Tilings of convex polygons

Kenyon, Richard

Annales de l'Institut Fourier, Tome 47 (1997), p. 929-944 / Harvested from Numdam

Résumé

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 $q(P)$, 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 ${\Bbb Q}[e^{2\pi i/n}]$.

Abstract

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 ${\Bbb R}^2$.

Using the above results, we show that a convex polygon $P\in {\Bbb C}$ 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 ${\Bbb Q}[e^{2\pi i/n}]$.

Détail

Publié le : 1997-01-01

MR 98h:52037 | Zbl 0873.52020

DOI : https://doi.org/10.5802/aif.1586

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