Trisections and Totally Real Origami

1. CONSTRUCTIONS IN GEOMETRY. The study of methods that accomplish trisections is vast and extends back in time approximately 2300 years. My own favorite method of trisection from the Ancients is due to Archimedes, who performed a “neusis” between a circle and line. Basically a neusis (or use of a marked ruler) allows the marking of points on constructed objects of unit distance apart using a ruler placed so that it passes through some known (constructed) point P . Here is Archimedes’ trisection method (see Figure 1): Given an acute angle between rays r and s meeting at the point O, construct a circle K of radius one at O, and then extend r to produce a line that includes a diameter of K . The circle K meets the ray s at a point P . Now place a ruler through P with the unit distance CD lying with C on K and D on the ray opposite to r . That the angle ODP is the desired trisection is easy to check using the isosceles triangles DCO and COP and the exterior angle of the triangle PDO. As one sees when trying this for oneself, there is a bit of “fiddling” required to make everything line up as desired; that fiddling is also essential when one does origami.