论文标题
具有相似三角形的常规多边形的不规则瓷砖
Irregular tilings of regular polygons with similar triangles
论文作者
论文摘要
我们说，如果可以将$ a $分为有限的许多非重叠三角形，类似于$ t $，则三角$ t $ t $ tiles a polygon $ a $。我们表明，如果$ n> 42 $，那么最多有三个非同类三角$ t $，因此$ t $的角度是$π$的理性倍数和$ t $ tiles tiles the普通$ n $ -gon。 如果这些零件具有两个角度，$ \ al $和$ \ be $，则将其铺成类似的三角形称为常规，使得在瓷砖的每个顶点的角度$ \ al $与$ \ be $相同。否则瓷砖是不规则的。众所周知，对于每一个常规的多边形$ a $，都有无限的三角形，这些三角形定期a $ a $。我们表明，如果$ n> 10 $，则仅当$ t $的角度是$π$的理性倍数时，常规$ n $ n $ n $ gon不规则。因此，每$ n> 42 $的三角形数量不规则地占用了常规$ n $ gon。
We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$, then there are at most three nonsimilar triangles $T$ such that the angles of $T$ are rational multiples of $π$ and $T$ tiles the regular $N$-gon. A tiling into similar triangles is called regular, if the pieces have two angles, $\al$ and $\be$, such that at each vertex of the tiling the number of angles $\al$ is the same as that of $\be$. Otherwise the tiling is irregular. It is known that for every regular polygon $A$ there are infinitely many triangles that tile $A$ regularly. We show that if $N>10$, then a triangle $T$ tiles the regular $N$-gon irregularly only if the angles of $T$ are rational multiples of $π$. Therefore, the numbers of triangles tiling the regular $N$-gon irregularly is at most three for every $N>42$.
