Geometric constructions using an unmarked straightedge and a compass have been studied for thousands of years. In these constructions, we can draw circles and lines starting with any two points, and we can create new points where they intersect. An n-gon is said to be constructible if can be constructed in a finite number of steps using these guidelines. We begin with constructions of several n-gons, and examine the field theory behind geometric constructions. Galois theory then provides a precise classification of which n-gons are constructible and which are not. Next is an exploration of origami construction, which examines a single-fold construction axiom, and establishes the classification of origami-constructible n-gons. For example, a heptagon is not constructible using traditional construction techniques, but it is constructible using origami. Finally, we investigate new axioms, which might allow for additional constructions, and examine their implications.