12 Alperin,

Roger & J. Lang, Robert. (2009). One,

Two, and Multi-Fold Origami Axioms. Origami4: 4th International Meeting of

Origami Science, Mathematics and Education, 4OSME 2006.

Accessed 29 Nov. 2017.

11 Robert

J. Lang, ‘Angle Quintisection’, Robert J. Lang Origami website, n.d.,

10 Robert

J. Lang, ‘Origami Diagramming Conventions’, Robert J. Lang Origami

website, n.d.,

Accessed 31 Oct. 2017.

9 Wikipedia,

‘Lill’s method’ website, n.d.,

Accessed 24 Nov. 2017.

8 Robert

J. Lang, ‘Origami Diagramming Conventions’, Robert J. Lang Origami

website, n.d.,

Accessed 31 Oct. 2017.

7 Robert

J. Lang, Origami 4, 1st edn, CRC Press, Boca Raton, 2009, pg. 371-394.

6

British Origami Society, Origami In Education And Therapy: Proceedings Of

The First International Conference In Origami In Education And Therapy (COET

’91), 2nd edn, CreateSpace

Independent Publishing Platform, n.p., 2016, pg. 37-70.

5

Jacques Justin, ‘Résolution Par Le Pliage De L’Équation Du Troisième Degré Et

Applications Géométriques’, L’ouvert online journal, 42, Mar. 1986,

pg. 9-19,

Accessed 31 Oct. 2017.

4 Robert

J. Lang, Origami Design Secrets, 1st edn, CRC Press, Boca Raton, 2003,

pg. 13.

3 Thomas

C. Hull, ‘Solving Cubics With Creases: The Work Of Beloch And Lill’, The

American Mathematical Monthly online journal, 118/4, Apr. 2011, pg. 307,

2

Tandalam S. Row, Wooster W. Beman & David E. Smith, T. Sundara Row’s

Geometric exercises in paper folding, 3rd edn, Open Court Pub. Co, Chicago,

1917, intro. xiv.

1 Joseph Wu,

‘Origami: A Brief History of the Ancient Art of Paperfolding’, Joseph Wu

Origami website, 15 July 2006, para. 1,

References

In summary, this report has laid

the foundations of origami and its link with geometry through presenting the

Huzita-Hatori axioms and the common folds from the outset. Subsequently, we

have demonstrated the power of origami with its application to a universally-known

mathematical problem of solving a cubic equation with the ideas of Beloch and

the geometric link between the Beloch Square and Lill’s method for solving

cubic equations. We concluded with Lang’s mind-blowing phenomenon in which a

quintic equation is actually possible to solve with the method of angle quintisection

and essentially challenging the work of Galois. To end, we proved Pythagoras’

Theorem using the Huzita-Hatori axioms and the common folds. We magnificently

removed 4 identical triangles from both pieces of paper, parting us with two

areas: and . As the triangles were identical, the same total area was

removed, which showed that .

Conclusion

(This diagram is a new revision of the

one which was made as a collective in the CRM group project.)

We begin this diagrammatic

proof with two square pieces of paper, and proceed as follows:

Proof:

a2 + b2 = c2

Let T be a right angled

triangle, with hypotenuse length c and remaining two sides length a and

b. Pythagoras’ theorem states that the 3 sides of T are related

by the following equation:

Theorem:

Section

5: The Pythagorean Theorem

Section

4: Quintic equations are solvable by Origami

The

seven Huzita-Hatori axioms which were mentioned in the first section define

what is physically feasible to construct by making successive single folds

formed by aligning groupings of points and lines. It has been mathematically

proven that there are only the seven axioms, and that those folds authorize the

construction of solutions to quadratics, cubics and even quartics, but it stops

there. 11

While

the Huzita-Hatori axioms only permit the solution of equations up to degree 4,

by elevating beyond the limitations that define that set, it is possible to

solve equations of much higher degree through origami. The crucial aspect of

the Huzita-Hatori axioms is they individually describe an action in which a

single fold is defined by positioning numerous combinations of points and

lines. 11

On

the other hand, it is also allowable to define two, three, or more simultaneous

folds by forming several alignment combinations. Some of these alignments can

be broken down into sequences of Huzita-Hatori folds, but others, which we call

non-separable, cannot be broken down into simpler units. These more complicated

alignments form an entirely new class of origami “axioms”, which

potentially can solve equations of considerably higher order. 11

Theorem:

