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= Origami-Constructible Numbers = An Origami-Constructible Number, aptly named after the Japanese art of paper folding, is a number that exists in the field of complex numbers C and can be created by the seven Huzita-Hatori axioms. In an origami construction, we can start with a sheet of paper that is considered to be infinitely large. The sheet of paper always starts with two points marked on it: 0 and 1. The real axis passes through the points 0,1 and the imaginary axis passes perpendicular to this at the point 0.

Origami numbers consist of simple series of folds in the paper; once unfolded the folds leave creases which become lines on our field C. In construction a point only exists at the intersection of two lines (excluding the original points 0,1)."'A complex number x is origami-constructible if, starting with a sheet of paper with 0 and 1 marked, we can make a series of folds such that two of the lines intersect at a point p that corresponds to x’s position on the complex plane.'"Many operations are possible using origami axioms, including but not exclusive to: addition, multiplication, inversion, square roots and cube roots.

The Seven Axioms of Origami Construction
The following seven Huzita-Hatori axioms, 1 to 6 rediscovered by Humiaki Huzita in 1991 and 7 rediscovered by Koshiro Hatori in 2001, are all that is needed to create any origami fold:


 * 1) Given two distinct points p1 and p2, there is a unique fold that passes through both of them.
 * 2) Given two distinct points p1 and p2, there is a unique fold that places p1 onto p2.
 * 3) Given two lines l1 and l2, there is a fold that places l1 onto l2.
 * 4) Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1.
 * 5) Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.
 * 6) Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.
 * 7) Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

Robert J. Lang has proven that this list of axioms completes the axioms of origami.

Constructing axis
We start with our complex plane and the two points 0 and 1. This means we need to construct the axis using the above axioms. If we apply axiom one to the points we will be given the real axis. We can then apply axiom 4 at the point 0 to give us the imaginary axis. Now we will need to find the point i on the imaginary axis as a reference point. To do this we can create a fold parallel to the imaginary axis at the point 1 and then use axiom 5 to create a new fold which will intersect our imaginary axis at the point i.

Once our axis are constructed we can use them as reference points and start using the axioms to create many different Origami-Constructible numbers.

A brief history of Humiaki Huzita
Humiaki Huzita, 1924-2005, was a Japanese-Italian Origami artist and mathematician. He studied nuclear physics at the University of Padua, Italy. Huzita formulated the first 6 axioms and reported these at the First International Conference on Origami in Education and Therapy in 1991. He was mostly inspired by Professor Margherita Beloch whom had announced the discovery that using paper folding it was possible to obtain roots of the fourth order, something which was impossible using Euclidean methods, with straight edge and compasses.

Applications of Origami Construction
The straight edge and compass, the study of constructible objects and the rules of construction are dated back the to ancient Greeks and Egyptians. For example, the famous Greek problems such as Trisecting the angle and Squaring the circle. Solving these problems and others alike led to the studying of geometric constructions with other tools such as Origami.

It has been shown that that trisecting and angle and doubling a cube are possible with origami constructions and also been proved that solving the general quintic equation is impossible with these constructions.

According to Jorge C Lucero, Professor of computer science in Brasilia, the application of origami can be found in materials science, computer science, robotics, civil engineering, biology, aerospace and automotive technology and acoustics.