@MISC{Su_tilingsof, author = {Zhanjun Su and Ren Ding}, title = {TILINGS OF PARALLELOGRAMS WITH SIMILAR TRIANGLES}, year = {} }

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Abstract

Abstract. We say that a triangle ∆ tiles the polygon P if P can be decomposed into finitely many non-overlapping triangles similar to ∆. Let P be a parallelogram with angles δ and pi − δ (0 < δ ≤ pi/2) and let ∆ be a triangle with angles α, β, γ (α ≤ β ≤ γ). We prove that if ∆ tiles P then either δ ∈ {α, β, γ, pi−γ, pi−2γ} or dimLP =dimL∆. We also prove that for every parallelogram P, and for every integer n (where n ≥ 2, n 6 = 3) there is a triangle ∆ so that n similar copies of ∆ tile P. AMS Mathematical Subject Classification: 52C20. Key words and phrases: Parallelogram, similar triangle, linear space, tiling. We say that a triangle ∆ tiles the polygon P if P can be decomposed into finitely many non-overlapping triangles similar to ∆. In [8] Szegedy considered the tilings of the square with similar right triangles and in [5] Laczkovich discussed the tilings of rectangles with similar triangles. In the present paper we consider the tilings of parallelograms with similar triangles. Let P be a parallelogram with angles δ and pi − δ (0 < δ ≤ pi2) and let LP = {q1δ + q2pi | q1, q2 ∈ Q}, where LP is a linear space over the rational field Q. Let ∆ be a triangle with angles α, β, γ (α ≤ β ≤ γ) and let L ∆ = {q1α+ q2β + q3γ | qi ∈ Q, i = 1, 2, 3}, where L ∆ is a linear space over the rational field Q.