Discovering the Geometric and Algebraic Foundations Behind Computational Origami

T h e U n i v e r s i t y o f A r k a n s a s

D e p a r t m e n t o f M a t h e m a t i c s

U n d e r t h e D i r e c t i o n o f D r . B . M a d i s o n

4 / 2 6 / 2 0 1 2

Todd J. Thomas

This paper provides the basic template for Research into the relativity new field of Computational Origami. In this paper we define Computational Origami based on the logic of the Euclidean Axioms and Algebraic Structures. We show that not only is the metric space in which this topology exists is well defined, but also that it is complete and efficient. But the truly most remarkable part is how we show that the real power behind Origami Folding is how it takes us out of the set of all Real Numbers and lets us fold our way in the set of all Complex Numbers.

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Contents

1.0  INTRODUCTION 2 

2.0 PRACTICAL APPLICATIONS 2 

2.1 Practical Applications in the Space Program .....................................................3 

2.2 Practical Applications to Biology and Medicine ...............................................4 

2.4 Current Real World Applications ......................................................................5 

3.0 FOUNDATIONS IN EUCLIDEAN GEOMETRY 5 

3.1 Axiomatic Systems ............................................................................................5 

3.2 Euclidean Axiom Set for Geometry ...................................................................6 

3.3 Huzita-Hutori Axiom Set for Origami ..............................................................8 

3.4 Linking Euclidean Constructions to Origami Folding .....................................11 

3.5 Linking Origami Folding to Euclidean Constructions .....................................13 

3.6 The Fundamental Difference ...........................................................................14 

4.0 ORIGAMI FOUNDATIONS IN ALGEBRA 15 

4.1 Definitions from Algebra .................................................................................15 

4.2 The Origami Pair..............................................................................................16 

4.3 Foundations of an Origami Constructible Set ..................................................17 

4.3 Origami Roots of Polynomials .........................................................................20 

5.0 BEYOND EUCLID 21 

5.1 Folding Cube Roots .........................................................................................21 

5.2 Solving the Classical Problem of Trisecting Any Angle .................................21 

6.0 CONCLUSION 23 

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BIBLIOGRAPHY 24 

1.0 INTRODUCTION

Origami is known to most as the ancient Japanese art of paper folding which was

started by Buddhist Monks in the sixth century. The word origami is actualy a mash of

two Japanese words “ori” for folding and “kami” which means paper. This art form has

largely gone unnoticed by the sciences for nearly 2000 years; however, starting in the

early part of the 21st century physicist, medical researchers, and of course mathematicians

started finding solutions to real world problems hidden deep in the folds of this ancient

art form. As scientist and mathematicians started probing the basic foundations of

origami they have found a world of wonder just as intricate as the ancient art itself.

2.0 PRACTICAL APPLICATIONS

Although computational and mathematical origami may be interesting to some

mathematicians, there is a practical aspect to this field of study as well. Real-world

problems that require large surfaces to be compacted into small spaces for transport, then

deployed reliably, are exactly the types of problems mathematical and computational

origami solves. Some of the most promising areas where this discipline can be applied

are the space program, medical sciences, and biology. Currently, the automobile industry

uses techniques from computational origami to keep people safer on the roads.

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2.1 Practical Applications in the Space Program

The Lawrence Livermore National Laboratory in Livermore, California has plans

to put a telescope into deep space, however; this is

not just any typical telescope. These forward-

thinking researchers are in the design phase of

engineering a deep space telescope with a 100-meter

aperture for deployment at some point in the future.

Just for reference, the Hubble Space Telescope has

an aperture of a paltry 2.4 meters by comparison.

