Tilings Of Regular Polygons And Right Triangles With Similar Triangles

We say that a triangle Δ tiles the polygon P, if P can be decomposed into finitely many non-overlapping triangles similar to Δ. The tiling is perfect if these triangles (?) are pairwise incongruent. Triangles Δ are called the tiles of the tiling.A tiling is called regular if there are two angles of the triangles, say α and β, such that at each vertex (?) of the tiling the number of triangles having (?) as a vertex and having angle α at (?) is the same as the number of triangles having angle β at (?). Otherwise the tiling is called irregular.Denote by Rn the regular n-gon and by (α,β,γ) the triangle of angles α,β,γ. In chapter 2 we look at tilings of Rn (≥5) with similar right triangles and get the following results:The right triangles (3/π,6/π,2/π) and (π/4,π/4,π/2) can not tile Rn, for every integer n (n≥ 5,n≠6).For every integer n (≥ 7), if right triangle (α,β,γ) can regularly tile Rn, then (α,β,π/2) = (n/2π,2/π-n/2π,2/π).For every integer n (≥7), if Rn has an irregular tiling with similar right triangles (α,β,2/π),then (α β)=(2/π-n/π,n/π),(m/2π,2/π-m/2π) or (3/π-3n-2π,6/π+3n/2π).In chapter 3, according to the similar tiling of rectangles and the perfect tiling of triangles, we consider the perfect tiling of right triangle(α,2/π-α,2/π)(0 <α<π/4) with non-right triangles. In 1990 Laczkovich [4] showed that if a non-right triangle A tiles a rectangle then its angles are given by one of the following triples: (4/π,3/π,12/5π),(6/π,6/π,3/2π), (8/π,4/π,8/5π),(12/π,4/π,3/2π). In 2005 Andrzej Z. [1] concluded that in any perfect tiling of right triangle with non-right triangles the number of tiles is at least 6. So we give the following results in this chapter:In the perfect tiling of right triangle(α,2/π-α,2/π) with triangle (6/π,6/π,3/2π), the number of tiles is greater than 6.In the perfect tiling of right triangle(α,2/π-α,2/π) with triangle (4/π,3/π,12/5π), the number of tiles is greater than 6.In the perfect tiling of right triangle(α,2/π-α,2/π) with triangle (8/π.4/π,8/5π), the number of tiles is greater than 6.In the perfect tiling of right triangle(α,2/π-α,2/π) with triangle (12/π,4/π,3/2π), the number of tiles is greater than 6.