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Tilings of polygons with similar triangles. (English) Zbl 0721.52013
The author’s abstract: “We prove that if a polygon P is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices of P. If P is a rectangle then, apart from four “sporadic” cases, the triangles of the decomposition must be right triangles. Three of these “sporadic” triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angle \(\alpha\) such that tan \(\alpha\) is a totally positive algebraic number.
Most of the proofs are based on the following general theorem: if a convex polygon P is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices of P belong to the field generated by the cotangents of the angles of the triangles in the decomposition.”

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
Full Text: DOI
[1] M. Dehn: Über Zerlegung von Rechtecken in Rechtecke,Math. Ann.,57 (1903), 314–332. · JFM 34.0547.02 · doi:10.1007/BF01444289
[2] H.Hadwiger:Vorlesungen über Inhalt, Oberfläche und Isoperimetrie., Springer,1957. · Zbl 0078.35703
[3] P. Monsky: On dividing a square into triangles,Amer. Math. Monthly,77 (1970), 161–164. · Zbl 0187.19701 · doi:10.2307/2317329
[4] L.Pósa:verbal communication, March1987.
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