Monsky’s Theorem, or the genius of bizarre thinking

I was shown this by the illustrious Nir Avni, and it was so beautiful and bizarre that I presented it at our undergrad math seminar that week.  Now a friend of mine wants to turn it into art, and I’m writing it up in service of that lofty goal.  Kate, may your project join the slim annals of awesome math-themed visual art.

The question is: for what can you cut a square into triangles of equal area?  By slicing it into rectangles and cutting each rectangle into two triangles, it’s easy to see that you can get any even number, as the pictures below show.

So can you do it with an odd number of triangles?  Think about it a bit, and after the jump, we’ll think about it together.  (This post will be at a lower level than most of my other writing here — it should be understandable by math undergrads and ideally even the Laity.  Let me know how I’m doing!)