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

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• 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

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