Complexity of the bisection method

The bisection method is the consecutive bisection of a triangle by the median of the longest side. In this paper we prove a subexponential asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. The relevant parameter is γ/σ, where γ is the biggest and σ is the smallest angle of the triangle. We get this result by introducing a taxonomy of triangles that precisely captures the behavior of the bisection method. We also prove that the number of directions on the plane given by the sides of the triangles generated is finite. Additionally, we give purely geometric and intuitive proofs of classical results for the bisection method.