Irregular Tilings of Regular Polygons with Similar Triangles

Laczkovich, Miklós (2021) Irregular Tilings of Regular Polygons with Similar Triangles. DISCRETE AND COMPUTATIONAL GEOMETRY. ISSN 0179-5376

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Abstract

We say that a triangle T tiles a polygon A, if A can be dissected into finitely many nonoverlapping triangles similar to T. We show that if N > 42, then there are at most three nonsimilar triangles T such that the angles of T are rational multiples of p and T tiles the regular N-gon. A tiling into similar triangles is called regular, if the pieces have two angles, alpha and beta, such that at each vertex of the tiling the number of angles alpha is the same as that of beta. Otherwise the tiling is irregular. It is known that for every regular polygon A there are infinitely many triangles that tile A regularly. We show that if N > 10, then a triangle T tiles the regular N-gon irregularly only if the angles of T are rational multiples of pi. Therefore, the number of triangles tiling the regular N-gon irregularly is at most three for every N > 42.

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