OrigamiLori Burns, Jess Barkhouse

History of OrigamiArt of paper making

originated in china in 102A

Origami is the Japanese word for paper folding

Japan developed origami as an art

Coincided with the development religion in Japan

Also used as toys, recycling and shapes

Origami axioms (Huzita 1992)1) Given points P and Q you can fold a line

connecting them2) Given points P and Q you can fold P onto Q3)Given lines L 1 and L2 you can fold line L1 onto L24) Given a point P and a line L1 you can make a

perpendicular fold to L1 passing through the point P5) Given points P, Q and a line L1 we can fold so that

P is placed onto L1 passing through point Q 6) Given points P and Q and lines L1 and L2 you can

make a fold such that P is placed onto L1 and Q onto L2

A 7th axiom that was overlooked7) given a point P and two lines L1 and L2

you can make a fold perpendicular to L2 that places P onto L1.

Topologically equivalent. If one shape can form the other without

tearing, attaching or creating holes.Euler characteristic = Faces- Vertices

+EdgesNo holes euler characteristic is 2

=

Now what about origami shapes with holes?Flat shapes: what we consider

filled finite connected planar graph with Euler characteristic 1- holes don't matter as long as it’s connected

3-d shapes: Torus has genus 1 and Euler

characteristic 0Shapes > 1 hole

E= 2-2G

i.e. Shape 3 holes = 2-2(3)= -4

Which are topologically equivalent?

Haga’s TheoremHaga's theorem lets paperfolders fold the

side of a square into thirds, fifths, sevenths, and ninths

Proof: by construction:Similar triangle AP/SA=BT/PB(1/2)/x=BT/(1/2)BT= 1/ 4. x

To find X, plug in AP=1/2X=3/8BT= 1/(4x(3/8))BT=2/3Therefore we can divide

the side of a square into thirds.

QED Using Haga’s general

formula you can generate 2/5, 6/7, 2/9 etc.

BT= 2AP/(1+AP)

Constructions unique to OrigamiConstructions that are not allowed in

Euclidean compass and straight edge constructions:Doubling the cube: Trisecting and angle

Lets trisect the angle!

Every point that is constructible using a compass and straightedge is constructible using origami.BUT- Origami makes things possible like trisecting an angle

What shapes can you make?Math shapes:

2-d shapesPlatonic solidsArchimedeanPrism/antiprismstellated

Fun shapes: AnimalsFlowersStrange shapes etc

Now lets make one!

Fun facts!Koryo Miura invented a celebrated Miura-Ori

folding method to more easily fold maps. Lawrence Livermore National Laboratory is

developing folding space telescopes using the math origami application

Radhika Nagpal is using this idea or biology and origami in artificial intelligence. Radhika is using Huzita’s orgami axioms to create a global self-organizing system language.

Exam questionUsing Haga’s Theorem what AP value do you

need to generate 2/5

BT= 2AP/(1+AP)

http://www.origamiwithrachelkatz.com/origami/origami.php

http://mathworld.wolfram.com/Origami.html http://en.wikipedia.org/wiki/Mathematics_of_

paper_folding

http://math.serenevy.net/?page=Origami-ApplicationLinks

http://origamido-en.blogspot.ca/2008/05/months-ago-when-i-get-envolved-with.html