February 6, 2018

The Limitations of Ruler-and-Compass Constructions

Most people know that the ancient Greeks excelled at geometry, producing the foundations of the Euclidean plane and solid geometry we learn today in school. Those geometers focused on their constructions being physically possible, and their tools consisted of an unmarked ruler and a compass (to draw circles). Some geometrical constructions plagued the ancient Greeks, seeming entirely out of reach. Among these:– Squaring the circle: drawing a square of area pi (equivalently, construct a line segment of length square-root-of-pi)– Doubling the cube: construct a line segment of length cube-root-of-2– Angle Trisection: given an angle, trisect it. It wasn’t until the 1800’s that these constructions were shown to be impossible using algebraic techniques.