Tilings Of Plane Convex Polygons With Congruent Triangles

Let Pυ be a plane convex polygon with υ vertices.We say that a triangle Δ tiles the polygon plane convex polygon with,if Pυ can be decomposed into finitely many non-overlapping triangles similar to Δ.The tiling is congruent if these triangles Δare congruent.Triangles Δare Called the tiles of the tiling.In 2012 Hertel,E.and Richter,C.studied the problem of tiling convex υ-gon(v= 3,4,5,6)with congruent equilateral triangles.In chapter 2,we considered the problem of tiling convex υ-gon(υ=3,4,5,6,7,8)with congruent isosceles right triangles and get the number of triangles. In chapter 3,we considered the problem of tiling convex υ-gon(υ=3,4,5,6,7,8,9,10,11,12)with congruent(π/6,π/3,π/2)triangles and get the number of triangles.In particular,we determine two sets L={(υ,κ):υ∈{3,4,5,...),κ∈{1,2,3,...), and there exists a convex υ-gon that can be tiled with κ congruent isosceles right triangles) for υ=3,4,5,6,7,8.T={(υ,κ):υ∈{3,4,5,...),κ∈{1,2,3,...),and there exists a convex υ-gon that can be tiled with κ congruent(π/6,π/3,π/2)triangles)for υ= 3,4,5,6,7,8,9,10,11,12.Then we let Lυ={κ:(υ,κ)∈L),Tv={κ:(υ,κ)∈T),so Lυ and L,Tυ and T are identifying.