Origami and Geometric Constructions.

Essay by lawrencefu April 2003

(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...