×

zbMATH — the first resource for mathematics

Tilings of convex polygons. (English) Zbl 0873.52020
Summary: 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 \({\mathbb{R}}^2\). Using the above results, we show that a convex polygon \(P\in\mathbb{C}\) 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 \(\mathbb{Q}[e^{2\pi i/ n}]\).

MSC:
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] C. BAVARD, E. GHYS, Polygones du plan et polyèdres hyperboliques., Geometriae Dedicata, 43 (1992), 207-224. · Zbl 0758.52001
[2] J.H. CONWAY, J.C. LAGARIAS, Tilings with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A., 53 (1990), 183-206. · Zbl 0741.05019
[3] M. LACZKOVICH, Tilings of polygons with similar triangles, Combinatorica, 10 (1990), 281-306. · Zbl 0721.52013
[4] R. KENYON, A group of paths in ℝ2, Trans. A.M.S., 348 (1996), 3155-3172. · Zbl 0896.52024
[5] W.P. THURSTON, Shapes of polyhedra, Univ. of Minnesota, Geometry Center Research Report GCG7. · Zbl 0931.57010
[6] W.T. TUTTE, The dissection of equilateral triangles into equilateral triangles, Proc. Camb. Phil. Soc., 44 (1948), 463-482. · Zbl 0030.40903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.