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I heard that straight edge + compass can solve up to quadratic equations. I've also heard the Origami/Paper-folding can solve cubic equations. But can it solve higher-degree polynomial equations (e.g. quartic, quintic) in general? If not, is there a geometric construction framework that could solve these higher degree polynomial equations?

My intuition says that degree 5 and higher can't be solved because of the Abel-Ruffini theorem proving that no general formula exists for degree-5 and higher polynomial equations that only involves arithmetic and radicals.

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    $\begingroup$ Notably, Mathologer published a video touching on this very topic a few days ago -- youtube.com/watch?v=IUC-8P0zXe8. He did say cubics were possible, which he showed with an example, but I forget if he touched on if that method generalized even higher up. $\endgroup$ – Eevee Trainer Apr 28 at 10:06
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Ordinary origami axioms only include single fold operations. It only allow one to solve generic cubic equations.

If a paper One-, Two-, and Multi-Fold Origami Axioms, Roger C.Alperin and Robert J.Lang has shown if one allow operations that involve multiple simultaneous folds, it is possible to use the it to solve higher degree equations. In general, one need operations that involve $n-2$ simultaneous folds to solve a polynomial equation of degree $n$.

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