Origami & Math
So, you're interested in origami and mathematics...perhaps you are a
high school or K8 math teacher, or a math student doing a report on the
subject, or maybe you've always been interested in both and never made
the connection, or maybe you're just curious. Origami really does have
many educational
benefits. Whether you are a student, a teacher, or just a casual
surfer, I have tried my best to answer your questions, so please read
on.
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So exactly how do origami and math relate to each other? The
connection with geometry is clear and yet multifaceted; a folded model
is both a piece of art and a geometric figure. Just unfold it and take a
look! You will see a complex geometric pattern, even if the model you
folded was a simple one. A beginning geometry student might want to
figure out the types of triangles on the paper. What angles can be seen?
What shapes? How did those angles and shapes get there? Did you know
that you were folding those angles or shapes during the folding itself?
For instance, when you fold the traditional waterbomb base, you
have created a crease pattern with eight congruent right triangles. The
traditional bird base produces a crease pattern with many more
triangles, and every reverse fold (such as the one to create the bird's
neck or tail) creates four more! Any basic fold has an associated
geometric pattern. Take a squash fold  when you do this fold and look
at the crease pattern, you will see that you have bisected an angle,
twice! Can you come up with similar relationships between a fold and
something you know in geometry? You can get even more ideas from this
presentation on Origami:
In Creasing Geometry in the Classroom.
On the other hand, if you are a person who likes puzzles, there
are a number of great origami
challenges that you might
enjoy trying to solve. These puzzles involve folding a piece of paper so
that certain color patterns arise, or so that a shape of a certain area
results. But let's continue on with crease patterns...
Origami, Geometry, and the Kawasaki Theorem
A more advanced geometry student or teacher might want to
investigate more in depth relationships between math and origami. You
can take a look at these geometry
exercises to get you
started. For instance, the traditional
crane (or another set
of diagrams) unfolded provides a crease pattern from which we can
learn a lot. Pick a point (vertex) on the crease pattern. How many
creases originate at this vertex? Is it possible for a flat origami
model to have an odd number of creases coming out of a vertex on it's
crease pattern? How about the relationship between mountain and valley
folds? Can you have a vertex with only valley folds or only mountain
folds?
How about the angles around this point? You can really impress
your teacher (or your students) with this...of course, you will need to understand
it first! There is a
theorem called KAWASAKI'S
THEOREM, which says that if the angles surrounding a
single vertex in a flat origami crease pattern are a_{1},
a_{2},
a_{3},
..., a_{2n},
then:
a_{1} +
a_{3} +
a_{5} +
... + a_{2n1} =
180
and
a_{2} +
a_{4} +
a_{6} +
... + a_{2n} =
180

In other words, if you add up the angle measurements of every other
angle around a point, the sum will be 180. Try it and see!
Can you see that this is true, or, even better, can you prove it?
Straight Edge and Compass vs Origami, and Huzita's Axioms
Although there is much to understand about crease patterns,
origami itself is the act of folding the paper, which mathematically can
be understood in terms of geometric construction. The most wellknown
construction is "straight edge and compass" construction, which refers
to the geometric operations that can be formed with only those two
instruments (note that the straight edge is not a
ruler with length markings). It is wellknown that SE&C constructions
can be encompassed (no pun intended) by four basic axioms, first defined
by Euclid, over 2000 years ago. It is also well known that there are
certain operations that are impossible given just a straight edge and
compass. Two such operations are trisecting an angle and doubling a cube
(finding the cube root of 2).
But back to origami construction...origami construction is defined
as those geometric operations that can be formed by folding a piece of
paper, using the raw edges and points of the paper, as well as any
subsequent crease lines and points created while folding. What is
fascinating is that origami construction, although at first may seem
less powerful than SE&C, is actually more powerful, allowing both the
trisecting of the angle as well as the doubling of the cube. The
mathematician Humiaki Huzita developed six axioms (and later a seventh)
based on origami construction. I will not go into the details here,
since below are links to a few sites which have very thorough
descriptions of Huzita's Axioms:

Huzita's Axioms, courtesy of Wikipedia

Tom Hull's overview of Origami
and Geometric Construction, which provides great descriptions
and exercises for the 5th and 6th axioms, as well as descriptions of
how to trisect the angle and double a cube via folding

Koshiro Hatori's page on Origami Construction describes
Huzita's Six Axioms as well as a seventh axiom that Hatorisan
discovered!

