Geometric Constructions

The traditional description of the theory of geometric constructions starts out by listing the different so-called constructional methods: ruler, compass, dividers, parallel ruler, permitted curves and the like, and aims to set down theorems on the possibility of applying these tools and their limitations. The best known are theorems of the impossible kind (trisection of the angle, doubling of the cube, etc.) and theorems about the substitutability or removability of constructional methods (construction with the compass alone). The first apply algebraic methods (Galois theory), the second, clever geometric constructions. The description of the theory of geometric constructions in algebra texts is certainly adequate from the algebraic viewpoint, but the basic concepts must be sharpened for foundational investigations. To this end the circle of ideas from programming languages — which I may here presume — can nowadays give useful service.