According to Dr. Robert Lang, an expert in Origami

Sekkei or Technical folding, the major problem these

researchers face is how to fit a lens that measures 100-

meters across into a shuttle or rocket transport whose

cargo bay is only a few meters wide (Wertheim, 2005). This is accomplished by treating

the lens like a laminar surface using hinges that follow along the crease lines. As shown

in Figure 1, the prototype of the lens was composed of 72 segments or panels made up of

sixteen rectangles, thirty-two right triangles, and twenty-four isosceles triangles. These

panels are then subdivided into eight petals each sweeping and area of 45 degrees. Each

petal consists of three isosceles triangles, four right triangles, and two rectangles (Heller,

2003). Likewise, Lang notes the solution to this problem and other similar problems

such as a 500-meter solar sail will require some type of folding on an unprecedented

scale. A first success using origami’s came in 1995 when the Space Flight Unit, a

Japanese satellite, was launched into low orbit. Its solar arrays deployed utilizing an

Figure 1: The prototype lens is composed of 72 segments which are then divided into eight petals. Each petal, one of which is highlighted, sweeps 45 degrees or one-eighth of the structure.

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origami technique known as Muira-ori which has also been found to occur naturally in

leaves.

2.2 Practical Applications to Biology and Medicine

Medicine has its own set of unique problems, and researchers at Oxford

University, U.K. wanted to develop an artificial stent that could easily travel through the

circulatory system of a patient, but be large enough to hold open the collapsed artery once

it was put into place (World Science, 2007) Using an origami pattern call the

Waterbomb Base, they were able to develop a stent that when folded was a mere 12mm

and would easily pass through small capillaries for long distances without damaging the

stent or the patient. Upon reaching its destination it would then expand to 23 mm when

unfolded to hold open a collapsed artery and restore blood flow.

Aside from the naturally occurring Muira-ori technique as

discussed above, researchers at the Dana-Farber institute have

combined origami with nanotechnology to fold sheets of DNA into

different shapes, such as octahedrons, smaller than the thickness of a

human hair (The Scripps Research Institute, 2004). In the near future

these folded objects could be used to transport medicines directly into

a cell. According to Dr. William Shih, the lead researcher on this

project and assistant professor in the Biological Chemistry

Department, his group was able to fold DNA to make several

different shapes to include a genie bottle, two kinds of crosses, a

square nut, and a railed bridge (Shih, 2008)

Figure 2. An artificial stent that uses the origami patter Waterbomb bases to save human lives.

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2.4 Current Real World Applications

At first thought the automobile industry may be a strange place to find the likes of

Dr. Robert Lang, an expert in Origami Sekkei, however; it seems that anyone who has

survived an automobile crash due to the deployment of the car’s airbags is deeply

indebted Dr. Lang. Working with researchers

at EASi Engineering in Germany, Dr. Lang

developed a crease pattern that allows for a

three-dimensional polyhedron (the airbag) to

be folded into a flat surface that deploys

easily, completely, and at the proper pressure

within microseconds. Using an algorithm

dubbed the Universal Molecule that was created by Dr. Lang, it shows that there is little

difference between folding a three-dimensional polyhedron onto a flat plane and origami

folding a flat sheet into another flat polygon (Lang, 2004-2012). EASi was able

incorporate this research into their airbag design system which has been used by

automotive manufacturers all across the globe.

3.0 FOUNDATIONS IN EUCLIDEAN GEOMETRY

3.1 Axiomatic Systems

In order to ensure that our origami system is consistent, we need to develop it as

an axiomatic system. Similar to the approach Euclid of Alexandria took with Geometry,

an axiomatic system is logical and will possess a set of axioms from which we can derive

other statements (Weisstein, 1999-2012). Furthermore, we shall strive for economy and

Figure 3. Using the Universal Molecule algorithm, Lang was able to identify a crease pattern that is highly efficient for deploying airbags.

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efficiency with our axioms, we want them to be independent, which means we do not

want to assume any axiom can be proven from the others. Lastly we want to make sure

that we can in-fact prove or disprove any statement about our system from the axioms

alone, that is we want to be able to say that our system is complete. The advantage to

having an axiomatic system is that once a set of fundamental axioms are determined, we

can then start deducing other properties (e.g., as lemmas and theorems) from within our

system, and thus construct a wholly consistent, independent, and complete origami

mathematical system.

