Angle trisection with Origami and related topics

One of the famous old problems from antiquity is to trisect a given angle by using only a compass and a ruler. In other words: By using only a tool to draw a straight line segment through any two points and a tool to draw circles and arcs and duplicating lengths, one wants to construct in a finite number of steps two half-lines that comprise one third of an angle that itself is already given in terms of two half lines comprising it. The proof that this venture is indeed impossible in general is a prime example of how axiomatic mathematics works. We start with algebraization of this problem. Let a subset M ⊆ A2(R) of the affine plane and two different points 0, 1 ∈ M be given. The fact that we have 0 and 1 is equivalent to the existence of a coordinate system including the unity length by setting (0, 0) = 0,