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I was first introduced to origami during a series of art and craft classes in kindergarten. Although I stopped doing it as a hobby as the years passed by, I got back into it during one of my CAS experiences, where I made origami orchids as part of a mixed media art piece I had created for an art competition. Considering my interest in Architecture and the rising prevalence of origami-inspired buildings, I figured that I could use my Extended Essay as an opportunity to learn about the mathematics behind paper folding. I also thought that it would be a fun and engaging way of learning new mathematical concepts because I would have to do the mathematics involved rather than just writing about it and origami is a rather unconventional way of analysing mathematics.
With this said, my Extended Essay aims to prove how it is possible to double a cube with origami when it is impossible to do so using a compass and straightedge. I have been able to do so by studying and proving how origami solves both quadratic and cubic equations, learning the origami technique of folding a piece of square paper into nths and proving Haga’s theorem. All of these are the mathematical concepts of origami that I needed to be familiar with to understand how to construct a cube with a volume twice as large as that of a given cube. The number of supposedly impossible problems that can be simplified by a single piece of paper is astounding.
How is the challenge of doubling a cube using a compass and straightedge overcome using origami?
Origami is the traditional art of folding paper which largely has its roots from China and Japan. The traditional style of origami involves folding one sheet of square paper into a finished sculpture without making any cuts or securing it in any manner; however, the art has now evolved to include several types with different tools and techniques. For example, rigid origami comprises of the folding of a single sheet of paper so that it minimises effortlessly with stiff movement, that is, without the areas between the creases being bent. This is the origami technique that a Japanese astrophysicist named Koryo Miura used in 1970 to design space satellites and solar panels that could automatically be packaged to small sizes after deployment.
Doubling a cube and trisecting an angle are two of the three ancient Greek geometric problems, the third being squaring a circle. According to Robertson (1999), there is a book titled ‘Platonicus’ written by a Greek astronomer named Eratosthenes that describes the rise of the problem of doubling a cube when a plague spread in Athens in 430 BC. The Athenians travelled to Delos to see the oracle of Apollo, which said that the gods would get rid of the plague only if they made the altar twice as large. However, the plague didn’t cease because the citizens doubled the sides of the altar instead of the size of the altar, which led to the altar having a volume that was eight times as large since (2x)3 = 8x3. Using their compasses and straightedges (rulers without any marks) the Greeks were never able to accomplish the task, which was ever since referred to as the Delian problem.
On this note, I shall investigate and prove why cube duplication is possible, just not through Euclidean construction.
Construction Using a Compass and a Straightedge
Euclid of Alexandria was a Greek mathematician who lived during the period of 300 BCE. He took interest in the possible constructions that could be made using a compass and a straightedge, which was how he was able to establish a set of postulates that outlined the possibilities that he discovered. A postulate, also referred to as an axiom, is a statement that is considered to be true without formal proof. This occurs because postulates are meant to be so simple and direct that the truth in it cannot be questioned by anyone, as it happens to be the case with the 5 postulates in geometry below.
- Postulate 1: Given two distinct points, P1 and P2, exactly one straight line can be drawn between them.
- Postulate 2: Any line segment can be drawn out continuously.
- Postulate 3: Given a point, P1, and a line segment that starts from P1, L1, a circle can be drawn with P1 as its centre and L1 as its radius.
- Postulate 4: All right angles are congruent.
- Postulate 5: If a straight line intersects with two other line segments in a way that the interior angles on one side are less than two right angles, then the two line segments will intersect on that same side if extended.
With the postulates stated above, it is apparent that the shapes that can be constructed through Euclidean construction are only lines and circles. Therefore, the points of intersection that are produced by two line segments, two circles or a circle and a line would be the only way to construct new points from existing points.
According to Spencer (1999), given existing points whose coordinates were all rational numbers and any of the axioms of Euclidean construction were applied, the coordinates of the new points formed would be numbers that can be obtained from the rational numbers if I did any of the following arithmetic operations: addition, subtraction, multiplication, division and extracting square roots. The arithmetic operations come in when solving for one of the coordinates of the point of intersection, in which one ends up with a linear or quadratic equation.
Knowing this, construction with a compass and straightedge thus limits our ability to double a cube because we would be required to construct point with coordinates that are irrational numbers, as proven using Field Theory.
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