Mathematics Investigatory Project

Submitted by:

CACNIO, Renaissa

CRUZ, John Robert

GUEVARRA, Jollien

VENDIVEL, Naomi Ricci

Introduction

Conceptual Framework

Related Literature

Conceptual Framework

Exploring Systematically

Conjectures

Hidden Learning

References
 CHAPTER 1

INTRODUCTION

Whenever people heard about origami, they often associated it with kids. It is an artwork

commonly created by children particularly those who are in kindergarten. It can be created using

a piece of paper and some folding. The 'origami' or art of paper folding was originated in China.

It is known as “Zhezhi”. And it later became popular in Japan and is now considered as Japanese

art associated with Japanese culture.

Traditional, modular, rigid are some of the various styles of origami. Traditional

origami remains in the idea that only a single sheet of paper can be used. It shouldn’t be cut or

glued in any way. Paper cranes and paper boats can be considered as part of this style. Rigid

origami can be folded flat with a rigid motion. It is the idea of folding a sheet of paper where-in

it collapses easily without bending the regions between creases. The rigid map fold that was

invented by Koryo Miura, a Japanese astrophysicist, was used as design pattern for solar panel

arrays use for space satellites.

Since using a single sheet of paper has its limitations, the modular origami is

invented. The modular origami models are made up of more than one sheet of paper. It is formed

by locking the paper together to form larger units. One defining characteristic of modular

origami states that the larger model must be made up of identical units. Although models made

with different units can be constructed and are often referred to as “modular”, they do not truly

meet the requirements of the modular definition.

On 1734, the first example of modular origami was presented. It was called as

“Magic Treasure Chest”, its shape is cube. Unfortunately, the traditional “kudusama”, paper
flower strung together into a sphere. It is considered as the precursor to modern day modular

origami. It was in the 1960’s where the modular origami became truly popular. Ever since it

became popular, mathematicians have discovered its uses in explaining a vast number of

mathematical models. By inventing new folds and models they are continually contributing

valuable information to the field of mathematics.

As the time goes by, people created various styles of origami and it is now expanded and

evolved. From paper cranes and paper boat, the origami today has a lot of improvement. Aside

from being an artwork, it is also a subject of interest in the field of mathematics. Although this

art is an art of paper folding, the studies reveal that it has many mathematical characteristics as

well. It can be used in geometry, calculus and even in abstract algebra.

This methodical investigation is expected to use the origami to provide the needs of the

students who are studying geometry, calculus and abstract algebra. The researcher’s wants to

provide a good foundation of mathematical support that may help the children and/or teenager on

solving the mentioned above field of mathematics. By researching about mathematical origami,

the researchers will help to increase the knowledge of each student in mathematics and art.
 CHAPTER 2

CONCEPTUAL FRAMEWORK

Geometrical shapes produced

Size of the Paper to use

Kind of paper to use

Paper size

Theoretically, you'll fold an oversized piece of paper a lot of typically than a tiny low

one. Considering the size of the geometrical shape you’re trying to make needs to be. Complex,

impressive pieces may benefit from added size, which means you should use a sheet of paper

that is larger and can be folded more ways. On the other hand, smaller paper forces you to work

meticulously, and results in stunning, delicate shapes. By calculating the surface one can

compare the results and see how the size of the paper affects the surface area to the origami.
Kind of paper

Almost any flat material can be used for folding; the only requirement is that it should

hold a crease.

Origami paper

Origami paper often referred to as "kami" (Japanese for paper), squares of various sizes

ranging from 2.5 cm (1 in) to 25 cm (10 in) or more Origami paper weighs slightly but copy

paper, creating it appropriate for a wider varies of models.

Normal copy

Normal copy of paper with weights of 70–90 g/m2 (19–24 lb) may be used for easy

folds,. Heavier weight papers of 100 g/m2 (approx. 25 lb) or more can be wet-folded. This

technique permits for a lot of rounded sculpting of the model that becomes rigid and durable

once it's dry.

Foil-backed paper

Foil-backed paper, as its name implies, is a sheet of thin foil glued to a sheet of thin

paper. Related to this is often tissue foil that is formed by gluing a skinny piece of paper to room

aluminum foil. A second piece of tissue may be affixed onto the reverse facet to supply a

tissue/foil/tissue sandwich. Foil-backed paper is out there commercially, however not tissue foil;

it should be hand sewn. Both sorts of foil materials area unit appropriate for complicated models.
Washi (和紙)

Washi (和紙) is that the ancient art paper employed in Japan. Washi is mostly harder than

normal paper made of pulp, and is used in many traditional arts. Washi is usually created

exploitation fibres from the bark of the gampi tree, the mitsumata shrub (Edgeworthia

papyrifera), or the paper mulberry but can also be made using bamboo, hemp, rice, and wheat.

Artisan papers

Artisan papers like unryu, lokta, hanji[citation needed], gampi, kozo, saa, and abaca

have long fibers and are often extremely strong. As these papers are floppy to start with, they are

often back coated or resized with methylcellulose or wheat paste before folding. In addition,

these papers area unit very skinny and compressible, allowing for thin, narrowed limbs as in the

case of insect models. Paper money from varied countries is additionally fashionable to make art

with; this is often far-famed multifariously as greenback art, Orikane, and cash art.
 CHAPTER 3

REALATED LITERATURE

3.1 Origami in Mathematics

In this section, it discussed how Origamis were used in Mathematics, and how Origamis

were used to connect it to Mathematics. A simple fold with or without a specific measurement

can create an artistic geometric shape. Moreover, whenever a person heard the word Origami, the

first thing that will come in one’s mind is “an activity for children”.

