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 on associe une forme quadratique , qui est positive semi-définie si est pavable par des polygones rationnels.
On démontre qu’un polygone convexe d’angles multiples de est pavable par des triangles d’angles multiples de si et seulement si est semblable à un polygone dont les sommets sont dans .
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 a quadratic form , which must be positive semidefinite if is tileable with rational polygons.
The above results also hold replacing the rationality condition with the following: a polygon is coordinate-rational if a homothetic copy of has vertices with rational coordinates in .
Using the above results, we show that a convex polygon with angles multiples of and an edge from to can be tiled with triangles having angles multiples of if and only if vertices of are in the field .
@article{AIF_1997__47_3_929_0, author = {Kenyon, Richard}, title = {Tilings of convex polygons}, journal = {Annales de l'Institut Fourier}, pages = {929--944}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {47}, number = {3}, year = {1997}, doi = {10.5802/aif.1586}, mrnumber = {98h:52037}, zbl = {0873.52020}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1586/} }
TY - JOUR AU - Kenyon, Richard TI - Tilings of convex polygons JO - Annales de l'Institut Fourier PY - 1997 SP - 929 EP - 944 VL - 47 IS - 3 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1586/ DO - 10.5802/aif.1586 LA - en ID - AIF_1997__47_3_929_0 ER -
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://www.numdam.org/articles/10.5802/aif.1586/
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