3.2 Euclidean Axiom Set for Geometry

Over 2000 years ago Euclid of Alexandria approached geometry from an

axiomatic stand point as mentioned earlier. By using five independent postulates, Euclid

constructed a logical and consistent geometry from which all other geometric lemmas,

and theorems would later be derived. In this section we will recall and briefly discuss

these first five postulates, and then in Section 3.3 we will compare them to the postulates

of Mathematical Origami. A quick search of the internet reveals thousands of websites

devoted to Euclid’s Elements; however, we refer to the website of Dr. David E. Joyce of

Clark University, and his translation for the remainder of our discussion. Note that the

designations E1 to E5 are not part of Dr. Joyce’s work, but will be used in future sections

as references to the specific postulate:

(E1) “Postulate 1. Let it have been postulated to draw a straight-line from any

point to any point.

(E2) Postulate 2: And to produce a finite straight-line continuously in a straight-

line.

(E3) Postulate 3: And to draw a circle with any center and radius.

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(E4) Postulate 4: And that all right-angles are equal to one another.

(E5) Postulate 5: That, if a straight line falling on two straight lines makes the

interior angles on the same side less than two right angles, the two straight lines,

if produced indefinitely, meet on that side which are the angles less than the two

right angles.” (Joyce, 1996)

Postulates three and four are self-explanatory so we will not go into detail about these,

however for the other postulates we shall give a brief

interpretation. Postulate 1 (E1) gives us our first

construction using a straight edge. In this postulate,

Euclid is simply telling us that for any two points in a

plane we can use a straight edge to construct a straight

line (AB) between the two points. What he does not

explicitly say, but it is implied, is that line is also unique. In

Postulate (E2), given line segment AB, we can construct and extend the segment AB to

CD. It is interesting to note that Euclid does not tell us how far CD can be extended.

Postulate 5 (E5) is also known as the parallel postulate. Thus if we have two lines l and

m and then a third line t intersects both l and m , the two lines will intersect on the side

where the angle each line makes with the traversal is less than ninety degrees (Joyce,

1996). Starting exclusively with one of these five postulates, a system is considered

geometrically constructible if we can show that by starting at a given point/line/circle we

end up at another point/line/circle that makes up whatever geometric object we are

seeking.

Figure 4: If two lines a and b are parallel, then angle theta is equal to angle beta.

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3.3 Huzita-Hutori Axiom Set for Origami

Origami is the art of paper folding, and as most of us know from our elementary

school days, there are certain folds that just seem fundamental. For instance, we can fold

a straight line quite easily, however trying to fold a curve, although possible, is quite a

difficult undertaking, and nearly impossible to control. Since we are trying to establish a

link between Euclid’s postulates and origami, we will consider the folding of a curve to

be non-fundamental and exclude it for our purposes (Geretschlager, 1995). Now consider

an origami construction such as a crane. We start with a flat sheet of paper, a plane if

you will analogous to the Euclidean plane in geometry constructions. Whereas in

geometric constructions we start with a point, in an origami construction we start with a

fold, and as we develop our folds, a more complex object is formed. Of course when we

fold an origami object, we are going from the two dimensional plane to a three

dimensional object; however, after we have created an origami object we can then unfold

object and return the paper back into a plane. What we will then consider are the creases

that are left behind from the unfolded origami object. Thus since we wish to find a

relation between Euclidean constructions and origami constructions, we must first define

a set of allowed operations, similar to five postulates that Euclid defined for plane

geometry. The first six postulates were presented by Humiaki Huzita at the First

International Meeting of Origami Science and Technology in 1991 and a seventh one was

found by Koshiro Hutori in 2002. Although Huzita and Hutori were the first to present

their axioms that bear their names, it is important to note that in 1989 Jacques Justin

published a paper entitled Resolution par le pliage de l’equation du troisieme degre et

applications geometriques, in which he accounted for seven combinations of alignments

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(Lang, Huzita-Justin Axioms, 2004). We will enumerate these postulates (O1),…(O7) to

distinguish them from the Euclidean Axioms (E1),…(E5).

