1. Tilings of Parallelograms with Similar Right Triangles.
- Author
-
Su, Zhanjun, Yin, Chan, Ma, Xiaobing, and Li, Ying
- Subjects
- *
TILING (Mathematics) , *PARALLELOGRAMS , *TRIANGLES , *POLYGONS , *MATHEMATICAL decomposition , *MATHEMATICAL models - Abstract
We say that a triangle $$T$$ tiles the polygon $$\mathcal A $$ if $$\mathcal A $$ can be decomposed into finitely many non-overlapping triangles similar to $$T$$ . A tiling is called regular if there are two angles of the triangles, say $$\alpha $$ and $$\beta $$ , such that at each vertex $$V$$ of the tiling the number of triangles having $$V$$ as a vertex and having angle $$\alpha $$ at $$V$$ is the same as the number of triangles having angle $$\beta $$ at $$V$$ . Otherwise the tiling is called irregular. Let $$\mathcal P (\delta )$$ be a parallelogram with acute angle $$\delta $$ . In this paper we prove that if the parallelogram $$\mathcal P (\delta )$$ is tiled with similar triangles of angles $$(\alpha , \beta , \pi /2)$$ , then $$(\alpha , \beta )=(\delta , \pi /2-\delta )$$ or $$(\alpha , \beta )=(\delta /2, \pi /2-\delta /2)$$ , and if the tiling is regular, then only the first case can occur. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF