lori burns, jess barkhouse. history of origami art of paper making originated in china in 102a...
TRANSCRIPT
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
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
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.
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