no. 3
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 ℚ[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 ℝ 2 .

Using the above results, we show that a convex polygon P∈ℂ 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 ℚ[e 2πi/n ].

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     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},
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     mrnumber = {98h:52037},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1586/}
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Kenyon, Richard. Tilings of convex polygons. Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 929-944. doi : 10.5802/aif.1586. https://aif.centre-mersenne.org/articles/10.5802/aif.1586/

[1] C. Bavard, E. Ghys, Polygones du plan et polyèdres hyperboliques., Geometriae Dedicata, 43 (1992), 207-224. | MR | Zbl

[2] J.H. Conway, J.C. Lagarias, Tilings with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A., 53 (1990), 183-206. | MR | Zbl

[3] M. Laczkovich, Tilings of polygons with similar triangles, Combinatorica, 10 (1990), 281-306. | MR | Zbl

[4] R. Kenyon, A group of paths in ℝ2, Trans. A.M.S., 348 (1996), 3155-3172. | MR | Zbl

[5] W.P. Thurston, Shapes of polyhedra, Univ. of Minnesota, Geometry Center Research Report GCG7.

[6] W.T. Tutte, The dissection of equilateral triangles into equilateral triangles, Proc. Camb. Phil. Soc., 44 (1948), 463-482. | MR | Zbl

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