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The Folding Mathematics

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The Special Issue of The Proceedings of Telangana Academy of Sciences
Vol. 01, No. 01, 2020, 26–37

The Folding Mathematics

∗
Archana S. Moryea
a
University of Hyderabad

Abstract: Origami is the art of paper folding, and it borrows its

name from two Japanese words ori and kami. In Japanese, ori means
folding, and the paper is called kami. While origami is just a hobby
to most, there is a lot more to it. If you fold a square sheet of
paper into any of the traditional origami model (for example the
flapping bird) and unfold it, you can see crease patterns. These
crease patterns tell us that there is a lot of geometry hidden behind
the folds.
In this article, we investigate the symbiotic relationship between
mathematics and origami. The first part of this article explores
the utility of origami in education. We will see how origami could
become an effective way of teaching methods of geometry, mainly
because of its experiential nature. Complex origami patterns can-
not be created out of thin air. They usually involve understanding
deep mathematical theories and the ability to apply them to paper
folding. In the second part of the article, we attempt to provide a
glimpse of this beautiful connection between origami and mathe-
matics.
Keywords: Origami, geometry, paper folding, fold-able numbers, tree-maker,
cubic polynomials.

1. Introduction

Origami is a technique of folding paper into a variety of decorative or representative

forms, such as animals, flowers etc. The origin of origami can be traced back to
Japan. Originally, the art of paper folding was called as orikata, the craft acquired
its current name in 1880 [5].
The early evidence of origami in Japan suggests that origami was primarily used
as a ceremonial wrapper called the Noshi. Noshi is a wrapper which is attached to
a gift, expressing good wishes (similar to greeting cards of today). A popular such
Noshi is a pair of paper butterflies known as Ocho and Mecho that were used to
decorate sake bottles (see Figure 1(a)1 ). Origami initially was an art of the elite,
mainly because the paper was a luxury item. As the paper became more accessible,
origami also became a well-practiced art.
The practice of origami can also be traced to Europe, the baptismal certificates

Email:sarchana.morye@gmail.com
1 Pic source: https://www.origami-resource-center.com/regular-mecho.html

26
 (a) Japanese Origami (b) European Origami (c) Wet Folding

Figure 1.

issued during the sixteenth century were folded in a specific way. (see Figure 1(b)2 ).
Here, the four corners of the paper was folded repeatedly to the center. Interestingly
this techniques is very different from the ones used in Japan. It is said that such
a crease pattern closely resembles an old astrological horoscopes. For this reason,
historians believe that folding in Europe developed more-or-less independently [5].
Some of the popular origami models from Europe are the Pajarita, the Cocotte
and the boat .

1.1. The modern origami

The modern era of origami can mainly be attributed to the grand master of origami,
Akira Yoshizawa. It was because of his relentless efforts that origami has trans-
formed into a living art, from a mere craft. Apart from contributing towards de-
veloping more than 50,000 origami models, he also pioneered the wet-folding tech-
nique (see Figure 1(c)3 ). This technique involves slightly dampening the paper
before using it for folding. This technique allowed the paper to be manipulated
more easily, resulting in models with rounded and sculpted looks. The famous
Yoshizawa-Randlett diagramming system is also his invention. Since the introduc-
tion of this system, origami has seen several advancements. Origami is now one of
the well-established topics of research in major universities across the world.
Origami as it exists today has several variations, we list some of them below.
(1) Pure Origami : This form of origami is arguably one of the oldest and well
studied form, the models here are made from a single square sheet of pa-
per, without the use of scissors and glue. Coloring the final model is also
strictly discouraged. Several models along with instructions can be found in
www.happyfolding.com. My personal favorites and recommendations include
the traditional crane, the swallowtail butterfly and the fawn models.
A more stringent variation is the Pureland Origami developed by John
Smith. This version disallows certain types of folds allowed by the pure
origami. Interestingly this type of origami is considered to be disable friendly.
(2) Action Origami : This form of origami involves developing models that can
be animated. The most famous among the models of this type is the flapping
bird. The bird flaps its wings when its tail is moved. Other interesting models
of this type include origami airplanes and the instrumentalist created by Prof.
Robert Lang.
(3) Modular Origami: In this form of origami, several identical units are folded
and then assembled into a more complex origami model. An of this type is the
Kusudama flower. In this model, sixty identical units are folded and arranged

