30 1. Trisecting Angles as permissible, where we slide the straightedge along the circle? To resolve this would take a more dedicated scholar, possibly one who knows ancient Greek as well as mathematics. My general sense of what the Greeks were about, however, leads me to think that such a placement of a ruler was not something they would have tolerated. There are several places on the internet where you can read about the trisection problem. One is Wikipedia, of course, http://en.wikipedia.org/wiki/Angle trisection another is http://www.jimloy.com/geometry/trisect.htm Also see George E. Martin, Geometric Constructions, Springer-Verlag New York (1997) and Nicholas D. Kazarinoff, Ruler and the Round: Classic Prob- lems in Geometric Constructions, Dover Publications (2003). The first book, in fact, contains thorough discussions of what can be done in constructions using a whole array of tools besides the compass and straightedge. 1.10. Some historical notes The exact origin and age of the trisection problem is unknown. As early as 430 B.C., the Greek Hippias of Elis sought, and was successful in find- ing, ways of trisecting angles by using tools other than the compass and straightedge. So the problem predates him and is over 2500 years old. This problem is connected to the problem of determining which regular polygons can be constructed by ruler and compass alone. An equilateral triangle can be constructed. A square can also be constructed, as well as a regular pentagon, though constructing a pentagon is more diﬃcult. Such constructions were done in the books of Euclid. To see the connection between trisecting angles and constructing regular polygons, suppose it is possible to construct, with straightedge and compass alone, a regular polygon with n sides. It is then possible to locate the center of the polygon by finding the point of intersection of the perpendicular bisectors of two of the sides. Once this is done, connect the center of the polygon to each of the vertices. This creates n isosceles triangles. All the angles at the center are equal and, since there are n of them, they must contain 2π/n radians. So if a regular polygon of n sides can be constructed, an angle of 2π/n radians can be constructed. Since a regular pentagon can be constructed, an angle of 2π/5 radians = 72◦ can be constructed. The converse of this statement is also true. That is, if an angle of 2π/n radians can be constructed, then a regular polygon of n sides can be con- structed using a compass and straightedge alone. To do this, pick a point O and draw a ray emanating from O. Using this ray as one side, construct

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