20 Constructions
Proving a negative is often difficult, but some of the greatest triumphs in mathematics
do just that. This means proving something cannot be done. Squaring the circle is
impossible but how can we prove this?
The Ancient Greeks had four great construction problems:- trisecting the angle (dividing an angle into three equal smaller angles),
- doubling the cube (building a second cube with twice the volume of the first),
- squaring the circle (creating a square with the same area as a particular circle),
- constructing polygons (building regular shapes with equal sides and angles).
To perform these tasks they only used the bare essentials:- a straight edge for drawing straight lines (and definitely not to measure lengths),
- a pair of compasses for drawing circles.
If you like climbing mountains without ropes, oxygen, mobile phones and
other paraphernalia, these problems will undoubtedly appeal. Without modern
measuring equipment the mathematical techniques needed to prove these results
were sophisticated and the classical construction problems of antiquity were only
solved in the 19th century using the techniques of modern analysis and abstract
algebra.