2 Pic source: https://www.origami-resource-center.com/history-of-origami.html

3 Pic source: https://en.wikipedia.org/wiki/Wet-folding

27
 (a) Fujimoto Hy- (b) Kusudama (c) knotology torus
drangeas

Figure 2.

into twelve flowers. These flowers are then arranged such that they form a
regular dodecahedron, the pieces are held in place using glue or a thread.
Many models have pocket and flap in each unit, so that units bind together
without a glue or a thread.
(4) Origami Tessellation: An origami tessellation is created by repeating a pat-
tern multiple times in all the directions, and this creates a mosaic. The kind
of folds in this process predominantly includes pleats and twists. The in-
vention of this technique can be attributed to Shuzo Fujimoto. This type of
origami has an additional feature, and they produce a beautiful effect when
they are backlit (when they are held against the light). Fujimoto Hydrangeas
(see Figure 2(a)) is an interesting model of this type.
(5) Strip Origami : Strip folding is a technique that involves both paper folding
and paper weaving. A fascinating model of this type is that of knotology
torus (see Figure 2(c)4 ) by Dáša Ševerová.
While origami certainly has evolved as an amazing art, its applicability has also
been phenomenal. Origami-inspired techniques are being sought after by almost
every engineering field, ranging from space science to medical equipment and even
automobile manufacturers. In medicine, origami techniques are often applied to
stent designs. Stents are collapsible tubes that can be inserted into a patient’s
veins or arteries. When deployed, the stent expands to open the veins or arteries to
improve blood flow. Origami design techniques are instrumental in developing thin
and small stents. NASA’s James Webb Telescope (JWST), the planned successor of
Hubble space telescope, is a rather sizable infrared space telescope with a primary
mirror of 6.5-meters. Origami techniques are being deployed to fold such a large
telescope compactly so that it can be airlifted to space, where it can be unfolded
again. Automobile manufacturers are pursuing efficient flat-folding techniques for
airbags so that it occupies less space and yet unfurls quickly enough when needed.
The use of origami techniques in the science and engineering field is endless.
Hence it is only prudent to study and understand origami as a science. One of the
aims of this article is to illustrate the deep connections of origami with mathemat-
ics. The article is divided into two parts; the first part deals with using origami as
a means to understand mathematics, geometry in particular. For example, a proof
for the Pythagoras theorem merely is folding a square sheet of paper. Trisecting
a line using just folds is possible. Interestingly, even trisecting an angle can be
achieved. The latter is of significant interest due to its impossibility within the
realms of Euclidean geometry. The second part of the article briefly dwells upon
the need for mathematics to design complex origami models.

4 Pic source:https://www.flickr.com/photos/dasssa/3426754850

28
 Before we delve into the technical details, I would like to share my experience
with the folding and origami community. While this subsection is unconventional,
the reason I write this is to get an opportunity to acknowledge individuals who
helped and inspired me during my journey. I also hope that this will nudge others
to start their journey into the world of folding.

