Tilings of convex polygons

Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon P a quadratic form q(P), which must be positive semidefinite if P is tileable with rational polygons. The above results also hold replacing the rationality condition with the following : a polygon P is coordinate-rational if a homothetic copy of P has vertices with rational coordinates in R-2. Using the above results, we show that a convex polygon P is an element of C with angles multiples of pi/n and an edge from 0 to 1 can be tiled with triangles having angles multiples of pi/n if and only if vertices of P are in the field Q[e(2 pi/n)].