## Tilings of polygons with similar triangles. II.(English)Zbl 0927.52028

Summary: Let $$A$$ be a polygon, and let $$s(A)$$ denote the number of distinct nonsimilar triangles $$\Delta$$ such that $$A$$ can be dissected into finitely many triangles similar to $$\Delta$$. If $$A$$ can be decomposed into finitely many similar symmetric trapezoids, then $$s(A)=\infty$$. This implies that if $$A$$ is a regular polygon, then $$s(A)=\infty$$. In the other direction, we show that if $$s(A)=\infty$$, then $$A$$ can be decomposed into finitely many symmetric trapezoids with the same angles.
We introduce the following classification of tilings: a tiling is regular if $$\Delta$$ has 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. We prove that for every polygon $$A$$ the number of triangles that tile $$A$$ irregularly is at most $$c\cdot n^6$$, where $$n$$ is the number of vertices of $$A$$. If $$A$$ has a regular tiling, then $$A$$ can be decomposed into finitely many symmetric trapezoids with the same angles.
[For part I see the author, Combinatorica 10, No. 3, 281-306 (1990; Zbl 0721.52013)].

### MSC:

 52C20 Tilings in $$2$$ dimensions (aspects of discrete geometry) 05B45 Combinatorial aspects of tessellation and tiling problems

### Keywords:

tilings of polygons; similar triangles; regular tiling

Zbl 0721.52013
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### Online Encyclopedia of Integer Sequences:

Numbers of smaller squares into which a square may be dissected.