According to the study of L.J Fei entitled Origami Theory and It’s Applications stated

that, Origami presents different fundamentals in the future Engineering industry. By means of

creating underlying principles in designing, one can create transformable that can show the

transition of Origami from a simple paper turning it into a new artistic material. Therefore,

origami is not just for fun but can be used as draft for practitioners in creating an artistic design

before coming up into one idea.

3.2 Origami in Daily Lives

Origami is very well-known as an art of folding papers. However, it is one of the trends

in fashion. For example, Zhong You and Weina Wu’s foldable grocery bag which was an

origami-inspired design. It allowed individuals especially women who are engaged in fashion to

use this as a new fashioned-style bag which was made from a rigid materials or any open-topped

paper bags (box style) that could be easily folded (depending on the preferred style) and should

not have an open bottom. This kind of bag is used also not only just for fashion but instead of

more on packaging gifts and groceries.

In addition, this origami-inspired paper bags are not only mathematically interesting

because of its wondrous various shapes, but also it has a lot of practical implications in which all

the consumer products can use.

Figure 3: Folding sequence of a bag with h=1.5d, w=2d and ϕ=75

STATEMENT OF THE PROBLEM

.
 CHAPTER 5

EXPLORING SYSTEMATICALLY

In this Chapter, this will show how to create origami using mathematics.

Origami models are basically done through folding, wherein, it will form a figure based

on desire. Folding a paper is standard but when it comes to origami there are patterns to consider.

There are problems in making an origami. Based on Maekawa's theorem it says that at any vertex

the number of valley and mountain folds always differ by two.

It follows from this that every vertex has an even number of creases, and therefore also

the regions between the creases can be colored with two colors. Also in Kawasaki's theorem it

says that at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even.

However a sheet cannot perforate a fold

Figure 4:
 However, we can make an origami step by step as shown below.

Activity 1 Construct a shape

1. Better understanding in making an origami will help you easily to make your own.

2. It’s best to analyze first before folding in order to produce desired shape.

3. You may do it again so that you won’t forget the process in making an origami.

4. Develop instructions so that others may do it as well.

Figure 5

Activity 2 Explore and Experiment with paper

1. Challenge yourself differ from the given shapes.

2. Create several shapes

3. Try different scales and type of papers.

1. Find a beginner shape that can easily copy.

2. Follow the instructions given.

3. Pay attention when following instructions.

4. Once it’s done color the open spaces with marker

5. Unfold the shape

6. Draw the folds with dotted lines

7. Consider the following: is it symmetrical? How do the color shaped relate to each

other?
 CHAPTER 6

CONJECTURES

From the example above, researchers made these following conjectures:

1.) The complexity of the origami produced will be based on the size of the paper.

2.) Mathematics and paper folding (origami) couldn't be related with one another.

Counterexamples:

1.) The modular origami is complex origami where-in the papers are connected through its

symmetrical pattern. The more papers are added, the more symmetrical foldings can be

made. The more symmetrical foldings, the complex origami can be.

2.) There are various styles of origami, flat and geometric. Mathematics and paper folding

(origami) is interrelated with one another. Through the penultimate module of Robert

Neale we can prove that it is related with one another.

Robert Neale, developed a system to a model equilateral polyhedra based on a module

with variable vertex angles. According to him, each module has two pockets and two tabs, on

opposite sides. The angle of every given tab can be changed independently of the other tab. Each

pocket can receive tabs of any angle.

The most angles form polygonal faces:

● 60 degrees (triangle)

● 90 degrees (square)
 ● 108 degrees (pentagon)

● 120 degrees (hexagon)

Each module joins others at the vertices of a polyhedron to form a polygonal face. The

tabs can form different angles on opposite sides of an edge. Whenever the internal angle

increases for squares, pentagons and so forth, the stability decreases. Many polyhedra call for

unalike adjacent polygons. For example, a pyramid has one square face and four triangular faces.

This requires hybrid modules, or modules having different angles. A pyramid consists of eight

modules, four modules as square-triangle, and four as triangle-triangle.

Further polygonal faces are possible by altering the angle at each corner. The Neale

modules can form any equilateral polyhedron including those having rhombic faces, like the

rhombic dodecahedron.
 CHAPTER 7

HIDDEN LEARNING

Origami is a paper folding as we all know. It is not all about paper folding, we can be

creative and it captivates not only children but all ages. It develops many aspects such as

sequencing skills, spatial skills, math reasoning, and patience skills. When it comes to

mathematics, origami is inclined with Geometry in a sense that the children will learn different

geometric figures. Set of Postulates are similar to some Euclidean geometry, proven that some

origami folds can solve quadratic and cubic equations. A scrutiny of various techniques and

models can enlighten how important origami is in the field of mathematics.

Bibliography

Yin, S. (2009, July 3). The Mathematics of Origami. Retrieved July 2019, from

sites.math.washington.edu:

https://sites.math.washington.edu/~morrow/336_09/papers/Sheri.pdf?fbclid=IwAR1Ct1qxfe1ecr

eSOYZvcOnCo9E6hDmg_cv39TUYMFdkz8Hd4v6F3AlVlOI

Cannella & Dai 2006; Dai & Cannella 2008