(A comparison between straight edge and compass constructions and origami)

In high school geometry students examine the types of geometrical operations that can be performed by using only a straight edge and a compass (SE&C). One learns how to draw a line connecting two points, how to draw circles, how to bisect angles, how to draw perpendicular lines, etc. In fact, you may remember that all SE&C constructions are a sequence of steps, each of which is one of the following:

Given two points we can draw a line connecting them.

Given two (nonparallel) lines we can locate their point of intersection.

Given a point p and a length r we can draw a circle with radius r centered at the point p.

Given a circle we can locate its points of intersection with another circle or line.

This list of axioms encompasses everything you can do with a SE&C.

That is, anything you do with a SE&C can be broken down into a sequence of the above operations. Using this axiom list, one can begin to talk about things that cannot be done using a SE&C. we can also make geometric constructions with origami, using the side of the paper as the straight edge and folding up to an angle to simulate a compass. Furthermore, trisecting angles and doubling cubes is possible with origami! Seeing this can lead to a greater understanding of why these things are impossible with SE&C, and is the main topic of this report.

Huzita's Origami Axioms:

Paper folding can be quite complex. There are many intricate paper folding exercises, and harnessing the power of origami through a list of axioms, like we did above for SE&C, is tricky. The Italian-Japanese mathematician Humiaki Huzita has formulated what is currently the most powerful known set...