Christian Lavoie developed an Origami Computational Model for a
computational geometry course at McGill University in 2002. As
background for this project her provides the axioms
of origami along with
graphs and equations.
What's especially great about all this is that these axioms are
not just theoretical  they have been put into real use! Robert Lang's
origami program, ReferenceFinder,
uses all seven of the axioms. ReferenceFinder is a program which finds
folding sequences to approximately locate any point on a square using a
small number of folds.
Origami and Topology
The study of origami and mathematics can be classified as
topology, although some feel that it is more closely aligned with
combinatorics, or, more specifically, graph theory. I will give an
example of an origami theorem which can be seen from both points of
view, but first a little about topology.
The connection with topology is less clear than the connection
with geometry, probably because most people are far less familiar with
this field. If you are a student doing a report on origami and math, you
can again impress your teacher by showing that you know what topology
is, and how it is related to origami.
So, what is topology? Here's a short
and idiosyncratic answer, but you should really read Neil
Strickland's thorough answer with pictures. It is important that you
understand that geometry and topology are very different. Topology is
sometimes called "rubber sheet geometry", meaning that in topology,
stretching an object or changing it's shape will not affect it (as long
as you do not create any holes or patch up any holes). To a topologist,
a coffee cup and a doughnut are the same, while a geometer sees them as
completely different.
If you read Neil's answer, you noticed that he mentioned a subway
map, which is just a network of points connected by lines, just like an
origami crease pattern! Studying origami crease patterns can help us
learn about networks such as subways and phone networks, and how to make
them faster and more efficient. But don't take my word for it. Thomas
Hull, an assistant professor of mathematics at Merrimack College in
North Andover, Massachusetts, is the expert in the field of origami and
topology. Tom is currently teaching a course in combinatorial geometry,
and you can view the course's syllabus
and assignments. If you are looking to do more in depth research in
this field, your first step should be to contact Tom. His
Web site was even
mentioned in a
story on ABCnews.com!
Now back to the origami theorem that I mentioned earlier, which
can be seen from two points of view.
Theorem: Every
flatfoldable crease pattern is 2colorable. 
In other words, suppose you have folded an origami model which lies
flat. If you completely unfold the model, the crease pattern that you
will see has a special property. If you want to color in the regions of
your crease pattern with various colors so that no two bordering regions
have the same color, you only
need two colors. This may remind you of the famous mapmaker's
problem: what is the fewest number of colors you need to color countries
on a map (again, so that two neighboring countries aren't the same
color)? This is known as the Four
Color Theorem, since the answer is four colors. As an interesting
aside, this theorem was proven in 1976 by American mathematicians Appel
and Haken using a computer to check the thousands of different cases
involved. You can learn
more about this proof, if you like.
But back to our theorem.
Can you see that you need only two colors to color a crease pattern? Try
it yourself! You will see that anything you fold (as long as it lies
flat) will need only two colors to color in the regions on its crease
pattern.
Here's an easy way to see it: fold something that lies flat. Now
color all of the regions facing towards you red and the ones facing the
table blue (remember to only color one side of the paper). When you
unfold, you will see that you have a proper 2coloring!
Warning...this section gets even more complicated! A more rigorous
proof goes as follows: first show that each vertex in your crease
pattern has even degree (the degree is the number of creases coming out
of each vertex  we discussed this earlier!). Then you know the crease
pattern is an Eulerian graph, that is, a graph containing a path which
starts and ends at the same point and travels along every edge (such a
path is called an Eulerian cycle). Don't
try to prove this unless you are an experienced mathematician!Finally,
it is well known that Eulerian graphs are 2colorable.
Hmmm...I started off by promising a result which can be seen as
both combinatorial and topological. Did we get it? Well, the result is
clearly combinatorial, since it is graph theory. How is the result a
topological one? Well, the 2coloring gives us an easy way of
determining the orientation of each region we color in. All regions
colored blue will be facing up (or down) while all regions colored red
