## Origami and Geometric Constructions## a comparison between straight edge and compass constructions and origamiIn 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 sequecne of steps, each of which is one of the
following (taken from What is Mathematics? by Courant and Robbins,
Oxford Univ. Press, New York, 1941):
- 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.
cannot be done using a SE&C. For example, students
are traditionally told that certain things are impossible to do with a
SE&C, like trisecting an arbitrary angle, or doubling the volume of a
cube (i.e., constructing the cube root of 2). Hearing this usually makes
students spend hours and hours trying to trisect an angle using SE&C (I
know I did!), but alas, this really is impossible. Unfortunately,
proving that it is impossible requires some high-level math, like
abstract algebra, which is beyond the reach of most sane adults.
However, 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 web tutorial.
## Huzita's Origami AxiomsPlease note that paper folding can be quite complex! There are many intricate paper folding maneuvers, and harnessing the power of origami through a list of axioms, like we did above for SE&C, is tricky. ( Footnote 1) The Italian-Japanese mathematician Humiaki Huzita has formulated what is currently the most powerful known set of origami axioms (from his article, "Understanding Geometry through Origami Axioms" in theProceedings of the First International Conference on Origami in
Education and Therapy (COET91), J. Smith ed., British Origami
Society, 1992, pp. 37-70):
- (O1) Given two points
**p1**and**p2**we can fold a line connecting them.
- (O2) Given two points
**p1**and**p2**we can fold**p1**onto**p2**.
- (O3) Given two lines
**l1**and**l2**we can fold line**l1**onto**l2**.
- (O4) Given a point
**p1**and a line**l1**we can make a fold perpendicular to**l1**passing through the point**p1**.
- (O5) Given two points
**p1**and**p2**and a line**l1**we can make a fold that places**p1**onto**l1**and passes through the point**p2**.
- (O6) Given two points
**p1**and**p2**and two lines**l1**and**l2**we can make a fold that places**p1**onto line**l1**and places**p2**onto line**l2**.
## Exercise for Axiom (O5)l1 and take
p1 to be a point in the middle and close to l1. Then
choose p2 to be anywhere on the left or right edge of the square
and perform axiom (O5).
Then choose a different p2. Repeat this 8 or 9 times. What do
you see?
Solution and moral of this exercise ## Exercise for Axiom (O6)Don't do this exercise until you've done the previous one! If you look closely, axiom (O6) is just like (O5) but times two: In (O5)p1 is the focus and l1 is the directrix of a parabola. In
(O6) we have this again, but also p2 is the focus and l2 is
the directrix of another parabola!
Thus axiom (O6) solves the following problem: Given two parabolas drawn
in the plane, find a line that is tangent to
The solution is left to the reader, but note that it might require some algebraic geometry.
## How to Trisect an Angle via FoldingThis method is by H. Abe (from "Trisection of angle by H. Abe" (in Japanese) by K. Fusimi, in Science of Origami, a supplement toSaiensu (the Japanese version of Scientific American), Oct.
1980, p. 8).
(1) Let the angle you want to trisect originate from the lower left
corner. Call this angle A. (Note that here we assume that A is acute,
but this method is easily extendable to obtuse angles.) Make two
parallel, equidistant horizontal creases at the bottom.
## How to double a cube via folding"Doubling a cube" means doubling the volume of a given cube. The real challenge for doing this is constructing a line with length equal to the cube root of 2. This is a solution of a cubic equation, and thus we need to be able to solve cubic equations to succeed. SE&C can't do it, but origami can.
Peter Messer has an amazingly elegant way to do this with paper folding.
(From Problem 1054, in
The proof of this is left to the reader! These are just a few examples of how axiom (O6) can be used to perform complicated constructions. There are many more, and readers encouraged to experiment and discover their own!
This tutorial is only an introduction to the subject of origami geometric
constructions. For a more rigorous development, see the original
articles by Humiaki Huzita and Robert Geretschlager (referenced in the
origami geometry section of the Origami Math
Bibliography). Although for a complete understanding of these concepts
the reader should investigate any of the numerous writings on SE&C
constructions, for example Chapter III in
There is a more historical context in which we can view origami
constructions. Although SE&C is not enough to trisect angles, one
Also, we do not know if Huzita's axiom list is complete! Might there
exist some other folding operation that would allow us to solve equations
of degree 5 or higher? The general consensus seems to be that no such
"7th axiom" exists, but no one has proven this.
Also, several researchers have been making advances into the question of whether or not there are other "origami axioms" that are yet undiscovered. Erik Demaine, Robert Lang, and Koshiro Hatori (see his web page) have been doing exciting things in this area. For a much more rigorous presentation of origami geometric constructions, see Roger Alperin's paper "A mathematical theory of origami constructions and numbers" which appeared in the New York Journal of Mathematics in 2000. It can be downloaded (in PDF and ps format) here. These pages Copyright 1997, 2003 Thomas Hull. Back to Origami Math pages |