(O1) Given two points and we can fold a line connecting them.

Figure 5

(O2) Given two points p andp we can fold p ontop .

Figure 6

(O3) Given two lines l andl we can fold linel ontol .

Figure 7

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(O4) Given as point p and a line l we can make a fold perpendicular to l that

passes through p .

Figure 8

(O5) Given two points p andp andlinel we can make a fold that places

p ontol andpassesthroughp .

Figure 9

(O6) Given two points p andp andtwolinesl andl we can make a fold that

places p ontol andp ontol .

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Figure 10

(O7) Given a point p andtwolinesl andl we can make a fold perpendicular

to l thatplacep ontol .

Figure 11

These seven postulates constitute all the allowed folds for origami constructions.

3.4 Linking Euclidean Constructions to Origami Folding

Now we wish to show a linkage between the Euclidean postulates and the Huzita-

Hutroi postulates. We must be able to show that each of the Euclidean postulates (E1)-

(E5) can be replaced by a series or combination of origami postulates (O1)-(O7). Note

that for ease of use and standardization, we will “draw” a Euclidean construction and we

shall “fold” an origami construction. Thus we define our lemma

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Lemma 1: The five Euclidean Postulates (E1)-(E5) can be replaced by

combinations of origami folding (O1)-(O7).

One can easily see that (E1) is identical to (O4), and similarly, (E2) is just an

extension of (O1).

(E3) Although we have chosen to not consider a circle in our origami

constructions, a circle is still be well defined if we know its center M and its radius r.

Thus this can be found by using several of our origami postulates.

Let’s designate the center as M, and radius as . If these parts of a circle are

known, then we can fold the perpendicular bisector of M . Suppose that the two lines

in (O3) intersect at M, then we can fold the perpendicular bisector at M that

passes through .

(E4) and (O4) both produce perpendicular lines which by definition have an angle

between them of ninety degrees, thus both right angles are equal. Consider that the two

lines in (E5) are not parallel; therefore we can find the unique point of

intersection of the two lines. Similarly (O3) we can fold a line through the point of

intersection of our two lines in which that passes also through . Thus in

summary, we refer to Dr. Robert Geretschlager to fully develop our first theorem of our

Origami System.

THEOREM 1. Every construction that can be done by Euclidean construction can

also be accomplished by elementary origami folding. Specifically, we can use (O1)-(O7)

either directly or in combinations of elementary origami folds, to replace the Euclidean

postulates (E1)-(E5). (Geretschlager, 1995)

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3.5 Linking Origami Folding to Euclidean Constructions

Now we must show that the origami folds (O1)-(O7) can be replaced by the

Euclidean postulates (E1)-(E5), with the exception of (O7) which we will need to look at

a little more closely.

Lemma 3.5.1: All of the origami folds (O1)-(O7) can be replaced by Euclidean

Constructions derived from the Euclidean postulates (E1)-(E5).

Recall from 3.4 that, (O4) is identical to (E1), and (O1) can be replaced with (E2).

Similarly, (O2) is replaced by (E1) as we may construct any point on a straight line. Now

we consider (O3), (O5), and (O6), which can easily be constructed by the mid-parallel for

(O3), and by basic construction of points for (O5) and (O6). Thus, since all of these

constructions are known by Geometrical methods, it follows that (O1)-(O6) can infact be

replace by (E1-E5).

