Origami Hyperbolic Paraboloid USD Math Days for Women

Origami Hyperbolic Paraboloid
Violeta Vasilevska
USD Math Days for Women
September 18, 2008
Origami math web sides and books
• Tom Hull - Merrimack College
g in North Andover,,
MA.
http://mars.wnec.edu/~th297133/
• Erik D. Demaine
http://erikdemaine.org/
• Project Origami – activities for exploring
mathematics by Thomas Hull
Origami and math
• Origami is a Japanese word which means paperfolding:
g
• Ori (oru) - meaning "folded," and
• Kami - meaning "paper."
• The art of paper folding (origami), has received a
considerable amount of mathematical study in the
recent years.
Project 1: Constructing Hyper
This
hi project
j
shows
h
h
how
to construct an
origami hyperbolic paraboloid-hyper.
Hyperbolic Paraboloid
• A hyperbolic paraboloid is an infinite surface
discovered in the 17th century.
Dr. Jose Flores and the Hyperbolic
Paraboloid Maplet
http://usd
http://usdmaplenet.usd.edu/maplenet/FloresJSP/HyperPar.jsp
Properties of Hyperbolic Paraboloid
• Mathematically,
y, a hyperbolic
yp
pparaboloid is defined byy the
equation
z/c=x2/a2 - y2/b2
• The name "hyperbolic paraboloid" comes from the property
that the
• the xy cross-sections are hyperbolas, and
• the yz cross-sections are translated copies of a common parabola P.
• Note also that the zx cross-sections are translated upside down
copies of the same parabola P.
Hypar
• The term hypar is used to mean a hyparbolic
paraboloid shape, or more formally a partial
hyparbolic paraboloid, cut from the full infinite
surface.
• The term hypar was introduced by the architect
Engel
(Structure Systems, Frederick A. Praeger Publishers,
New York, 1967).
Hypars in Architecture
• Hypars and joining hypars have been used
extensively in architecture since the 1950's.
A few examples can be found in Curt Siegel's 1962
book Structure and Form in Modern Architecture :
• Philips
ps ppavilion
v o at thee 1958
9 Brussels
usse s eexhibition;
b o ;
• The roof of the Girls
Girls'ss Grammar School in London.
London
Project 2: Constructing a Hyparhedra
In this project you will be joining 5 hypars together
to get a “face” of a hyparhedra, and then use 12
“faces” to construct a hyparhedra
yp
that corresponds
p
to the dodecahedron.
Hyparhedra
Demaine at el. developed algorithms for building hypar
"sculptures" based on polyhedra. They called these structures
hyparhedra.
Hyparhedra that corresponds to Dodecahedron
Project 3: The Miura Map Fold
In this project you will be constructing a rigid
origami model.
model
Rigid Origami
"Which origami
g
models can be opened
p
(to
( a
completely flat, unfolded state) and closed (to the
completely folded state) in a rigid manner?"
By rigid origami we mean origami that can be
folded while keeping all regions of the paper flat
and all crease lines straight (i.e. the regions of
paper between crease lines do not bend or twist in
the folding process).
Why care about rigid origami?
• Those seeking to use origami in industrial designs
often want to know that their chosen fold is a rigid
fold before devoting the resources for
manufacturing, say, stiff cardboard boxes that you
hope will fold up properly.
• The problem of rigid origami, treating the folds as
hinges joining two flat, rigid surfaces, such as sheet
metal, has great practical importance.
The Miura Map Fold
• The Miura mapp fold is a famous rigid
g fold that has
applications to space science – it has been used to
deploy large solar panel arrays for space satellites.
• Koryo Miura invented this fold while searching for
a way to collapse a large solar panel into a package
that could be attached to a space satellite and fit
inside a rocket capsule.
The Miura Map Fold
• Also
Also, Miura discovered that since this model opens
and closes so easily, it makes an ideal map fold.
In fact
fact, one can buy Tokyo subway maps that are
folded in this way.
Video presentation of the fold
http://www.math.lsu.edu/~verrill/
Reference:
• E. D. Demaine, M. L. Demaine, and A. Lubiw,
“Polyhedral Sculptures with Hyperbolic
Paraboloids”, in Proceedings of the 2nd Annual
Conference of BRIDGES: Mathematical
Connections in Art, Music, and Science
(BRIDGE’99) Winfield,
(BRIDGE’99),
Wi fi ld Kansas,
K
July
J l 30–August
30 A
t
1, 1999, pages 91–100.
• E. D. Demaine, World Wide Web.
http://erikdemaine org/hypar/
http://erikdemaine.org/hypar/
Reference:
• T. Hull, Project Origami
Origami-Activities
Activities for exploring
mathematics, A. K. Peters, Ltd., 2006.
• T. Hull, World Wide Web.
http://kahuna.merrimack.edu/~thull/rigid/rigid.html
p
g
g
• H.. Ve
Verrill,, Algebraic
geb c geometry
geo e y origami,
o g
, World
Wo d
Wide Web. http://www.math.lsu.edu/~verrill/