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**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)].

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 |

### Citations:

Zbl 0721.52013
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\textit{M. Laczkovich}, Discrete Comput. Geom. 19, No. 3, 411--425 (1998; Zbl 0927.52028)

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