1. Mathematics of Origami
By: Lizzie Hammerberg, Anna Fronhofer, Kassandra Perantoni, & Brianna DaSilva

2. History of Origami
Origami is the Japanese art of paper folding. ORI means to fold and KAMI means paper.
The origin of origami is unknown and disputed.
The earliest form of origami is found in Japan used for religionist and ceremonial purposes
By the 17th century play-origami was prevalent in Japan
Origami as well began to develop in Europe
Origami then spread from Europe to South America and then North America
In 1950 a standard set of Origami symbols was developed and is still used today
Today, there are thousands of origami books, videos, and other resources and the art form continues to evolve and develop

3. Math & Origami
Origami creates not only a folded piece of art, but as well a geometric shape figure.
Each origami shape has a geometric crease pattern, which can determine how the paper should be folded and what shape it should form.
You as well would be able to tell at each fold if it was a mountain or valley crease.
In the 1970’s Dr. Koryo Miura invented the Miura map fold.
Modular origami requires two or more pieces of paper and is used to create a “module.”

4. Terms in Origami
Crease Patterns –the pattern of creases found when an origami is completely unfolded
Mountain Crease- a crease that looks like a mountain
Flat Fold- an origami which you could place flat and compress without adding an new creases.
Valley Crease- a crease that look like a valley or a trench
Vertex – a point on the interior of the paper where two or more creases intersect.

5. The Axiom of Origami “We can fold a line connecting any two points.”
“We can fold any two points onto each other.”
“We can fold any two lines onto each other.”
“Given a point P and a line L, we can make a fold perpendicular to L passing through P.”
“Given two points P and Q and a line L, we can make a fold that passes through P and places Q onto L.”
“Given two points P and Q and two lines K and L, we can make a fold that places P onto line K and places Q onto line L.”
“Given a point P and two lines K and L, we can fold a line perpendicular to K placing P onto L.”

6. Maewaka’s Theorem
The difference between the number of mountain creases and the number of valley creases intersecting at a vertex is always 2.
Let M be the number of mountain creases at a vertex S.
Let V be the number of valley creases at a vertex S.
So, the Maekawa’s Theorem states that at the Vertex S.
M-V= 2 or V-M=2
But how do we prove this ?!!
Definition of a Polygon: the sum of the interior angles of a regular polygon is (n-2) * 180 degrees. {n is the number of sides of polygon}
AND
Interior Angles of Mountain Creases is 0˚ of flat origami
Interior Angles of is Valley Creases is 360˚ of flat origami.

7. Now Let’s Prove Maewaka’s Theorem
Proof: Let n be the number of creases, then n=M+V. Each interior angle 0˚ of mountain crease and each interior angle of 360˚ is a valley crease.
Then the sum of the interior angles is 0M+ 360V. We know that by the definition of a polygon, the sum of the interior angles is 180n-360˚ or 180(n-2).
When you equate both of these expressions 180 (n-2)=0M+360V and note that n=M+V.
Then,
180(M + V – 2) = 0M + 360V
180M + 180V – 360 = 360V
180M – 180V = 360
M – V = 2, as desired.

8. Other theorems
Kawasaki's Theorem: states that a  given crease pattern can be folded to a flat origami if and only if adding and subtracting the angles of consecutive folds around the vertex gives an alternating sum of zero.
Haga’s Theorem: states that the side of a square can be divided at an arbitrary rational fraction in numerous ways.

9. How does this connects to class?
For the Maekawa’s theorem, you can use a direct proof base on the information that is given.
We were able to proof this theorem by using the given and goal method with basic algebraic steps to solve the equation.
Kawasaki’s Theorem connects by using a Riemann sums, repeating addition of the angles
In other origami theorems functions and sets are used to deconstruct and create crease patterns.

10. How does Origami apply to real world?
self-assembling robots
foldable mirrors and solar panels
air bags
heart stent
architecture
nanobots

11. Let’s Do Origami

13. What is the Maekawa Theorem?
Answer: The difference between the number of mountain creases and the number of valley creases intersecting at vertex particular is always 2.

14. References
Denne, Elizabeth. (n.d.): n. pag. Folds and Cuts: Mathematics and Origami. 1 Apr Web. 23 Apr
"History of Origami." History of Origami from past T. N.p., n.d. Web. 23 Apr
Hull, Thomas C. "Maekawa and Kawasaki Revisited and Extended." N.p., n.d. Web. 22 Apr
Kroll, Seth. "From Robots To Retinas: 9 Amazing Origami Applications." Popular Science. N.p., n.d. Web. 22 Apr
Legner, Philipp. "60 Rhombic Dodecahedron." Wolfram Demonstrations Project (2007): n. pag. Mathematical Origami. Web. Apr.-May 2017.
"Modular Origami." Create Amazing Geometric Shapes through Unit Origami. N.p., n.d. Web. 22 Apr
Yin, Sheri. The Mathematics of Origami. N.p.: n.p., n.d.  pdf. 3 June Web. 22 Apr