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.

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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, Accessed 25 Oct. 2017.

 

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,
Accessed 25 Oct. 2017.

 

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