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, , 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'