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# Origami Mathematics

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by

## Caroline Noonan

on 13 June 2013

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#### Transcript of Origami Mathematics

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
~7 Axioms
~Origami saves lives!
Where did it come from?
~"Ori"=to fold
~"Kami"= Paper
~Started in China

~"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.