**Are There Mathematics Behind Origami?**

**By Caroline Noonan**

What is Origami?

What is Mathematical Origami?

Crease Patterns

Idea+Square of paper =origami figure

Huzita-Justin Axioms

Why should we care?

~Art Form

~Japanese art of paperfolding

~"Origami Mathematics is the subset of mathematics that describes the underlying laws of origami. As part of a deep, consistent (albeit incomplete) logical structure."

~Robert Lang

~Additional set of laws

~7 Axioms

~Origami saves lives!

Where did it come from?

~"Ori"=to fold

~"Kami"= Paper

~Started in China

~Spread to Japan

~"Ori"=to fold

~"Kami"=paper

Where did it come from?

~No written instructions

~1797 "How to Fold a

Thousand Cranes"

~1845 "Window on Midwinter"

~"Blueprint"

~Four Laws

Law #1:

~Two-colorability

Law #2:

~At any vertex, the number of valley folds vs. mountain folds differs by two.

Law #3

~No matter how many times you stack folds and sheets, a sheet cannot penetrate a fold.

Law #4

~Every other angle around a vertex equals 180 degrees

Kawasaki Theorem:

~In a flat-foldeable figure, every other angle must add up to 180 degrees.

I decided to test this out myself.

Why?

Axiom #1

Given two points P1 and P2, we can fold a line connecting them.

Axiom #2

Given two points P1 and P2, we can fold P1 onto P2.

Axiom #3

Given two lines L1 and L2, we can fold L1 onto L2.

Axiom #4

Given a point P1 and a line L1, we can make a fold perpendicular to L1, passing through the point P1.

Axiom #5

Given two points P1 and P2 and a line L1, we can make a fold that places P1 onto L1 and passes through the point P2

Axiom #6

Given two points P1 and P2 and two lines L1 and L2, we can make a fold that places P1 onto L1 and places P2 onto line L2.

Axiom #7

Given a point P1 and two lines L1 and L2, we can make a fold perpendicular to L2 that places P1 onto L1.

GEOMETRIC CONSTRUCTIONS

How can I create origami?

~Play around with things and hope

~Math!

How can I used Math to make origami?

~ Circle/river packing!

But how does that have anything to do with origami?

Flaps and Legs

~Flap takes up a circle

~To make multiple flaps, need multiple circles

~Circle packing!

Circle Packing

~"Placing circles on a square (or other shape) so that they do not overlap and their centers are inside the square."

How do I create origami?

~Draw "tree" (stick figure)

How do I create origami?

~Create crease pattern

(base)

~With artful folding, you get object.

Airbags

Heart Stent

THANK YOU!

Treemaker!

Callie and Jeremy. “History of Origami.” Pieces ‘n Creases.

Thinkquest Junior, March 1999. Web. 3 June 2013.

Hatori, Koshiro. “Origami Construction: Origami versus Straight-

Edge-and-Compass. K’s Origami. Koshiro Hatori, n.d. Web. 3 June 2013.

Lang, Robert. Robert J. Lang Origami. Robert Lang, 2004.

Web. 3rd June 2013.

Lang, Robert. “Robert Lang: The Math and Magic of Origami.”

TED Talks. North American West Coast. Lecture. July 2008.

“Origami and Math.” Paperfolding.com (Origami).

Paperfolding.com. 2012. Web, 3 June 2013.

Vihart. "Origami Proof of the Pythagorean Thereom." Video.

Youtube. Youtube, 22 June 2011. Web. 3 June 2013.

Bibliography

Axiom Application