1.2. My Origami Journey

My first exposure to origami was like any other kid when I learned to fold simple
models like box, flower, purse, and so on in school. The first complicated model I
folded was that of a fish, from a magazine I found in my relatives’ place. Learning
this model gave me a huge sense of accomplishment and satisfaction. My first formal
tryst with origami was during my undergraduate days when I chanced to find a
Marathi and English origami series written by Indu Tilak [16]. This series is part
of the textbooks prescribed by Maharashtra board for primary education. Even
though these are school books, they are quite interesting and extensive. The book
includes the necessary foundations to start creating basic and complex structures
in origami.
While I was helped and inspired by many in the origami community, one who
was extremely helpful and kind to me was Dáša Ševerová. Papers are the lifeline for
origami artists. Some particular models require individual papers. Dáša Ševerová
was kind enough to help me obtain certain papers that were difficult to get in
India. She has also helped me fold some intricate origami tessellations.
These are some other origami books that one may wish to read.
• Origami Tessellations: Awe-Inspiring Geometric Designs, by Eric Gjerde [3].
• Origami Boxes by Tomoko Fuse [2].
• Origami Butterflies by Micheal LaFusse [11].
• Origami Journey: Into the Fascinating World of Geometric Origami, by Dáša
Ševerová [15].
• Origami Inspiration by Meenakshi Mukerji [14].
The website www.happyfolding.com, owned by Sara Adam, is a one-point source
of abundant information, beginners or otherwise.

2. Origami for Mathematics

Mathematics has unfortunately attained the notoriety of not just dreading the
youngsters but also the adults. How many times have we not heard the phrase,
“Thank god, I do not have to study mathematics anymore.” This feeling has been
captured aptly by the following Marathi couplet.
BolAnAT u@yA aAh
 gEZtAcA pp
r
poVAt mA$yA k y
Un dKl kA r Yopr
Bholanath is a mystical, mythological, and benevolent Ox. The children pray
to him for ill-health, for it is their mathematics paper the next day. It is not
very surprising that mathematics is viewed so unfavorably, as it involves many
abstract concepts challenging to comprehend. Even a simple definition such as
an area or volume is very abstract, for it is a measurement irrespective of the
shape. One reason why mathematics lacks the popularity of the other sciences
is that it lacks visual appeal. Prof. John J Hopfield (the recipient of the ICTP

29
Dirac medal, 2001) in one of his articles [6] wrote that the reason for him being
a scientist was because of the encouragement he received to do experiments. Labs
and experiments are never associated with mathematics. While one could still argue
that mathematics does provide the same experience through puzzles and problems,
my personal experiences and memories negate such claims. Solving mathematical
problems in the current day scenario has degenerated to learning to apply formulas
to score high. Here is where origami can fill in to provide the missing fun.

Example 2.1 Consider a simple problem of proving that the sum of interior angles
of a triangle adds up to 180◦ . While there are many ways to prove this, the folding
in Figure 3 demonstrates this clearly and crisply. Here the desired rectangle is

C
β
h
b/2

h/2

β
α γ α γ
A b B A, B, C

Figure 3.

identified, and the cones of the triangle are folded so that their tips meet. Since the
angles α, β, γ covers a straight line, this proves that their sum is indeed 180◦ . Now
the same folding also provides us with the argument of why the area of a triangle
is 1/2 × base × height. Notice that the fold covers the rectangle with height h/2
and length b/2 two times. So the area of the triangle is two times the area of this
rectangle. This immediately provides us with the relation

Area(4ABC) = 2 × (b/2) × (h/2) = 1/2 × b × h.

This demonstrates that every folding has a deep mathematical connection, pos-
sibly many. Discovering them depends on the creativity of the folder. After all,
creativity is in the eye of the beholder, beautifully summarized by our beloved
Dr. A. P. J. Abdul Kalam as
“Creativity is seeing the same thing but thinking differently.”
We will demonstrate the deep connection of origami with mathematics through
another example, this time, the famous Pythagorean Theorem. The Pythagorean
theorem is one of the oldest known theorems and was studied by Babylonian,
Egyptian, Indian, and Greek mathematicians centuries earlier. It states that the
square of the hypotenuse of any right-angled triangle is equal to the sum of the
squares of the other two sides. This geometric theorem probably has the most
number of proofs. The standard proof which is given in the most high-school books
is using similar triangles. Another exciting proof is as follows. Let 4ABC be any
right-angle triangle with c as its hypotenuse and its other two sides being a, b. Let
C be a square-shaped bucket with length and height being c-units and its width
1-unit. Similarly, let A ( respectively B) be a square-shaped bucket with length and
height a (respectively b) and with width exactly 1-unit. It can be demonstrated that
the C bucket can be filled using the water in buckets A and B, respectively. While
this is undoubtedly a fun proof, the knowledge of volumes is needed to understand
the proof. Further conducting such an experiment in a class is cumbersome.

