The set of constructible numbers can be understood as the smallest field extension to the rational numbers that's closed on square roots. Equivalently, it can be understood as the set of coordinates that can belong to points that can be marked out with nothing but a compass and an unmarked straightedge.

But when we consider the set algebraically, why is it closed only on *square* roots specifically? Why roots of degree 2? Could it be because any angle that can be constructed with a compass can be contained in a 2D space?

Does this perhaps mean that if we somehow invented a "3D compass," capable of somehow "drawing" arbitrary solid angles out into 3D space the same way you can draw flat angles on paper with a compass, then we could have "hyperconstructible" numbers that extend the constructible numbers to include coordinates that can belong to points that can be marked out with our new tool?

Pic unrelated