Origami & Mathematics:
Fold tab A to flap B?
By Joseph M. Kudrle
Department of Mathematics/Statistics
University of Vermont
Content
I. Introduction
II. History of Origami
III. Origami and Mathematics (Some neat
theorems)
IV. Constructing Polygons (Yet another
neat theorem)
V. Constructing Polyhedra (Modular
Origami)
Objectives: What I want you
to get out of this talk.
An appreciation of the art of origami.
An appreciation of the mathematics
that are found in the art of origami.
Some “wicked cool” origami models that
you can show off to all of your friends.
History of Origami
History of Origami
Origami – In ancient Japanese ori
literally translates to folded while gami
literally translates to paper. Thus the
term origami translates to folded paper.
History of Origami
Origami has roots in several different
cultures. The oldest records of origami
or paper folding can be traced to the
Chinese. The art of origami was
brought to the Japanese via Buddhist
monks during the 6th century.
The Spanish have also practiced origami
for several centuries.
History of Origami
Early origami was only performed during
ceremonial occasions (i.e. weddings,
funerals, etc.).
Traditionally origami was created using
both folds and cuts, but modern rules
established in the 1950’s and 1960’s
state that only folds shall be allowed.
History of Origami
Early origami was simple in form and
fold. A classic example is the crane.
History of Origami
History of Origami
Modern origami still utilizes the same
ideas found in the traditional models;
however, the folds are becoming
increasingly more difficult.
Some modern origami model’s folds are
highly kept secrets and can take hours
and hours for an experienced folder to
complete.
Origami & Mathematics:
Some neat theorems
Terms
FLAT FOLD – An origami which you
could place flat on the ground and
compress without adding new creases.
Terms
CREASE PATTERN – The pattern of
creases found when an origami is
completely unfolded.
Terms
MOUNTAIN CREASE – A crease which
looks like a mountain or a ridge.
VALLEY CREASE – A crease which looks
like a valley or a trench.
Terms
VERTEX – A point on the interior of the
paper where two or more creases
intersect.
Maekawa’s Theorem
(1980’s)
The difference between the
number of mountain creases and
the number of valley creases
intersecting at a particular vertex
is always…
2
Example of Maekaw’s Theorm
The all dashed lines represent mountain
creases while the dashed/dotted lines
represent valley creases.
Maekawa’s Theorem
(1980’s)
Let M be the number of mountain
creases at a vertex x.
Let V be the number of valley creases
at a vertex x.
Maekawa’s Theorem states that at the
vertex x,
M–V=2
or
V–M=2
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
Note – It is sufficient to just focus on
one vertex of an origami.
Let n be the total number of creases
intersecting at a vertex x. If M is the
number of mountain creases and V is the
number of valley creases, then
n=M+V
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
Take your piece of paper and fold it into
an origami so that the crease pattern
has only one vertex.
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
Take the flat origami with the vertex
pointing towards the ceiling and cut it
about 1½ inches below the vertex.
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
What type of shape is formed when
the “altered” origami is opened?
POLYGON
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
How many sides does it have?
n sides : where n is the number of
creases intersecting at your vertex.
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
As the “altered” origami is closed, what
happens to the interior angles of the
polygon?
Some get smaller – Mountain Creases
Some get larger – Valley Creases
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
When the “altered” origami is folded up,
we have formed a FLAT POLYGON
whose interior angles are either:
0° – Mountain Creases
or
360° – Valley Creases
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
Recap – Viewing our flat origami we have
an n-sided polygon which has interior
angles of measure:
0° – M of these
360° – V of these
Thus, the sum of all of the interior
angles would be:
0M + 360V
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
What is the sum of the interior angles
of any polygon?
SIDES SHAPE ANGLE SUM
3 180°
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
What is the sum of the interior angles
of any polygon?
SIDES SHAPE ANGLE SUM
180°(4) – 360°
4 or
360°
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
What is the sum of the interior angles
of any polygon?
SIDES SHAPE ANGLE SUM
180°(5) – 360°
5 or
540°
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
What is the sum of the interior angles
of any polygon?
SIDES SHAPE ANGLE SUM
180°(6) – 360°
6 or
720°
Proof of Maekawa’s Theorem
(Jan Siwanowicz – 1993)
This completes our proof!
Corollary to Maekawas Theorem
(Unknown Date)
The number of creases at a particular
vertex on a CREASE PATTERN of a
FLAT ORIGAMI must always be:
EVEN
Proof of Corollary
(Thomas Hull – 1990’s)
Let M be the number of mountain
creases and let V be the number of
valley creases at a vertex x.
Maekawa’s Theorem states that,
M–V=2
or
V–M=2
Proof of Corollary
(Thomas Hull – 1990’s)
Let n be the total number of creases
intersecting at a vertex x. If M is the
number of mountain creases and V is the
number of valley creases, then
n=M+V
Proof of Corollary
(Thomas Hull – 1990’s)
Using some tricky algebra we get:
n=M+V
n = (M – V ) + 2V
Proof of Corollary
(Thomas Hull – 1990’s)
Now apply Maekawa’s Theorem.
n = (2) + 2V = 2(1 + V )
or
n = (-2) + 2V = 2(-1 + V )
Proof of Corollary
(Thomas Hull – 1990’s)
Both 2(1 + V ) and 2(-1 + V ) are even
numbers. This completes the proof.