30
Example 2.2 Now consider the folding in Figure 4, it clearly demonstrates the
proof of Pythagoras Theorem. In the figure, we take a square sheet of paper and

c
a

Figure 4.

mark out four right-angled triangles of equal sizes along the four corners. Each of
these triangles has c as its hypotenuse and a, b as the size of its other sides. Each
of these triangles has its hypotenuse in the inner part of the square, touching each
other. Dotted lines in the figure denote these. Folding along the hypotenuse gives
us a square of length c. Notice that the area of such a square is c2 , the area of the
original square we started with was (a + b)2 . What was folded in were 4 triangles
each of area 1/2 × a × b. With this, we get the following equation, which also proves
the Pythagoras Theorem.

c2 = (a + b)2 − (4 × (1/2 × a × b)) = a2 + b2 .

While we could go on demonstrating the utility of origami in proving theorems

involving simple properties (for example see [4]), one may ask whether origami can
also be used to solve more involved problems and theorems. For this, we need to
formalize folding, i.e., make precise what kind of folds are allowed and what are
not. This would allow us to investigate the constructible geometric objects through
origami. This is in the same lines as the classical ruler-and-compass construction.

2.1. Huzita-Hatori Axioms

The formal axioms for origami is given by the Huzita–Hatori axioms. We briefly
recall them here and direct the interested readers to [7, 10] for a comprehensive
coverage on the subject.
(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 are folds 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 are folds (possibly empty)
that places p1 onto l1 and passes through p2 .
(6) Given two points p1 and p2 and two lines l1 and l2 , there are folds (possibly
empty) that places p1 onto l1 and p2 onto l2 .
(7) Given a point p, and two lines l1 and l2 , there are folds (possibly empty)
perpendicular to l2 that places p onto line l1 .

31
Axiom 1 Axiom 2 Axiom 3 Axiom 4 Axiom 5 Axiom 6 Axiom 7

p p2 p2 p1 l1
p
p1 l1
p1 p2 l p1
p2
l2 l1 l2 l1 l2

Figure 5.

Remark 2.3 The relevant question here is, what can these postulates achieve in
comparison to the classical ruler-and-compass. While the above set of operations
are termed as axioms, they do not necessarily indicate that such a fold is achievable
or unique (see [8, 9] for details). For example, in Axiom 5, it is impossible to obtain
a fold when p1 = p2 . Similarly in Axiom 6, it is impossible to obtain a fold if p1 = p2
and `1 6= `2 two parallel lines. Rest of the article only deals with the scenarios where
the relevant fold is achievable.

Towards understanding the axioms, we note that the first four postulates are
self-explanatory. We will examine the fifth and sixth postulates more closely. In-
terestingly the fifth postulate can be used to solve a quadratic equation and the
sixth a cubic equation. We will demonstrate the latter in the sequel. We first prove
the following lemma which states that the dotted line in the Axiom 5 of Figure
5 is actually a tangent to a parabola with its focus on p1 and the directrix on l1 .
This proof is an adaptation from [10]. Recall that a parabola is those set of points
(called the locus) that are equidistant from a fixed point (called the focus) and a
fixed-line (called the directrix ).

Notation 2.4 If d(x, y) denotes the Euclidean distance between two points x and y,
then the distance between the point x and the line l is equal to min{d(x, y)|y ∈ l},
and is denoted by d(x, l). Notice that, d(x, l) = d(x, y) where y is the foot of a
perpendicular drawn from the point x to the line l. This distance d(x, l) is called
the perpendicular distance between x and l.

