Abstract: If you want to build a square out of triangles, what restrictions are there on the areas of the triangles? For example, the triangle areas must sum to the area of the square. Are there other restrictions? Spoiler alert: yes. It turns out that for each combinatorial type of triangulation, there’s exactly one additional polynomial relation that must be satisfied by the areas of the triangles in any triangulation of the given type. My talk will describe recent discoveries and current mysteries surrounding this polynomial invariant.