User:AmritAndAsh

= Origami Numbers = When we consider origami in a mathematical sense, it is the idea of being able to construct a line of any length we like by using a finite number of folds on a piece of paper. Cutting the paper is traditionally not allowed in the construction of these numbers, instead you rely on creating reference points by making intersections using folds and creases. This leads to the formal definition;

A real number r is origami-constructible if one can construct in a finite number of steps two points which are a distance of |r| apart.

History
Origami is the art of folding paper, originating in Japan. You begin with a square piece of unmarked paper and using reference points created through creases and folds you are able to construct different shapes as well as lines of differing lengths. These constructed lines and shapes were traditionally made using Greek methods of an unmarked ruler and compass.

The word origami comes from the Japanese words oru (to fold) and kami (paper).

Despite the long history of using a compass and straight edge to construct numbers; commonly referred to as Euclidean Constructions due to their relationship to Euclid's Elements dating back to ancient Greece ; the method of origami has only taken off in the past century - starting with the Huzita Axioms.

While a 'constructible ' number is a real number, r, given a line segment of unit length, a line segment of length |r| can be constructed with a compass and straightedge (a ruler with no markings) in a finite number of steps. However, there was restriction in this method, there were three constructions that could not be done via Euclidean Construction: Squaring the Circle (Constructing a square with area pi); Doubling the Cube (Constructing a line segment of length $$\sqrt[3]{2}$$ ); and Trisecting an Angle - whereas these can be done easily by the use of origami.

Huzita Axioms
The Huzita–Hatori or the Huzita–Justin axioms, are a set of principles that are used to describe the operations that can be made when folding a piece of paper. The operations are assumed to be used on a plane with linear folds. There are seven axioms in total. The first six were initally discovered in 1986, before being published by Humiaki Huzita in 1991. Then the seventh axiom was discovered in in 1989 by Jacques Justin, and then rediscovered in 2001 by Koshiro Hatori.

Axiom 1
Given two points p1 and p2, there is a unique fold that passes through both of them, connecting them.

Axiom 2
Given two points p1 and p2, there is a unique fold in order to fold p1 onto p2

Axiom 3
Given two lines l1 and l2, there is a fold in order to fold line l1 onto l2

Axiom 4
Given a point p1 and line l1, there is a unique fold perpendicular to l1, passing through the point p1

Axiom 5
Given two points p1 and p2 and line l1, there is a unique fold in order to fold p1 onto l1 and passes through the point p2

Axiom 6
Given two points p1 and p2 and two lines l1and l2, there is a fold that places p1 onto l1,and places p2 onto l2

Axiom 7
Given one point p and two lines l1 and l2, there is a fold that places p onto line l1 and is perpendicular to l2 == Constructions == Using the Huzita-Hatori axioms, it is possible to solve some of the Greek problems that could not be solved. This section will cover how to use Origami to solve Trisecting an Angle and Doubling the Cube.

Trisecting an Angle
Using Origami constructions, it is possible to trisect an angle. This was not possible when using Euclidean Constructions. This can be done using any angle, but the following method is for angles below 90°. For this walk-through, we're going to be trisecting 90° itself. Following the diagram above;


 * 1) First, make a horizontal fold anywhere across the square, creating line Bh2. (This is a mixture of Axiom 2, 3 and 4).
 * 2) Now fold line Ah0 up to meet line Bh2, creating line h1. (Axiom 3)
 * 3) Next, fold the corner of A up so that it touches line h1 and so that B stays where it is, creating point A'. This also creates line p. (Axiom 2).
 * 4) With the corner still folded, follow the crease created by line h1, folding along it to create line d.
 * 5) Unfold the corner and extend line d down to point A.
 * 6) Finally, fold line Ah0 up until it is on top of line d, creating fold line t. (Axiom 2).
 * 7) The angle has then been trisected.

It is important to note that if you wish to trisect an angle that is less than 90°, choose a point C along the top of your square paper in order to create that angle between that point, A and h0. Then, in Step 3, fold it so that point B touches the line AC and A touches h1. This creates the two points A' and B' respectively.

Doubling the Cube
The idea of "Doubling the Cube" means that suppose you have a cube with side length S1 and volume V; you are to find the side length S2 that makes the cube with volume 2V. Following the image above;


 * 1) First, make a small fold half-way up the right hand edge of the paper. We will name this point p3. (Again, a mixture of Axiom 2, 3 and 4).
 * 2) Next, apply Axiom 1 to connect the top left corner and the bottom right corner with a line. Do the same with the bottom left corner and point p3.
 * 3) Fold the top edge down to meet the intersection of these two lines. This creates line l2. (Axiom 2 or Axiom 3)
 * 4) Unfold that, and then fold the bottom edge up to meet with the new crease that you just created. The right edge point of this new crease we will name p2. (Axiom 3)
 * 5) Fold point p1 up until both p1 lies on l1 and p2 lies on l2. (Axiom 6)
 * 6) Point p1 then divides up line l1. Work out the ratio X/Y where X is the length of the top corner down to p1 and Y is the length of p1 down to the bottom right corner. Multiply S1 by this ratio and this gives you the length of S2.