Lemma 2.5 Given two points p1 and p2 and a line l1 , assume that there exists a
fold ` that places p1 onto l1 and passes through p2 . Then ` is tangent to the parabola
P defined by the focus p1 and the directrix l1 .

Proof. To prove the lemma, we prove that there is a unique point x on the line `
which is equidistant from p1 and l1 . By definition, this point will lie on the parabola.
Since this point is unique, no other points of the line will lie on the parabola. This
will prove that, the line will be tangential to P.
For this, we prove two things. Firstly we prove that there is a point x in `, which
is equidistant from p1 and l1 . Then we will prove that for any point y 6= x in `,
d(y, p1 ) is not equal to d(y, l1 ). Without loss of generality, we will assume that the
line l1 is the bottom edge of a square and that the point p2 lies on the left edge
of the square. Observe that given any l1 , p1 , p2 , one could arrange them in this
manner in an appropriate large enough square. By assumption, the line ` folds l1
in such a way that a point in it coincides with p1 . Let this point on l1 be p¯1 (see
Figure 6 for an illustration). Now draw a line perpendicular to the line l1 , starting
at p¯1 , let this line intersect ` at x. We claim that this intersection point x is the
required point. This is easy to observe since p¯1 coincides with p1 when the paper
is folded along `. This immediately implies that d(x, p1 ) = d(x, p¯1 ).
For proving the second part (uniqueness), we again use the fact that p¯1 coincides
with p1 when the paper is folded along `. This immediately also tells us that
for any point y on the line `, d(y, p1 ) = d(y, p¯1 ). Let py denote the foot of the

32
perpendicular drawn from y to line l1 . Hence d(y, l1 ) = d(y, py ). Now for any point
y 6= x, consider the right angle triangle 4ypy p¯1 . In this triangle since y p¯1 is the
hypotenuse, it follows that d(y, p¯1 ) > d(y, py ) = d(y, l1 ). This proves that y ∈/ P.
Hence ` is tangent to P. 

l2 : x = 2 ` : y = 2x − 4

`
p2
y p1 = (0, 1)
x
p1
p¯1 = (4, −1)
o l1 : y = −1

p2 = (−2, −3)

¯ −5)
p2 = (2,
py p¯1 l1

Figure 6.: Illustration for Lemma 2.5 Figure 7.: Illustration of Theorem 2.7 on Eqn:x3 + 0x2 − 3x − 2

Remark 2.6 Notice that in the proof above, the role of p2 is non-existent. The fact
that ` is a tangent to the parabola P is invariant to p2 . Indeed, any fold obtained
by placing the point p1 onto the line `1 creates a tangent to the parabola P. Also,
notice that there are infinite tangent lines to the parabola, one for every point on
it. This collection of tangent lines is called the tangent bundle. The point p2 picks
out a set (of size ≤ 2) of tangent lines from this tangent bundle.

Using the same technique of Lemma 2.5, one could also prove that the fold `
obtained in Axiom 6 is a line tangent to two parabolas P1 , P2 , determined by the
focus and directrix p1 , l1 and p2 , l2 respectively. We will instead illustrate how
to use Axiom 6 to solve a cubic equation. More specifically, we will show how to
obtain a solution for equations of the form x3 + ax2 + bx + c = 0 where a, b, c ∈ R.
The idea here is to use the points p1 = (a, 1), p2 = (c, b), and the lines l1 given by
y = −1 and l2 given by x = −c in Axiom 6 and show that the slope of the fold
obtained is a solution to the equation. The idea behind the Theorem is illustrated
in Figure 7.

Theorem 2.7 Let x3 + ax2 + bx + c = 0 be any cubic equation with a, b, c ∈ R.

