One can create an axiomatic system of geometry through such coincident folds (as an alternative to straight-edge and compass) and it turns out to be more powerful than the Euclidean system.

One can construct cube roots, trisect angles.

Depending on the choice of paper folding axioms one can go beyond cube roots and k-secting angles to the entire set of algebraic numbers.

https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms