Origami Template

							  Paper Folding
          By:
Eric Gold and Becky Mohl
Definition
• Origami (1603-1867) is the Japanese art of paper
  folding, and is the most commonly known (origami-
  wikipedia)

• Origami is a form of visual / sculptural representation
  that is defined primarily by the folding of the medium
  (usually paper). (origami history)

• “Ori” is the Japanese word for folding

• “Kami is the Japanese word for paper (history of origami)
Definition


• The goal of origami is to create a representation of an
  object using geometric folds and patterns without cutting
  or gluing additional pieces of paper.

• Other countries such as China, Korea, Germany, and
  Spain have developed similar arts. (origami-wikipedia
   • Not certain whether started in Japan or China.
   • China 1st Century AD
   • Japan 6th Century AD
Origin
• Started as historical backgrounds and fashionable
  wrappings
• As an aspect of Japanese history during the Heian
  period
      - Samurai warriors would exchange gifts, noshi, as a
      token of good luck which were paper folded with
      strips of dried fish or meat.
      -Origami butterflies were used to celebrate Shinto
      weddings by wrapping glasses of sake or rice wine
      folded like a butterfly to represent the bride an
      groom. origami-wikipedia, histry of origami)
 Paper Folding in Geometry
• A straight line becomes a crease or a fold

• A point is defined as the intersection between tow folds

• Folding paper is analogous to mirroring one half plane
  in a crease

• Folding means both drawing a crease and mapping
  one half of a plane onto another
Paper Folding In Geometry

• Folding is an isometry of the part of the plane on one
  side of the fold to another, the fold bearing the curve of
  fixed points of the isometry. The curve is straight
  because it has zero curvature.

• In spite what was said, one can fold paper along an
  arbitrary smooth curve. 1. portions of the paper on two
  sides of the curve meet at a zero angle 2. flattening out
  period in order to apply a crease in paper folding to
  geometry
Paper Folding in Geometry

• Paper folding in Geometry was introduce 1983 by T.
  Sundar Row and R.C Yates who listed the axioms which
  plane Eclidean constructions were based on

• In 1191, The Italian-Japanese mathematician Humaiki
  Huzita added 6 more axioms of paper folding in terms of
  straight edges and compasses
Axiom O1: Yates

• There exists a single fold connecting two distinct
  points.
Axiom O2: Yates

• Given two points P1 and P2, there exists a
  unique fold that maps P1 onto P2.
Axiom O3: Huzita

• Given two creases L1 and L2, there exists a
  unique fold that maps L1 onto L2.
Axiom O4: Huzita

• Given a point P and a crease L, here exists a
  unique fold through P perpendicular to L.
Axiom O5: Yates

• For given points P1 and P2 and a crease L,
  there exists a fold that passes through P1 and
  maps P2 onto L2.
• Does not work for all points P1 and P2. If P1P2
  is shorter than P1L does not exist.



                                   Intersection of a
                                   circle and a line
Axiom O6: Huzita

• Given two points P1 and P2 and two creases L1
  and L2, there exists a unique fold that maps P1 into
  L1 and P2 onto L2.
• Works for all points.
    Constructing a Triangle
• Begin with a new sheet of wax paper.
• Construct 2 intersecting lines (an angle)
• Construct the angle bisector
• Construct a segment perpendicular to the angle
  bisector—you have constructed an isosceles
  triangle.
• Construct the midsegment of the triangle.
• Constrict segments connecting the midpoints of
  the sides to the midpoint of the base.
Constructing a Triangle: Points to ponder
• How do you know that you have constructed an
  isosceles triangle?
• How could you construct an equilateral triangle?
• What is the easiest way to construct the
  midsegment (what mapping do you use?)
• What is the ratio of the area of the triangle created
  between the midsegment and the vertex to that of
  the original triangle?
• What other similar and congruent elements have
  been constructed?
• How did you use the axioms in this construction?
Coordinate Geometry

• The algebraic study of
  geometry through the use
  a coordinate plane or
  system.
• A Coordinate plane is a
  grid used to locate a point
  by its distances from 2
  intersecting straight line
Parabola: Definition
• Parabola: is the set of
  all points in the plane
  equidistant from a
  given line L (the
  directrix) and a given
  point F not on the line
  (the focus).



Weisstein, Eric W. "Parabola." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/Parabola.html
GSP Construction

• Parabola




Animation
Equation of a Parabola




                         Animation
Folding the Parabola

• Take a sheet of wax paper and fold a line
  perpendicular to the sides of the paper. This is
  your directrix
• Construct a point not on the line. This is the
  focus.
• Fold points on the line onto the focal point.
  Make at least 50 folds.
Points to ponder-the parabola
• What happens as you change the distance from F to
  the directrix?
• Using the geometry of the parabola, explain how a
  headlight (or any of the various objects that are
  parabolic) works.
• Where else do we see these kinds of curves in the
  world?
• How do you relate the construction to the locus
  definition?
• How can you get to the equation of the parabola when
  F is not on the y-axis and the directrix isn’t the origin?
 Ellipse: Definition
• Ellipse: an ellipse is a
  curve that is the locus of all
  points in the plane the sum
  of whose distances r1 and
  r2 from two fixed points F1
  and F2 (the foci) separated
  by a distance of 2c is a
  given positive constant.



 Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource.
 http://mathworld.wolfram.com/Ellipse.html
GSP Construction

• Ellipse:




• Animation
The equation of the ellipse




                              Animation
Folding the ellipse
• Construct a circle on a clean sheet of wax paper.
• Construct a fixed point F inside the circle.
• Fold points on the circle onto F, making at least 50 folds
Points to ponder-the ellipse
• What happens as you move the fixed point around in
  the circle? Why?
• Where else do we see these kinds of curves in the
  world?
• How do you relate the construction to the locus
  definition?
• How can you get to the equation of the ellipse it is not
  conveniently placed in the plane?
Circle: Deifnition


• Circle: The degenerate
  case of an ellipse where
  the focal point F
  collapses onto the center
  of the circle C.