Consider a large enough square paper with origin at its center, let p1 and p2 be two
point in it with the coordinates given by (a, 1) and (c, b) respectively. Let l1 and l2
be lines defined by the equation y = −1 and x = −c respectively. Then the slope t
of the line ` that folds the point p1 onto l1 and p2 onto l2 is a solution to the given
cubic equation.

Proof. We wish to prove that any solution to the slope of the line ` obtained as
a result of applying Axiom 6 to the points and lines specified below, is a solution
to the given cubic equation x3 + ax2 + bx + c = 0. The points are specified as
p1 = (a, 1) and p2 = (c, b) and the lines given by the equation l1 : y = −1 and
l2 : x = −c. Let the equation defining the line ` be ` : y = tx + u. Recall that, t is
the slope and u is the y intercept here.
Firstly notice that in both Axiom 5 and 6, the point p1 is placed onto the line l1 .
By Lemma 2.5 and Remark 2.6, we obtain that the line ` is tangent to the parabola
defined by its focus on p1 and the directrix on the line l1 . Let this parabola be P.
Using the focus given by p1 and the directix given by line l1 , we obtain the equation

33
of P as
1
y = (x − a)2 . (1)
4

Since the line ` is tangent to the parabola, we obtain the equation of its slope as
∂y
t = ∂x = 12 (x − a). Let (x1 , y1 ) be the point on ` where it makes contact with the
parabola. Evaluating the equation of slope at this point provides us with:

1
t = (x1 − a). (2)
2

Since we know the slope of ` and also that (x1 , y1 ) lies on `, we obtain the equation
of ` as follows.

y = tx − tx1 + y1 . (3)

From this we obtain that u = −tx1 + y1 . Further from Equation 1, we know that
y1 = t2 . We also know the equation of x1 in terms of t ( x1 = 2t + a from the
Equation 2), as a result we get the following equation.

u = −t2 − ta. (4)

Notice that Axiom 6 mandates that the line ` is tangential to another parabola P 0 ,
defined by the focus p2 = (c, b) and the line and l2 : x = −c. The equation of this
parabola is given by

1
x= (y − b)2 .
4c

Applying the technique seen earlier, we obtain that the equation of the slope of `
2c
in this case to be t = (y−b) . Using this, we get u = b + ct . Equating the value of u
obtained in Equation 4 with this, we obtain the following equation.
c
b+ = −t2 − ta. (5)
t

Notice that this translates to the equation t3 + at2 + bt + c = 0, matching with the
original equation that we started with, this completes the proof. 

3. Mathematics for Origami

While in the previous section, we saw that origami could be very handy to learn
mathematical concepts, in this section effectively, we will see that the relationship
is symbiotic. Many of the origami models require deep mathematical insights and
techniques, and we will survey some of them.
The complexity of an origami model depends on the number of attachments it
has. For example, an origami model resembling a bird with a head, a tail, and two
wings is less complex than a model resembling a beetle with a head, two horns, and
six legs. One approach to building an origami model is to come up with what is
called a base. A base is a geometric object which roughly resembles the end product.
Historically most of the origami models were constructed by trial and error basis.

34
 head
1
1

1 2
1

1
3

tail

Figure 8.

That is, the paper is folded roughly in the direction of the base until the desired
objective is achieved. One of the most difficult and important tasks in coming up
with an origami construction is to identify how and when to create a fold. This
translates to identifying the crease pattern required for the construction. While for
simpler models, the trial and error approach is possible, it is highly inefficient for
complex models. One important and pertinent question in this regard is whether
one can come up with the crease patterns that can be folded into the desired base.
The computer programs TreeMaker and Origamizer achieve this objective. Here,
we survey the techniques used in the Tree Maker program. We closely adopt the
definitions and language used in the TreeMaker manual [13]. To keep the article
simple, we are making our exposition is very brief and informal. We direct the
interested readers to [1, 12] for an extensive exposition on the subject. TreeMaker
is a computer program developed by a famous origami artist and scientist Robert
Lang. The software has been used to develop several complex origami models, see
Figure 95 .

Figure 9.

Given any base, let its tree diagram be the graph obtained by shrinking its
skeleton to straight lines. In reality, the edges of such a tree diagram can have
weights; these weights correspond to the size of foldings required in the base. For
example, Figure 86 is the tree diagram of the lizard model next to it. Notice that
it has a head, a tail, two forelegs, and two hind legs, not all have the same size. A
tree diagram is said to be uni-axial if firstly it is tree-like (i.e., it is connected and
has no cycles) and further, it has a main stem, and every branch only originates
from this stem. The TreeMaker algorithm works when the required base is of type
uni-axial.
The tree diagram is crucial to obtaining the desired algorithm. Every vertex (in
a graph sense) of such a tree diagram, which has no outgoing edges, is called the
terminal node. In the figure, these are represented by circular nodes. Every other
vertex is called the internal node; in the figure, these are the rectangular nodes.
An important mathematical property that allows a given uni-axial tree diagram to
be folded to its base is as follows:

5 Pic source: https://langorigami.com/artworks/

6 Pic source: https://origami.me/lizards/

35
 If one can find in a square, a set of points, each corresponding to a vertex in the
tree diagram such that the distance between any two points is greater or equal to the
distance between the corresponding vertices in the tree diagram, then it is possible to
fold such a set of points into the required base.
Notice that the property is existential and does not immediately provide a way
to recover the folding pattern. To obtain the crease pattern, the first observation
is as follows:
If the distance between any two terminal points is equal to the distance between their
corresponding vertices in the tree diagram, then the line between them is definitely a
crease in the final base.
The algorithm crucially identifies points in the square so that maximal such creases
occur, partitioning the square into polygons. This already is a computationally hard
problem and has deep connections to a famous graph theoretical problem called
the cycle packing problem.
The algorithm then depends on yet another mathematical property that: creases
for each polygon partition of the square can be identified separately. The handling
of a triangle and quadrilateral is well known, i.e., there are well-known methods to
find creases for such simple polygons. However, other complex polygons can create
complexities. One way to get around this is to simplify these polygons by adding
extra terminal vertices to the tree diagram. This roughly translates to refining the
creases and hence, partitioning of the complex polygons into simpler polygons. The
algorithm can now be summarized as follows:

• Obtain the tree diagram of the desired base.

• Obtain points on the square such that the points have appropriate distances.
• Mark all the creases with minimum lengths.
• Identify the polygons partitioning the square, based on the crease marking
obtained earlier.
• Process the crease pattern for the polygons individually. Simplify the poly-
gons if needed.

While this algorithm provides the necessary crease pattern to obtain the base,
identifying whether a crease is an inside fold (valley fold) or an outside fold (moun-
tain fold) is another problem entirely. While there are effective heuristics for this,
the problem, in general, is still open.
Another interesting problem that one encounters in origami is that of identifying
whether a given crease pattern can be flat-folded. An origami model is said to be
flat-folded if it can be compressed without making any additional creases. The
question is whether it can be determined automatically, just by looking at the
crease pattern, if it can be flat-folded. While there are results for sub-classes of the
crease pattern due to Maekawa Jun and Kawasaki Toshikazu, the general problem
still remains open. Thus, there are very many mathematical properties of origami
that are being actively studied, some of which are still waiting to be solved. This
demonstrates the profound impact that mathematics has on the folding art.

4. Conclusion

In this article, we investigated the connections between origami and mathematics.

We first surveyed different types of origami models that are in vogue. We then
showed how origami can effectively be used to prove mathematical theorems. We

36
 also explored briefly the use of mathematics in constructions of complex origami
models.

Acknowledgment
The author would like to thank UGC SAP (DSA I) for providing financial support.
The author would like to thank the anonymous referee for helpful comments and
suggestions.

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