close to the Arctic circle, the length of the days is essentially a zigzag, straight up from the winter solstice all the way to the summer solstice and back down again."

I had noticed this too and wondered if it was exactly true, with the "zigzag" being straight lines - I thought there might be a simple proof of this fact based on some trig identities. There's not, because it's not true - the lines aren't exactly straight, even if you ignore solar refraction - but it's a very good approximation.