# 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 $n$ can you cut a square into $n$ 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!)

# Arithmetic Topology

One of my favorite viewpoints on math is that it is the study of metaphor.  By abstracting an existing metaphor, we invent new objects (groups were invented because Galois realized that ‘polynomials are like plane figures’); by inventing a new metaphor, we can import techniques from one field to another (the Italian school of algebraic geometry got a lot of mileage from ‘varieties are like manifolds’; the functor-of-points school, from ‘schemes are like functors on the category of rings’).  A poet might compare the evening to an etherized patient and leave it there, and in poetry this is fine, for in poetry we revel in mystery, allusion, in half-knowledge.  But in math, we can’t stand these things, and so we must grab our things and run to the nearest hospital, examining all the gurneys we can in the hope of better understanding the twilight.

One particular metaphor that’s captivated me recently goes by the unassuming name of ‘arithmetic topology,’ essentially due to Barry Mazur in an unpublished paper from 1963 (though as always, neverendingbooks has a more precise history of the idea).  The one-sentence idea, which should be all the convincing anyone needs to do math, is that the integers are like three-space and prime numbers are like knotted circles.  I know, I know.  And yes, it does go a lot further than that — more generally, there’s a fascinating story about Galois-group-looking things relating to the algebraic topology of honest-to-God topological things.  In the interest of always doing something that’s slightly too hard for you, I’m trying to understand this right now, and in doing so will hopefully learn a lot of algebraic geometry, algebraic number theory, and knot theory ‘by accident.’  I’ll write about it here as I do so, and I invite you to learn with me if you don’t know what I’m talking about / correct my errors and give me references if you do.  Below the fold, an introduction to what seems to be going on, and a rough outline of what I’ll probably be writing about.

# The Goodstein Sequence, or, using infinity to prove finitary results

I’d like to inaugurate this blog by demonstrating a wacky and somewhat radical proof technique.  If you feel threatened as you wade through a world of skeptics of infinity, I present to you your weapon.  It’s a cute little sequence of natural numbers called a Goodstein sequence.