Mathematics in Origami

The Japanese art of origami is not only fun and cool, but it is mathematically powerful too. Using only simple origami axioms and a little bit of creativity and time, we can fold truly amazing shapes. The possible applications of these origami shapes are just beginning to be explored. 

The relationship between mathematics and origami has been around for centuries, but the exploration of this relationship and the use of origami to teach both lower and higher level mathematics is quickly growing. And while origami is being used increasingly often to teach mathematics, mathematics is also being used to prove a wide variety of theorems about origami. For instance, in this paper we will show how mathematics can be used to help us perform a flat vertex fold of a square using an interesting twist method. Applying mathematics to paper folding allows us to narrow down how many and what type of folds we should use to properly create flat folding models. At the same time, mathematics can also be used to do the opposite; it helps us explore how many different folds we can do – and just how far the limits of origami reach.

History of origami mathematics

1. 1893: Geometric exercises in paper folding by row

2. 1936: Origami first analysed according to exam by Beloch

3. 1989: present

4. Huzita - Hatori axioms

5. Flat folding Theorems: Maekawa, Kaw Applications Justin, Hull

6. Rigid origami

7. Applications from large to very small

Types of Patterns and Folds

Flat fold: an origami which you could place flat on the ground and compressed without adding new creases.

Crease pattern: the craziest pattern found when an origami is completely unfolded. 

Mountain crease: a crease which looks like a mountain or a ridge.

Valley crease: a crease which looks like a valley or a trench.

Vertex: a point on the interior of the paper where two or more creases intersect.

Euclid's elements

Euclid, a Greek mathematician over 2000 years ago, is known as the father of geometry. His book, The Elements, is a renowned textbook in math history, focusing on geometry. Euclid’s unique contribution was his systematic approach to geometry. He derived all geometric constructions and results from five basic assumptions, including the use of a straight-edge and compass. These assumptions allowed for operations like drawing lines, extending line segments, creating circles, and defining right angles. Euclid’s method revolutionised geometry by providing a logical and step-by-step framework for geometric proofs and constructions.

Euclid’s axioms form the basis for complex geometric proofs, yet limitations exist in Euclidean geometry. Problems like angle trisection and cube doubling were unsolvable with straight-edge and compass alone. Legend has it that Delos citizens faced this challenge to double their altar’s volume. Surprisingly, origami can solve these problems, suggesting origami geometry surpasses Euclidean geometry’s limitations. 

Before we explore the twist method for performing a flat vertex fold on a square, we are going to have to get a bit more familiar with origami, its guiding principles, and some key definitions.

First, let’s begin with a definition of Origami.

Origami

The Japanese art of folding paper into decorative shapes and figures.

It is important to realize that this is an extremely broad definition of Origami. We will use a much more restricted view of Origami for the rest of this paper; We will abide by seven axioms. The first six were presented by Humiaki Huzita in 1991 and are referred to as Huzita’s Axioms, and the seventh axiom was written by Koshiro Hatori in 2001. These seven axioms are commonly accepted as the purest form of origami and we will use them to govern the ways in which we can fold paper in order to create origami forms.

The seven axioms are as follows:

1. Given two points p1 and p2, there is a unique fold that passes through both of them

2. Given two points p1 and p2, there is a unique fold that places p1 onto p2.

3. Given two lines l1 and l2, there is a fold 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 is a fold that places p1 onto l1 and passes through p2.

6. Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.

7. Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

The seven axioms of Euclidean geometry offer vast origami possibilities but limit techniques like wet origami for curved lines, modular origami for combining pieces, and Kirigami for cutting paper. These restrictions showcase the unique boundaries of Euclidean geometry in the world of Origami. Origami diagrams are detailed guides with numbered steps, ideal for beginners but restrictive for intricate designs. For more complex origami, artists turn to crease patterns, which reveal fold locations and types without a set sequence. Unlike diagrams, crease patterns challenge folders to interpret and execute the design, fostering creativity and problem-solving skills. This intricate crease pattern shared demands a solid grasp of origami fundamentals and folding techniques to transform into a stunning origami creation. While origami diagrams are user-friendly, crease patterns offer a deeper level of origami artistry, requiring a keen eye and understanding of folding principles. The transition from diagrams to crease patterns signifies a shift towards advanced origami craftsmanship, where precision and skill play pivotal roles in creating intricate paper sculptures. Exploring the properties that govern flat foldability is a task that has been explored extensively as of late.

The following four properties must hold true in order for a given single vertex crease pattern to be flat foldable.

1. The crease pattern must be two colourable. This is similar to the concept of the four colour map problem, but instead of using four colours, we may only use two to make sure no two touching sections of our crease pattern are the same colour. Below is an example of a two colourable crease pattern.
   

2. Kawasaki’s Theorem: Kawasaki’s Theorem : Given a vertex in a flat origami crease pattern, label the angles between the creases as α1, α2,… , α2n , in order. Then we must have: α1 + α3 +… + α2n-1 = α2 + α4 +… + α2n = 180°. Applying Kawasaki’s Theorem once again to Figure 1 on the previous page, we can conclude that in order for the model to be flat foldable, α1 + α3 = 180° and α2 + α4 = 180.

3. Maekawa’s Theorem: Given a vertex in a flat origami crease pattern, if M is the number of mountain creases and V is the number of valley creases at the vertex, then we must have M-V=±2. For example, if we refer back to Figure 1 on the previous page, we can see how Maekawa’s theorem can be applied to a crease pattern. Since we have four creases, we cannot have all mountains or all valleys, or else we would be violating Maekawa’s theorem because the number of mountain and valley creases would differ by four. We also violate the theorem if we have two of each type of fold. Thus, the only valid choices we have are either three mountain folds and one valley  fold, or three valley folds and one mountain fold. 

Fin.

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