Origami-Constructible Numbers as a Subfield
The origami-constructible numbers form a subfield of $$\mathbb{C}$$. That is to say that the field of origami-constructible numbers is contained within $$\mathbb{C}$$ and satisfies the field axioms with the same operations as $$\mathbb{C}$$.

This can be proved by applying the Huzita Axioms!

Before we can do this however, we first need to define two more Axioms that are made by composing the Huzita Axioms; which will help prove the operations. These will be referred to as the Elementary Operations;

E1. Given a point p and a line l, fold a line parallel to l that goes through p

This is constructed using two applications of Axiom 4, i.e. the first application provides line l1 , which is a line perpendicular to our original l. Applying the Axiom once more to l1 creates line l2 which is perpendicular to l1 , meaning it is also parallel to l.

E2. Given a point p and a line l, reflect p across l.

This is constructed by applying Axiom 4 and then Axiom 5 to the result.

Now that we have these two Axioms and the Huzita Axioms, we can prove that the field of origami-constructible numbers form a subfield of $$\mathbb{C}$$. We do this by checking Addition, Multiplication and Inversion. This is to say, given origami-constructible numbers $$\alpha$$ and $$\beta$$, $$\alpha$$≠0 in $$\mathbb{C}$$, $$\alpha$$+$$\beta$$, $$\alpha$$$$\beta$$ and $$\alpha^{-1}$$ are also in $$\mathbb{C}$$.

Regular Parallelograms
The proof starts out with three points: 0, p1 and p2. Using Axiom 1 it is easy to create the lines l1 and l2.

From here, use E1 to construct l3. That is, use E1 in regards to p1 and l2.

Similarly, use E1 to construct l4. This construction involves p2 and l1.

It is clear then that p3 is the point of intersection of the two constructed lines l3 and l4.

Degenerate Parallelograms
If the points p1 and p2 are scalar multiples of each other (i.e. lie on the same line from 0) this method cannot be used.

As before, begin by using Axiom 1 to create the starting line l1, this time going through 0, p1 and p2. Then, apply Axiom 2 to fold p2 onto p1 and leave the paper folded over. Thus, p3 is now on top of 0 (in the same spot). To create the intersection crease, apply Axiom 4 to 0 and the original line l1. This creates the intersection marking p3.

Multiplication
One important thing to note about Multiplication is that because origami-constructed numbers are being shown to be a subfield of the Complex field, they can either be multiplied by a Real number or $$\imath$$.

Real Number Multiplication
To start, we are going to assume that p1 is not a real number. If it is, add $$\imath$$ to it and then apply Axiom 4 to project it onto the Real axis.

We begin with four points; 0, 1, r, and p1. It is obvious that r is a scalar multiple of 1 (multiplied by r) so they begin along the same line from 0. When multiplying a number p1 by a real number r we use the following steps;

First; apply Axiom 1 to 0 and p1 to create our first line l1 through both those points.

Similarly, apply Axiom 1 to p1 and 1 to create our second line l2 through both of these points.

From here, apply E1 to r and l2 to create l3, the line that goes through r, parallel to l2.

Here it is obvious that p3 is created using the intersections of lines l1 and l3.

Multiplication by $$\imath$$
Multiplying a number by $$\imath$$ means to rotate it counter-clockwise around the point 0, $$\pi/2$$ radians. To do this, we simply need to follow these steps;

Start with 2 points 0 and p1. Apply Axiom 1 to create the first line l1 through these two points.

Next, apply Axiom 4 on the point 0 and the line l1 to create l2, the line perpendicular to l1 through the point 0.

Now apply Axiom 3 on the lines l1 and l2 to create line l3, the line that bisects the angle between them.

Finally apply E2 to point p1 and line l3 to create point p2.

Inversion
To find the inverse of a complex number in terms of origami-constructible numbers, we simply divide by a real number, making the process almost identical to multiplying.

Start with four points 0, 1, r and p1. First, create line l1 by applying Axiom 1 to 0 and p1, creating the line through them.

Similarly, apply Axiom 1 to r and p1 to create line l2, the line through those two points as well.

Next, apply E1 to the point 1 and line l2 to create line l3, the line parallel to l2 through the point 1.

Now the intersection of l1 and l3 is clear, which we mark as point p3.

Future of Origami and Further Uses
People have found the mathematics of origami to have many practical uses in various fields, such as space travel and observation; robotics and medicine. For example, engineers have designed a 'star shade' to help observe exoplanets for use in space telescopes. There have also been origami usage in medicine. For example, compact devices such as oriceps for use in robotic surgery.

Some are changing the way we look at origami in mathematics. By fusing origami with the Ancient Greek method of using a compass, three more axioms were discovered in 2011 by researchers at University of Tsukuba. These new axioms allowed the construction of more structures such as ellipses and hyperbolas. They also allow another method of trisecting an arbitrary angle.

Some are trying to explore further extensions in origami, for example, allowing use of cutting and tracing. Typically, in origami these things are not permitted, however they can have significant mathematical impact. For example, if cutting is used as well as paper-folding, then we would possibly be able to superimpose points and lines by cutting the paper into two seperable parts and placing one on top of the other. This idea can also be seen if we allow the use of tracing. These ideas can allow many more constructions to be possible. Another idea that can be explored is using non-flat surfaces for the origami, such as spheres and paraboloids.