(O7) gives us a little difficulty, but if we suppose that is the directix of a

parabola and that is the focus of the parabola, and then we make tangent to the

parabola. Thus if we know the Focus ( and we know the directix ( , and also if we

know any point on the parabola T, we know that by the definition of a parabola that the

length from the focus ( to any point on the parabola (T) equals the distance from T to

the directix intersecting at (B). If we now construct B , this is the diagonal of a

rhombus, Thus by constructing the other diagonal, starting at T and intersecting at a

right angle, continuing until it intersects the directix at S, then ST is tangent to the

parabola at point T. It is easy to see that ST is the perpendicular bisector of at K, thus

and since a ST is tangent is to the parabola, if we fold along ST then we will

place on the directix. This particular Euclidean construction is nicely laid out by Dr.

William Harter of the University of Arkansas in his book entitled, Modern Physics and

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its Classical Foundations: A Geometric Introduction to Analysis of Quantum Momentum,

Energy and Action (Harter, 2012). Now we are ready to state our next theorem:

THEOREM 2. Any construction that can be completed using origami folds (O1)-

(O7), can also be constructed with straightedge and compass according by way of the

Euclidean postulates (E1)-(E5) (Geretschlager, 1995).

Since all of the Euclidean postulates can be replaced with origami folds, and all

origami folds can be replaced by Euclidean postulates, then the two systems must be

equivalent. Thus the set of all possible Euclidean constructions must be a subset of the

set of all possible constructions of origami folds.

3.6 The Fundamental Difference

We have shown that in order to analyze and compare Geometric Construction

with origami folding we had to define the procedures that were allowed. This set of

fundamental folds we have defined for our system are consistent, independent, and

complete. This assures us that origami constructions are in fact an axiomatic system. An

important fundamental difference between Euclidean constructions and origami folding is

the fact that for Euclidean constructions, the point is the most basic entity of the

construction, however in origami constructions the basic entity is the straight line. Thus

we have also shown that just as a point in Euclidean construction can be drawn anywhere

in the plane, so too can a straight line be folded anywhere on an origami space.

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4.0 ORIGAMI FOUNDATIONS IN ALGEBRA

4.1 Definitions from Algebra

In order to approach origami construction from an algebraic point of view, we

need to recall some standard definitions from Abstract Algebra. The following

definitions are taken directly from David S. Dummit and Richard M. Foote’s textbook

Abstract Algebra (Dummit & Foote, 2004) Definition 4.1.1. A Group is an ordered pair ,∗ where G is a set and ∗ is a binary operation on G satisfying the following axioms:

(i) ∗ ∗ ∗ ∗ ∀ , , ∈ . . ,∗ is associative. (ii) There exist an element in , called the identity of G, such that for all

∈ we have ∗ ∗ , (iii) For each ∈ there is an element of called an inverse of , such

that ∗ ∗ . (2) The group ,∗ is called abelian (or commutative) if ∗ ∗ for all , ∈ . Definition 4.1.2. Let Ω be any nonempty set and let be the set of all bijections from Ω to itself, then symmetric group is the set of all permutations of Ω. Thus if we consider , , ∈ then we can express this group using set notation as , , , , , , , , , , , , , , , , , This set consist of all the permutations of , , . It should also be noted for future reference that the order of ! Definition 4.1.3. A ring R is a set together with two binary operations and (called addition and multiplication) satisfying the following axioms:

(i) , is an abelian group, (ii) is associative : for all , , ∈ , (iii) The distributive laws hold in : for all , , ∈

and . (2) The ring R is commutative if multiplication is commutative. (3) The ring R is said to have identity (or contains 1) if there is an element 1 ∈ with 1 1 for all ∈ . Definition 4.1.4. A field is a set F together with two commutative binary operations on F such that , is an abelian group (with identity called 0) and

0 , is also an abelian group, and the following distributive law holds:

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for all , , ∈ . Definition 4.1.5. The element ∈ is said to be algebraic over F if is a root of some nonzero polynomial ∈ . If is not algebraic over F (i.e., not the root of any nonzero polynomial with coefficients in F) then is said to be transcendental over F. Definition 4.1.6. A polynomial in any field is said to be irreducible over F if its degree is less than or equal to one, and given a factorization,

, with , ∈ , then deg 0. Now that we have reviewed some necessary basics of Algebra we can continue our look at origami constructions from an algebraic point of view.