will be facing the opposite way. Try it! This time, fold a model, unfold
it, and color the crease pattern regions with red and blue. Now, refold
the model and see for yourself!
Other Resources
Most people don't realize how much information there is on the subject
of mathematics and origami. There are books and papers published on the
subject, presentations given, as well as an international conference
called "The International Meeting of Origami Science and Technology". In
fact, many of the origami creators and authors here in the US and abroad
are mathematicians, physicists, and other scientists. I am thinking of
people like Robert Lang, Jun Maekawa, Toshikazu Kawasaki, and Thomas
Hull. Another origami math figure with no formal mathematical
training is Chris
Palmer (also see this
article about Chris), the creator of the flower
tower (see picture).
As for books, the first ones that come to mind are a set of books
by Rona Gurkewitz and Bennett Arnstein: 3D
Geometric Origami: Modular Polyhedra, Modular
Origami Polyhedra, and just published in 2003, Multimodular
Origami Polyhedra: Archimedeans, Buckyballs and Duality. You can learn
more about Professor Gurkewitz's work, as well as view galleries,
links, and more.
A few other origami books which discuss the connection with
mathematics are Origami
Omnibus by Kunihiko
Kasahara (ISBN 087040699x), Origami
from Angelfish to Zen by
Peter Engel (ISBN 0486281388), Easy
Origami, by Gay Merrill Gross and Tina Weintraub, and Math
in Motion: Origami in the Classroom K8 by
Barbara Pearl (ISBN 0964792435). The Web site for Math
in Motion even includes a
sample lesson plan for educators. If you teach grades K8, then this
book may be what you have been looking for.
The first book I mentioned, Origami
Omnibus discusses topics
such as isoarea folding, the golden rectangle, folding a regular
pentagon from a square, and the Kawasaki Theorem. The book is out of
print, but you can probably get it at a local library or thorugh
interlibrary loan.
The next one, "Origami from Angelfish to Zen" has a wonderful
introduction called "Crossing the Divide" which discusses the
relationship between origami and topics like MC Escher, Zen philosophy,
fractals and chaos, music, and art. An excellent introduction to origami
and mathematics. This book should be available at places like Amazon.com, Sasuga
Bookstore, and Fascinating
Folds.
Easy Origami, by
Gay Merrill Gross and Tina Weintraub (ISBN 0590535498), is a workbook
of projects for teachers in K6, many with basic math content.
I have also heard great things about Paperfolding,
a Fun and Effective Method for Learning Math. It has a nice Web
site, which is worth checking out.
Another recent book is Mathematical
Origami, by David Mitchell (ISBN 189961818X). Wellknown for his
modular designs and sculptures, David provides in this book instructions
for the platonic solids and their variants, as well as other geometrical
shapes such as rings, stars, rhombic shapes, and skeletal and outline
solids.
You can find a list of more origamimath books at Tom
Hull's Origami Bibiliography
There are also a number of great places on the Web to start
looking. Tom
Hull's Origami Math Web site, which I've already mentioned a few
times, is the definitive resource on the Web for origami and
mathematics. Tom Hull has done some quite advanced work in the subject.
His origamimath papers have appeared in the in MAA
(Mathematical Association of America) journals.
He has written on topics like origami constructibility vs ruler/compass
constructibility and origami and topology.
I can also tell you about Helena
Verrill's Origami Page, which has information on modular origami,
origami tesselations and origamimath tesselation problems, as well as
other origamimath problems like angle trisection and the perimeter
problem. Great stuff for a high school project!
Another interesting page (especially if you like fractals) is Jeannine
Mosely's business card Menger Sponge project, the goal of which is
to build a Menger sponge of depth 3 origami modular out of 66,048
business cards.
A recent addition to the Web is Krystyna
Burczyk's Origami Gallery, which displays modulars, the regular
polyhedra, and the semiregular polyhedra. You can learn a lot here; how
can a school report with the word "rhombicosadodecahedron" get a bad
grade?
Wolfram Research has a page
about origami and math which
has pictures, lists Huzita's axioms, and provides many further
references.
Another great site is the Geometry
Junkyard, which provides a number of links to various topics in
origami and mathematics.
Finally, if you're still hungry for more, check Tom
Hull's Origami Math links. By the time you're done with these sites,
you should have lots of great ideas for that report, project, or lesson
plan!
I hope this gives you a good overall view
of the topic of origami and mathematics. I will continue to update this
page as more Web sites on this topic appear. If you have any comments or
suggestions for me, please drop me a line.
