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# A Comparison Between Straight Edge and Compass Constructions and Origami

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 SEC 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 SEC 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 SEC That is anything you do with a SEC 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 SEC 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 SEC and is the main topic of this report Huzitas 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 SEC is tricky The Italian-Japanese mathematician Humiaki Huzita has formulated what is currently the most powerful known set of origami axioms O1 Given two points p1 and p2 we can fold a line connecting them O2 Given two points p1 and

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