Math and Origami

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Math and Origami
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What is Origami?

• Origami is the art of paper folding.
• Comes from the Japanese word for paper folding.
  
• ORI Paper
  KAMI Folding

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Origami Axioms

• Today the most accepted set of axioms are the
  axioms given by Humiaki Huzita and are known as
  Huzitas Axioms.
• Understanding Geometry through Origami Axioms"
  in the Proceedings of the First International
  Conference on Origami in Education and Therapy,
  1992

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O1

• Given two points p1 and p2 we can fold a line
  connecting them.

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O2

• Given two points p1 and p2 , we can fold p1 onto
  p2

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O3

• Given two lines l1 and l2, we can fold line l1
  onto l2

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O4

• Given a point p1 and a line l1, we can make a
  fold perpendicular to l1 passing through the
  point p1

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O5

• 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.This axiom has 0, 1 or 2
  solutions

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O6

• Given two points p1 and p2 and two lines l1 and
  l2, we can make a fold that places p1 onto l1 and
  p2 onto l2. This axiom has 0, 1, 2 or 3
  solutions

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O7 (Hatoris Axiom)

• Given a point p1, and two lines l1 and l2, we can
  make a fold perpendicular to l2 that places p1
  onto line l1

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Find the length after fold

• In ABCD,AB8,BC10
• Make a fold that places D onto BC and passes
  through the point A.
• (By O5)
• Whats the length of EC?

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Angles

• Can you from an equilateral triangle with a sheet
  of paper?

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Steps

• 1. Fold the paper, places the bottom side (BC)
  onto the top side (AD) by O3
• 2. Fold through the left bottom (B), places the
  top left (A) onto the fold line in step 1 by O5
• call the intersection of fold line and AD point
  E
• 3. Fold through E and A by O1, call the
  intersection with BC point F

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Possible Exam Problem

• In rectangle ABCD, ?AEB ?AEB, AD//MA//BC,
  BMAM. Prove ?BEF is an equilateral triangle.

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Proof

• ? ?AEB ?AEB
• ? ?A ?EAB 90 and AB AB
• ?AEB ?AEB, ?ABE ?ABE
• ? BM AM ? 2BM AB AB
• ? ?A 90 and AD // MA
• ? ?AMB ?A 90
• ? In ?AMB, ?AMB 90
• ? sin?MAB BM / AB 0.5 that is ?MAB 30
• ? MA // BC ??ABF ?MAB 30
• ? In ?AMB, ?BAF 180-?EAB 90, ?ABF
  30, so ?BFA 60
• ? AD // BC ??AEB ?AEB ?BFA 180, so 2?AEB
  60 180
• ? ?AEB 60 so we have ?BEF ?BFE 60
• ? In ?BEF, ?BEF ?EBF ?BFE 180, so ?EBF 60
  
• ? ?BEF ?EBF ?BFE 60, ?BEF is an equilateral
  triangle?

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Applications Hip Fracture

• Dynamic Hip screws are often used to help heal a
  fracture of the hip.

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• The hip screw must be inserted at either 135 or
  150 degree angles to the shaft of the femur in
  relation to the head of the femur

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Problem

• The Problem is that during the operation there is
  rarely a protractor on hand. (Auyeung Thomas)
• Using Origami this can easily be solved.

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135?
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150?
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The Dragon Curve

• The Dragon Curve is the end result of infinitely
  many folds
• Each new generation is double the previous
  generation.

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Other Fractals

• Sierpinskis triangle
• Sierpinski Tetrahedron

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Mengers Sponge
This is made from 2400 pieces of paper
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References

• Burger, E. Starbird, M. Coincidences, Chaos,
  and All That Math Jazz.(2005). W,W. Norton
  Company, Inc. 500 Fifth Avenue, New York, New
  York.
• Auyeung, J. Thomas, O. Origami in Dynamic Hip
  Screw Surgery.(2004). Injury, International
  Journal Care Injured. 35, 1039-1041.
• http//hektor.umcs.lublin.pl/mikosmul/origami/fra
  ctals.html
• http//www.math.okstate.edu/mathdept/dynamics/lecn
  otes/node17.html
• http//www.merrimack.edu/thull/omfiles/geoconst.h
  tml
• http//www.ganymeta.org/darren/origami2.php