Transcript Indezine Template - University of Georgia

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 (origamiwikipedia)
• 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
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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
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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
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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=SMN2002Smith, 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/educationaltutorials/presentation/177-Coordinate-Geometry
Sources
•
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Howstuffworks video center. (1198). Retrieved December 7, 2008
Origami Resources. (2008). Retrieved December 7, 2008, from http://www.origami-resourcecenter.com/origami-resources.html
Free Origami Instructions. (2008). Retrieved December 7, 2008
Boakes, N. (2008). Origami-mathematics lessons: Paper folding as a teaching tool. Mathitudes,
1(1), 9.
Demaine, E. (October 20, 2008). Erik Demaine. Retrieved December 7, 2008, from
http://erikdemaine.org/
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). 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
•
•
•
•
•
•
Fuse, T. (1990). Unit Origmi: Multideminsional Transformation: Japan Publications.
Hull, T. (1998). Pentagon-hexagon zig-zag (PHIZZ) unit. Retrieved December 7, 2008, from
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
Mukerji, M., & Hull, T. (2008). Modular Origami. Retrieved December 7, 2008, from
http://www.origami-resource-center.com/modular-origami.html
Serra, M. (1994). Patty paper geometry Berkeley: Key Curriculum Press, 1 edition