TRISECTIONS AND TOTALLY REAL ORIGAMI ROGER C. ALPERIN Summary: TRISECTIONS AND TOTALLY REAL ORIGAMI ROGER C. ALPERIN 1. Constructions in Geometry The study of methods that accomplish trisections is vast and extends back in time approximately 2300 years. According to Knorr, [9], even Plato had a favorite method. My own favorite method of trisection from the Ancients is due to Archimedes. He performed a neusis between a circle and line. Basically a marked ruler method 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. The standard marked ruler method allows only neusis between lines; a trisection method using a neusis between lines is due to Apollonius. Here is Archimedes' trisection method. Given an acute angle between rays r, s meeting at O, construct a circle of radius one at O; the ray r is extended to give a line which is a diameter of the circle; the circle meets the ray s at the point P. Now place a ruler through P with the unit distance CD lying on the circle at C and diameter at D, on the opposite ray to r. The angle ODP is the desired trisection; you can easily check this using the isosceles triangles DCO, COP and the exterior angle of the triangle PDO. As you see when you try this for yourself, there is a bit of `fiddling' Collections: Mathematics