Every

polynomial equation of degree with real solutions can be solved by simultaneous folds. 12

The

following figure below shows the key step in performing an origami angle

quintisection (division into fifths with the use of folds only):

11

If you would like more

information on angle quintisection then refer to the appropriate reference

which directs to Robert Lang’s website at the end of this paper where you can

find his detailed diagrammatic folding sequence for the angle quintisection in

PDF form.

Section

3: Cubic equations are solvable by Origami

3.1: The Beloch Fold (Axiom A6) 3

A way to notice

what the Beloch Fold is demonstrating is to consider one of the point-line

pairs. If we fold a point to a line , the resulting crease line is tangential to the

parabola with

focus and directrix

(the

equidistant set of lines from and ). This can be demonstrated by the following activity:

Proof: (See Step 1 above) 3

After

folding a point on to , draw a line perpendicular to the folded image of , on the folded flap of paper from to the crease

line. If is the point

where this drawn line intersects the crease line, then we see when unfolding

the paper that the point is equidistant

from the point and the line . Any other point on the crease line will be

equidistant from and and thus will

not have the same distance to the line . Therefore, the crease line is tangent to the

parabola with focus and directrix 3.

This

essentially means that folding a point to a line implies locating a point on a

certain parabola and is comparable to solving a quadratic equation. The Beloch

Fold discovers a common tangent to two parabolas. Two parabolas drawn in the

plane can have a maximum of three distinct common tangents, hence this origami

fold is equivalent to solving a cubic equation. Ruler and compass constructions,

on the other hand are only capable of solving quadratic equations.

3.2: The Beloch Square 3

Given two

points A and B and two lines and in the plane,

construct a square with two

adjacent corners and lying on and , respectively, and the sides and , or their extensions, passing through and , respectively. The Beloch Square is shown below:

3

3

3.3: Constructing

Take to be the -axis and to be the -axis of the plane.

Let and . Then we construct the lines to be and to be . Folding onto and onto using the

Beloch fold will make a crease which crosses at a point and s at a

point . Consulting the Beloch square diagram, if we let be the origin,

then notice that , , and are all similar

right triangles. This follows from the fact that is

perpendicular to and . 3

So

we have , where denotes the length of the segment. Put and yields and so ), (Discovered by Martin).

3.4: Solving cubic

equations using Lill’s method 9

Lill’s method 9:

“To

employ the method a

diagram is drawn starting at the origin O. A line segment is drawn rightwards

by the magnitude of the first coefficient (the coefficient of the highest-power

term) (so that with a negative coefficient the segment will end left of the

origin). From the end of the first segment another segment is drawn upwards by

the magnitude of the second coefficient, then left by the magnitude of the

third, and down by the magnitude of the fourth, and so on. The sequence of

directions (not turns) is always rightward, upward, leftward, downward, then

repeating itself. Thus each turn is counterclockwise. The process continues for

every coefficient of the polynomial including zeroes, with negative

coefficients “walking backwards”. The final point reached, at the end

of the segment corresponding to the equation’s constant term, is the terminus T

9.”

“A line is then launched from the origin at some angle , reflected off of each line segment

at a right-angle (not necessarily the “natural” angle of reflection),

and refracted at a right angle through the line

through each segment (including a line for the zero coefficients) when the

angled path does not hit the line segment on that line. The vertical and horizontal

lines are reflected off or refracted through in the following sequence: the

line containing the segment corresponding to the coefficient of , then of , etc. Choosing ? so that the path

lands on the terminus, the negative of the tangent of ? is a root of this

polynomial. For every real zero of the polynomial there will be one unique

initial angle and path that will land on the terminus. A quadratic with two

real roots, for example, will have exactly two angles that satisfy the above conditions

9.”

