Proving the non-degeneracy of the longest-edge trisection by a space of triangular shapes with hyperbolic metric

Abstract

From an initial triangle, three triangles are obtained joining the two equally spaced points of the longest-edge with the opposite vertex. This construction is the base of the longest-edge trisection method. Let Δ be an arbitrary triangle with minimum angle α. Let Δ′ be any triangle generated in the iterated application of the longest-edge trisection. Let α′ be the minimum angle of Δ′. Thus α′≥α/c with c=π/3arctan3/11 is proved in this paper. A region of the complex half-plane, endowed with the Poincare hyperbolic metric, is used as the space of triangular shapes. The metric properties of the piecewise-smooth complex dynamic defined by the longest-edge trisection are studied. This allows us to obtain the value c.