So I was inspired by Numberphile to learn about these construction. I find it amazing that folding alone is stronger than compass and ruler because it can solve the problem of trisecting an angle and doubling a cube with ease. In the video they briefly explain that Origami construction involves solving cubic equations while those of compass and ruler is only quadratic. As I read more, I found the key lies at the 6th axiom of Origami. It involves finding tangent of two parabolas, and something called "neusis". I read wiki but still have little clue on them. So now I have some questions that need answering:
What is neusis and what does it have to do with higher degree equations than quadratic?
What is the general formula for parabola (I mean when its axis is kind of diagonal, not the popular cases like $y = ax^2 + bx + c$ or $y = \sqrt{x}$)? And how to find the tangent of two parabolas? (also why the equation is cubic)
How can neusis be applied on compass and ruler construction? If so will it have the same potent as Origami construction?
What are the limitations of Origami construction (things that cannot be constructed using Origami) ? And what constructions are stronger than this?