Polygons and Paper
Folding
Polygon Theorem
(Author - Date Unknown)
Any polygon drawn on a sheet of paper
can be extracted from the paper by
only one cut, provided the paper is
folded into a proper flat origami.
Polygon Theorem
(Author - Date Unknown)
Regular Polygon – A convex polygon
where all sides have equal measures and
all interior angles have equal measures.
Test the theorem on your set of regular
polygons.
Polygon Theorem
(Author - Date Unknown)
Challenge Question: What is the least
number of folds that it takes to extract
a regular n sided polygon?
Contact me if you get an answer…I will
keep working on it myself.
jkudrle@cem.uvm.edu
Constructing Polyhedra
Terms
POLYHEDRON – A solid constructed by
joining the edges of many different
polygons. (Think 3-Dimensional polygon.)
Terms
STELLATED POLYHEDRON – Created
by taking the edges of a polyhedron and
extending them out into space where
they eventually intersect with each
other and close in a new polyhedron.
Terms
SONOBE – A flat origami which when
pieced together with identical SONOBE
units can be used to modularly construct
polyhedra.
Terms
Note: There is no distinct way that a
SONOBE must be formed. According to M.
Mukhopadhyay only three properties must
be met in order for an origami to be
considered a SONOBE.
(1) The unit must have two points.
(2) The unit must have two pockets.
(3) A point must be able to fit within a pocket.
Creating a SONOBE
(1) Start with the white side up and do a
valley fold horizontally in the middle
of the paper. Open the paper back up
after the fold.
Creating a SONOBE
(2) Now take the paper and valley fold
horizontally in-between the last valley
fold and the ends of the paper.
Creating a SONOBE
(3) Mentally label the corners 1, 2, 3, and
4 as I have done. Valley fold corners 1
and 3 as shown below.
Creating a SONOBE
(4) Now valley fold the corners once more
as shown…think of the type of fold
used to make certain paper air-planes.
Creating a SONOBE
(5) Now refold the paper along the valley
folds done in step (2).
Creating a SONOBE
(6) Now valley fold the corners mentally
labeled 2 and 4.
Creating a SONOBE
(7) You can tuck the corner flap
underneath the “air-plane” fold.
Creating a SONOBE
(8) Now flip the paper over and valley fold
the two points in to make a square.
Creating a SONOBE
(9) Finally mountain fold the paper down
the diagonal found between the two
flaps.
Creating a SONOBE
When unfolded the final SONOBE
should look like this.
BUILDING A CUBE
This requires 6 SONOBE pieces.
(1) Fit a point of one piece into a pocket
of another piece.
BUILDING A CUBE
(2) Fit a point of another piece into the
inside pocket of the previously added
piece.
BUILDING A CUBE
(3) Fold the flap of the original piece over
and tuck it into the pocket of the
newly added piece. A pyramid should
be formed.
BUILDING A CUBE
(4) Using two more pieces and one of the
existing points, attach another
pyramid.
BUILDING A CUBE
(5) Fold the pyramids along the common
side and tuck in the appropriate points.
It should basically resemble a cube
with two flaps at the top.
BUILDING A CUBE
(6) Now add in the last piece. Again, place
points into pockets and pockets over
points. Here’s our cube!
BUILDING A STELLATED
OCTAHEDRON
This requires 12 SONOBE pieces.
(1) Do steps 1 to 4 that were done in the
construction of the cube.
BUILDING A STELLATED
OCTAHEDRON
(2) Now add another pyramid by using one
of the existing pieces and two new
pieces.
BUILDING A STELLATED
OCTAHEDRON
(3) Using one more piece and two of the
existing pieces form another pyramid
so that 4 pyramids are clustered
around a common apex.
BUILDING A STELLATED
OCTAHEDRON
(4) Flip the structure over and using one
of the existing flaps and two new
pieces add yet another pyramid. Do
the same for the opposite flap.
BUILDING A STELLATED
OCTAHEDRON
(5) Now just close up the structure.
Match points with flaps so that around
every apex you have 4 pyramids.
BUILDING OTHER POLYHEDRA
If you cluster 5 pyramids around a common
apex then you will create a STELLATED
ICOSAHEDRON (made up of 20 pyramids!).
How many pieces will you need? Why?
BUILDING OTHER POLYHEDRA
You can mix and match the number of pyramids that
you cluster around a common apex. By doing this you
can create several really strange polyhedra. Here’s a
construction consisting of three pieces. I call it
“FRANKENSTEIN’S CUBE”.
GO NUTS!
Try building a strange polyhedron, or try
constructing a stellated icosahedron. It’s fun and
very relaxing…I tend to fold while listening to
soothing music.
If you make something really neat send me a picture
and I can put it up on my webpage. Again, my email
is jkudrle@cem.uvm.edu.