Two

examples of Lill’s method being applied to a quintic is shown below for a

visual understanding 3:

Problem:

Given a polynomial , locate a real root of .

Claim:

is a root of 9

Proof: 3

Let

be the length of the side opposite the angle in the triangle whose side adjacent to is part of the segment length.

Case

1: All coefficients are positive –

,

=,

=, …

,

but

is indeed a root of

Case

2: All coefficients are negative –

,

) =)),

, …

,

but

is indeed a root of

Case

3: All coefficients are zero (we prove for completeness, but it is trivial that

for this case) –

,

,

, …

is indeed a root of

In

the cubic case, the line of refraction (angled path) for the general cubic

polynomial will have four sides, so our line of refraction (angled path) will

have three sides. If we think of as the point , and as the point , and we think of the lines

containing the -side and the -side as the lines and respectively, then a Beloch square with

adjacent corners on and and opposite passing through and will give us a line of refraction. This is

shown in the figure 3 below. We have

finally demonstrated that paper folding can be used to perform Lill’s method in

the cubic case and thus solve general polynomials of degree three.

3

Origami and Geometry: Solving Polynomial Equations &

proving Pythagoras’ Theorem using Origami

Faculty

of Mathematical Sciences

University

of Southampton

Student

ID: 28229428

Abstract

Origami is defined to mean

“fold paper” in Japanese and is renowned as a delicate art form, however it

also comprises of geometric concepts which relate to a variety of fields in

mathematics. This report demonstrates an understanding of Margharita P.

Beloch’s and Lill’s work in which we can solve general cubic polynomials and

extend this to higher order polynomials to also solve the quintic equation

using origami which was explored by Robert Lang. We conclude with a visual

proof of the Pythagorean Theorem to illustrate the contrast how origami can be

used not only to solve algebraic problems such as polynomials but also

geometric problems.

Introduction

The art of paper folding has

been around since the 6th Century 1, however the first

mathematical publication on Origami was released in 1893 when Tandalam S. Rao published his book

“Geometric Exercises in Paper Folding”. In this paper, Rao used paper folding

to demonstrate proofs of geometric constructions. 2 Roughly 40

years later, Margharita P. Beloch then went on to show that it was possible to

use Origami to solve the general cubic equation. 3

Section 1 of this report

introduces the seven Huzita-Hatori axioms, which are the key foundations for

paper folding. The common folds of Origami appear in Section 2, along with the

Yoshizawa-Randlett system. The purpose of this system is to define the

different folds in a way that straightforwardly exemplifies the necessary

notions of paper folding and has since become the default universal system.

4 Section 3 of this report discusses Beloch’s use of Lill’s method for

solving cubic polynomial equations. Section 4 extends to solving the quintic

equation with the assistance of 3-fold origami.

To end, we take advantage of

origami and geometry to prove a well-renowned mathematical theorem. Section 5 of this

report contains an illustrative proof for the Pythagorean Theorem, which states

that in a right angled triangle, the square of the hypotenuse is equal to the

sum of the squares of the two remaining sides. Our approach method involved

using Pureland Origami, Kirigami and simple geometry to transform a square

piece of paper into a visual representation of the theorem.

Section

1: Huzita-Hatori Axioms

The Huzita-Hatori axioms, occasionally called the Huzita-Justin axioms,

were first discovered in 1986 by Jacques Justin 5,

then rediscovered and reported at the First International Conference on Origami

in Education and Therapy by Humiaki Huzita in 1991. 6

These axioms describe, using a combination of pre-existing points and lines,

every possible situation in which a single fold can be carried out. 7

Section

2: Yoshizawa-Randlett System and the Common Folds

2.1: The Yoshizawa-Randlett System

Origami comprises of

transforming a 2D flat sheet of paper into an arbitrary 3D structure, so it is

assistive to make use of symbols 8 and annotations to describe the

iterations. The Yoshizawa-Randlett system below is universally

recognised as the universal standard origami diagramming system. These

notations are commonly used in step by step origami guides, along with some of

origami’s common folds. Additionally, we use a straight line to represent a

crease.

2.2: Common Folds