4.2 The Origami Pair

Recalling our original origami axioms (O1)-(O7), we will now formalize these

folds in order to define an origami pair in the plane. We note that creases on the paper

are isomorphic to lines in a plane, and the corners of our paper are simply points where

our line creases shall meet (Auckly & Cleveland, 1995).

Definition 4.2.1. Let ℘ be a set of points and suppose that is any collection of

lines such that ℘, ∈ . Then ℘, is an origami pair if

(Pi) For any two intersecting lines and , the point of intersection

℘ ∈ ℘.

(Pii) Given any two points ℘ ,℘ ∈ ℘ there is a unique straight line that

passes through both points.

(Piii) Given a line segment with endpoints ℘ and ℘ in , then the

perpendicular bisector of that segment is also in .

(Piv) Let , ∈ then a third line that is equidistant from both

is also in .

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(P6) If , ∈ then there exist an ∈ , such that is a mirror

reflection of about .

Thus for any subset of the origami plane containing at least two points, there is at most

one collection of lines which will pair with it to become an origami pair (Auckly &

Cleveland, 1995).

4.3 Foundations of an Origami Constructible Set

Similar to how we can construct a set of Euclidean constructible numbers, we

want to be able to define set for origami constructible pairs. This is our end-goal, to

algebraically define a set that contains all the

origami constructible numbers using only the

allowed folds from our definition of origami pair,

which was derived from our origami postulates

(O1)-(O7). Let us first define the framework our

origami constructible set as a subset of the

complex numbers. Let us first construct the

complex plane using our origami folds. We will

suppose that our origami plane is infinitely large

and contains only two points 0 and 1. Now we

first apply (O1) to the point 0 and 1 to construct the real axis, then by applying (O4) to 1

we create perpendicular to the real axis. Similarly we apply (O4) to point 0 and the

real axis to construct the imaginary axis. Lastly if we apply (O5) to point zero, the we get

a line that interests the imaginary axis and passes through point 1. This intersection is the

number i.

0 1

i

Figure 12. An application of (O1), (O4), and (O5), to create the complex plane.

18

Next we want to show that addition is possible through our origami constructions. Let us

start with our origami plane and three points 0, , .

Using the following series of steps we find that the

point .

⟵ 1 0,

⟵ 1 0,

⟵ 1 ,

⟵ 1 ,

⟵ ,

Multiplication is the next logical operation that is to be shown. Since we are in the

complex field, we know that

, thus by using properties of

similar triangles we can multiple a real number

r by as shown below

⟵ 1 0,

⟵ 1 1,

⟵ 1 ,

⟵ ,

We run into a problem if is a real number. We get around this issue by adding i to ,

then multiply by r then project this product onto the real axis using (O4). Another issue

that arrives if we wish to multiply a number by i then we must employ a mathematical

trick and rotate it by radians counterclockwise about zero, then we can employ the

following steps.

0

0

1 r

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⟵ 1 0,

⟵ 4 0,

⟵ 2 ,

⟵ 2 ,

Now that we can perform the basic binary operations of addition, subtraction,

multiplication by real number and multiplication by i, we can now combine these

operations to multiply by two complex numbers (King, 2004).

Inversion is the last of these operations that we need to define for our system. The

inverse of a complex number (a,b) is defined by . One will notice that inversion is

a very similar process to multiplication, and we will once again use the properties of

similar triangles to complete the inversion

process.

⟵ 1 0,

⟵ 1 ,

⟵ 1 1,

⟵ ,

Just like with multiplication, we run into a

problem when is a real number; however, we

solve this problem in the same manner that solved it for multiplication.

