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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 \(\mathbb{Q}[e^{2\pi i/ n}]\).

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
Full Text: DOI Numdam EuDML
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