One Possible Solution
This simple solution really requires no words and is why I love mathematics. It is clear when I sketch in that dashed line, that each piece is now cut in half by the diagonal of the triangle. Therefore, there is just as much area inside the triangle as outside. Thus, the triangle takes up half of the box!
This is what real mathematics feels and looks like; it is an art that consists of asking simple and elegant questions from our imaginations and finding even more beautiful solutions.
To quote Lockhart's book:
This is what real mathematics feels and looks like; it is an art that consists of asking simple and elegant questions from our imaginations and finding even more beautiful solutions.
To quote Lockhart's book:
This is really what keeps me in the math game - the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.
To me, this kind of mathematical experience goes to the heart of what it means to be human. And I'll go even further and say that mathematics, this art of abstract pattern-making - even more than story-telling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.