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Lori Burns, Jess Barkhouse

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History of Origami Art 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

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Origami axioms (Huzita 1992) 1) Given points P and Q you can fold a line connecting them 2) Given points P and Q you can fold P onto Q 3)Given lines L 1 and L2 you can fold line L1 onto L2 4) Given a point P and a line L1 you can make a perpendicular fold to L1 passing through the point P 5) 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

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A 7 th axiom that was overlooked 7) given a point P and two lines L1 and L2 you can make a fold perpendicular to L2 that places P onto L1.

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Topologically equivalent. If one shape can form the other without tearing, attaching or creating holes. Euler characteristic = Faces- Vertices +Edges No holes euler characteristic is 2 =

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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 0 Shapes > 1 hole E= 2-2G i.e. Shape 3 holes = 2-2(3)= -4

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Which are topologically equivalent?

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Haga’s Theorem Haga'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

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

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Constructions unique to Origami Constructions that are not allowed in Euclidean compass and straight edge constructions: Doubling the cube: Trisecting and angle

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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

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What shapes can you make? Math shapes: 2-d shapes Platonic solids Archimedean Prism/antiprism stellated Fun shapes: Animals Flowers Strange shapes etc

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Now lets make one!

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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.

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Exam question Using Haga’s Theorem what AP value do you need to generate 2/5 BT= 2AP/(1+AP)

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http://www.origamiwithrachelkatz.com/origami/origa mi.php http://www.origamiwithrachelkatz.com/origami/origa mi.php http://mathworld.wolfram.com/Origami.html http://en.wikipedia.org/wiki/Mathematics_of_paper_f olding http://en.wikipedia.org/wiki/Mathematics_of_paper_f olding http://math.serenevy.net/?page=Origami- ApplicationLinks http://math.serenevy.net/?page=Origami- ApplicationLinks http://origamido-en.blogspot.ca/2008/05/months- ago-when-i-get-envolved-with.html http://origamido-en.blogspot.ca/2008/05/months- ago-when-i-get-envolved-with.html

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