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Tilings of convex polygons. (English) Zbl 0873.52020
Summary: 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 $${\mathbb{R}}^2$$. Using the above results, we show that a convex polygon $$P\in\mathbb{C}$$ 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 $$\mathbb{Q}[e^{2\pi i/ n}]$$.

##### MSC:
 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry)