The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be made by adjoining successive quadratic extensions, and we have to adjoin the root of $4x^3 - 3x - \alpha = 0$
In Origami we can trisect the angle and I am trying to understand the proof, here reduce to 4 figures. (pdf from MIT's 6.885)
I found myself challenged by computing the fold described here.
In order to reproduce these figures on my own computer, what is the shape of the grey right triangle in Figure 3? (Still needs and angle $\phi$).
Even more generally, is a question about about the fold in figure $2$. Given
- line segment $\overline{AB}$ and two lines, $\ell_1, \ell_2$.
How to compute the fold with so that $A' \in \ell_1$ and $B' \in \ell_2$. In the Huzita Axioms of Origami #6.