Tilings of convex polygons
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(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 π/n est pavable par des triangles d’angles multiples de π/n si et seulement si P est semblable à un polygone dont les sommets sont dans [e 2πi/n ].

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 2 .

Using the above results, we show that a convex polygon P with angles multiples of π/n and an edge from 0 to 1 can be tiled with triangles having angles multiples of π/n if and only if vertices of P are in the field [e 2πi/n ].

     author = {Kenyon, Richard},
     title = {Tilings of convex polygons},
     journal = {Annales de l'Institut Fourier},
     pages = {929--944},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {3},
     year = {1997},
     doi = {10.5802/aif.1586},
     zbl = {0873.52020},
     mrnumber = {98h:52037},
     language = {en},
     url = {archive.numdam.org/item/AIF_1997__47_3_929_0/}
Kenyon, Richard. Tilings of convex polygons. Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 929-944. doi : 10.5802/aif.1586. http://archive.numdam.org/item/AIF_1997__47_3_929_0/

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