Results on the Convergence of the Longest-Edge Trisection Method for Triangles

Abstract

Let ABC be a triangle with vertexes A, B, and C. The longest-edge trisection of ABC is as follows: choose the longest side (say AB) of ABC, let D and E be the points which divide in three AB, then replace ABC by three triangles ACD, CDE and BCE. If longest-edge trisection is iteratively applied to an initial triangle, then it is proved that the diameters of the resulting triangles are between two sharped experimental decreasing functions. This paper responds to the question of how fast do the diameters of a triangle mesh tend to zero, as repeated trisection is performed.