0

1 r

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4.3 Origami Roots of Polynomials

What we have shown up to this point is that our set of origami constructible

numbers contains the integers and the rational numbers. Similarly, the set of Euclidean

constructible number also contain the integers and rational numbers (Dummit & Foote,

2004). Thus the next logical step is to see what irrational numbers our origami set

contains. In Euclidean geometry we use the geometric mean to create square roots;

however this involves constructing a circle, which we have excluded for being a non-

elementary operation. We do have a method for a solution, recall that we can define a

unique circle if we know its origin and its radius. Thus before we take the square root of

a real number, consider the following equation solving gives us

y=√ . Now consider a circle centered at 0, ) with radius . Notice that it will

intersect at √ , 0 . Thus instead of

having to construct the circle we only need to

construct where it crosses the real axis, and

we do this by the following steps.

⟵

⟵

⟵ 5 , ,

⟵ 2 ,

Suppose we wish to take the square root of a complex number , then we must first

bisect the angle between and the positive real axis obtaining the angle bisector then

we rotate onto the positive real axis using (O5) and (P2) to get the magnitude that is

both positive and real, and we will denote this point our new r. Now we can rotate back

0 1 r

‐i

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to using (O5) and (P2) to obtain ,which is the complex square root of (King,

2004).

5.0 BEYOND EUCLID

5.1 Folding Cube Roots

We know that if we abide by the rules of Euclidean construction, we cannot

construct any roots of a higher degree than two. However, with origami construction we

will show that it is possible to construct the cube roots of a polynomial. Hatori Koshiro

method for solving for cubic equations of the form using origami is

quite elegant and relies on using two parabolas to solve the equation. Let us imagine that

for a moment that the real axis is the x-axis and the imaginary axis is the y-axis. Then in

his construction Koshiro first constructs the point , 1 and , , and we

also construct the lines and by defining as equal to 1,and as .

We then use origami axiom (O6) and apply it to , and and thus we obtain a new

line denoted as . This line is denoted by where m is a solution to the

original cubic equation (Koshiro, 2010).

5.2 Solving the Classical Problem of Trisecting Any Angle

One of the most famous problems of Greek mathematics is that of trisecting any

angle using Euclidean construction methods. It is a well-known fact among

mathematician that this is an impossible task since it requires us to be able to construct

the cube roots of an equation with just the compass and straightedge. However now that

we have shown that our set of origami constructible numbers includes cube roots we

should be able to trisect any angle by solving the equation 3 3 0

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where and tan . For this construction we will refer to Thomas Hull’s

method and show his step by step folding process for trisecting an angle.

Step 1: Starting at the corner

of the paper fold any triangle

with hypotenuse .

Step 2: Fold any horizontal line

with a height of h.

Step 3: Fold the paper along and then fold over one more time (like rolling a burrito),

then unfold the entire sheet back to the plane. You should now have lines , , .

Step 4: Mark a point d in the bottom corner where the intitial fold was made, and then

mark point b on at the edge of the paper on the same side as d.

B

D

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Step 5: Now fold the paper so that d lies on line and b lies on .

Step 6: Unfold to starting position.

Step 7: Fold point B on top of point D.

You have now successfully trisected an

angle (Hull, 1996).

6.0 CONCLUSION

We have shown that origami mathematics is a complete, independent, and well

defined axiomatic system. We have also shown that our origami postulates can be

replaced by Euclidean postulates and vice versa. This shows that both systems share

certain properties. We defined the binary operations of addition, negation, and

multiplication on our origami system, and also showed that we can construct the inverse

as well, thus defining our set of origami constructible numbers as a subfield of the

complex numbers. It is important to note that the Euclidean constructible number are a

subset of the Reals, and thus the reason that we cannot construct the solution to cubic

equations using compass and ruler. However, as we have seen we can in-fact use our

origami solve for both quadratic and cubic equations, which leads us to solving the

problem of trisecting any angle. As our technological knowledge increases, we may

someday find the solutions to technical problems in the fields of physics, astronomy, and

medicine folded away in our mathematics of origami.

24

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