Tilings of convex polygons

Kenyon, Richard

Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 929-944.

Un polygone est appelé rationnel si les rapports des longueurs d’arêtes sont rationnels. On démontre qu’un polygone convexe est pavable par des polygones rationnels si et seulement s’il est lui-même rationnel. À tout polygone $P$ on associe une forme quadratique $q\left(P\right)$, qui est positive semi-définie si $P$ est pavable par des polygones rationnels.

On démontre qu’un polygone convexe $P$ d’angles multiples de $\pi /n$ est pavable par des triangles d’angles multiples de $\pi /n$ si et seulement si $P$ est semblable à un polygone dont les sommets sont dans $\mathbb{Q}\left[e^{2\pi i/n}\right]$.

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 $q\left(P\right)$, 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 $P\in ℂ$ 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 $\mathbb{Q}\left[e^{2\pi i/n}\right]$.

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