Weisstein, Eric W. "Circle." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/Circle.html
GSP construction
• Circle




• Animation
How is the circle an ellipse?
• When the focal point F becomes the center of the circle,
  we have a circle that is constructed.
Hyperbola: Definition
• Hyperbola: the locus
  of all points P in the
  plane the difference
  of whose distances
  r1=F1*P and r2=F2*P
  from two fixed points
  the foci F1 and F2
  separated by a
  distance of 2c.

 Weisstein, Eric W. "Hyperbola." From MathWorld--A Wolfram Web Resource.
 http://mathworld.wolfram.com/Hyperbola.html
GSP Construction

• Hyperbola




• Animation
Equation of a hyperbola




                          Animation
Folding the hyperbola
• Construct a circle on a clean sheet of wax paper.
• Construct a fixed point F outside the circle.
• Fold points on the circle onto F, making at least 50 folds
Points to ponder: the hyperbola
• How does the hyperbola relate to the ellipse?
• How do you relate the construction to the locus
  definition?
• How can you get to the equation of the ellipse it
  is not conveniently placed in the plane?
References
•   Origami. (2008, December 4, 2008). Retrieved December 7, 2008, from
    http://en.wikipedia.org/wiki/Origami
•   Anderson, E. M. (1999, February 13, 2004). Origami history. Retrieved December 7, 2008, from
    http://www.paperfolding.com/history/
•   Bogomolny, A. (1996). Paper folding geometry. Retrieved December 7, 2008, from
    http://www.cut-the-knot.org/pythagoras/PaperFolding/index.shtml
•   Demaine, E. (2004). Retrieved December 7, 2008, from http://courses.csail.mit.edu/6.885/fall04/
•   Demine, E., & Benbernou, N. (2007). Geometric folding algorithms: Linkages, origami, polyhedra.
    Retrieved December 7, 2008, from http://courses.csail.mit.edu/6.885/fall07/
•   Fuchs, D., & Tabachnikov, S. (1999). More on paperfolding. The American Mathematical Monthly,
    106(1), 9.
•   Hatfield, L. L. Fold, plot, simulate, do algebra: Using technology to help students understand the
    parabola. Unpublished Prepublication draft journal article. University of Georgia.
•   Hatfield, L. L. ConicsTE.gsp. Unpublished Geometer's Sketchpad File. University of Georgia.
•   Jeremy, C. a. (1999). History of Origami. Retrieved December 12, 2008, from
    http://library.thinkquest.org/5402/history.html
•   01-1a
References
•   Joyce, D. E. (1997). Euclid's Elements. Retrieved December 7, 2008, from
    http:/aleph0.clarku.edu/~djoyce/elements/toc.html
•   Koschitz, D., Demaine, E. D., & Demaine, M. L. (2008). Curved Crease Origami. Paper presented
    at the Advances in Architectural Geometry.
•   Masunaga, D. (2002). Origami: It's not just for squares. NCTM Student Math Notes Retrieved
    December 7, 2008, 2008, from
    http://my.nctm.org/eresources/article_summary.asp?uri=SMN2002-
•   Smith, S. G. (2003). Paper folding and conic sections. Mathematics Teacher, 96(3), 6.
•   Weisstein, E. (1999, December 4, 2008). Parabola. Retrieved December 7, 2008, from
    http://mathworld.wolfram.com/Parabola.html
•   Weisstein, E. (1999, December 4, 2008). Ellipse. Retrieved December 7, 2008, from
    http://mathworld.wolfram.com/ellipse.html
•   Weisstein, E. (1999, December 4, 2008). Hyperbola. Retrieved December 7, 2008, from
    http://mathworld.wolfram.com/hyperbola.html
•   Weisstein, E. (1999, December 4, 2008). About Eric Weisstein, creator of mathworld. Retrieved
    December 7, 2008, from http://mathworld.wolfram.com/about/author.html
•   Wesstein, E. (1999, December 4, 2008). Circle. Retrieved December 7, 2008, from
    http://mathworld.wolfram.com/circle.html
•   WizIQ. (2008). Retrieved December 9, 2008, from, http://www.wiziq.com/educational-
    tutorials/presentation/177-Coordinate-Geometry
Sources
•   Howstuffworks video center. (1198). Retrieved December 7, 2008
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    center.com/origami-resources.html
•   Free Origami Instructions. (2008). Retrieved December 7, 2008
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    1(1), 9.
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    http://erikdemaine.org/
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•   Demaine, E. (2004). Folding and unfolding in coputational geometry. Retrieved December 7,
    2008, from http://courses.csail.mit.edu/6.885/fall04/
•   Demaine, E. (2004). Retrieved December 7, 2008, from http://courses.csail.mit.edu/6.885/fall04/
•   Franco, B. (1999). Unfolding mathematics with unit oragami. Berkeley: Key Curriculum Press.
•   Fuchs, D., & Tabachnikov, S. (1999). More on paperfolding. The American Mathematical Monthly,
    106(1), 9.
Sources
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    http://kahuna.merrimack.edu/~thull/phzig/phzig.html
•   Koschitz, D., Demaine, E. D., & Demaine, M. L. (2008). Curved Crease Origami. Paper presented
    at the Advances in Architectural Geometry.
•   Masunaga, D. (2002). Origami: It's not just for squares. NCTM Student Math Notes Retrieved
    December 7, 2008, 2008, from
    http://my.nctm.org/eresources/article_summary.asp?uri=SMN2002-01-1a
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