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Origami constructions
Violeta Vasilevska
Origami math web sides
• Tom Hull - Merrimack College in North Andover,
MA.
http://www.merrimack.edu/~thull/
• Erik D. Demaine
http://theory lcs mit edu/%7Eedemaine/
http://theory.lcs.mit.edu/%7Eedemaine/
Origami Book
• “Project Origami – activities for exploring
mathematics” by Thomas Hull
mathematics
Origami
• Origami is a Japanese word which means paperfolding:
g
• Ori - meaningg "folded,"
, and
• Kami - meaning "paper."
History of Origami
• Several conjectures:
j
• 2000 years ago in China;
• Heian era in Japan.
• Japanese classic origami:
• The oldest unequivocal document of origami is a short
poem composed by Ihara Saikaku in 1680;
• Origami was included in the manners of the samurai
class
l (Muromachi
(
hi era );
)
• Mainly ceremonial origami.
History of Origami
• European
p
Classic Origami
g
• 15th century – origami boat;
• The baptismal certificate of 16th to 17th century.
• Traditional origami has been born and brought up
in the cultural exchange between East and West
(a hybrid of Japan and Europe).
History of Origami
• Modern origami
g
are regarded
g
as "models" "designed"
g
byy
"origami creators." (20th century).
• Mathematical Origami (1930-1940)
• the aspect of origami as the puzzle is more and more emphasized in
mathematical origami.
g
That is, theyy compete
p in designing
g g realistic or
complex models under the rule of one sheet of square with no cut. In
addition, they regards the crease pattern as an important part of the
model besides the final shape and the sequence.
Origami and Geometric Constructions
• A comparison
p
between straight
g edge
g and compass
p
constructions (SE&C) and origami
• Certain things are impossible to do with a SE&C,
SE&C like:
• trisecting an arbitrary angle,
• doubling the volume of a cube (i.e., constructing the cube root of 2).
• We can make geometric constructions with origami, using
the side of the paper as the straight edge and folding up to
an angle
l to simulate
i l a compass. Furthermore,
h
trisecting
i
i
angles and doubling cubes is possible with origami!
Folding Equilateral Triangle in a Square
• This task asks,, "How do yyou fold a pperfect,, equilateral
q
triangle in a square piece of paper?”
Exploring Basic Origami Move
• Given two points p1 and p2 and a line L, fold p1
onto L so that the resulting crease line passes
through p2.
Exploring Basic Origami Move
•
•
•
•
•
Take a square piece of paper.
Let the bottom edge be the line L.
Choose a point p somewhere on the paper.
paper
Fold p onto L.
Repeat this 8 or 9 times.
times
• What do you observe?
• What are the crease lines forming?
• How does your choice of the point p and the line L fit
into this?
Other Origami Projects
• Folding Five Intersecting Tetrahedra
• The units of this model are easy to make, but putting it
all together is a real puzzle. Building this stunning model
offers
ff a chance
h
to study
d numerous symmetries
i off the
h
dodecahedron.
Folding Five Intersecting Tetrahedra
http://www.merrimack.edu/~thull/gallery/modgallery.html
p
g
y
g
y
http://www.merrimack.edu/~thull/origamimath.html
Other Origami Projects
• Making Origami Buckyballs
• Buckyballs or sometimes fullerenes, named after the
artist-architect Buckminster Fuller.
• A Buckyball is a polyhedron with two properties:
• every face is either a pentagon or a hexagon, and
• each vertex has degree 3.
3
Making Origami Buckyballs
http://www.merrimack.edu/~thull/gallery/modgallery.html
p
g
y
g
y
Making Origami Tori
http://www merrimack edu/%7Ethull/combgeom/tori/torusnotes html
http://www.merrimack.edu/%7Ethull/combgeom/tori/torusnotes.html
Making Origami Tori
http://www.merrimack.edu/%7Ethull/combgeom/tori/torusnotes.html
p
g
THANK YOU!