I. History of Origami
II. Terms related to origami
III. Origami and Mathematics (Some
neat theorms )
IV. Constructing Polygons (Yet
another neat theorem)
V. Constructing Polyhedra (Modular
VI. Other ways how maths is used in
•In ancient Japanese ori literally
translates to folded while gami literally
translates to paper.
Thus the term
origami translates to folded paper
Origami has roots in several different
cultures. The oldest records of origami
or paper folding can be traced to the
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.
Early origami was only performed
during ceremonial occasions (i.e.
weddings, funerals, etc.).
FLAT FOLD – An origami which you could
place flat on the ground and compress
without adding new creases.
CREASE PATTERN – The pattern of creases
found when an origami is completely
MOUNTAIN CREASE – A crease which
looks like a mountain or a ridge.
VALLEY CREASE – A crease which looks
like a valley or a trench.
VERTEX – A point on the interior of the
paper where two or more creases
The difference between the
number of mountain creases and
the number of valley creases
intersecting at a particular vertex
• The all dashed lines represent
mountain creases while the
dashed/dotted lines represent
Let M be the number of mountain creases
at a vertex x.
Let V be the number of valley creases at a
Maekawa’s Theorem states that at the
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
1. Take your piece of paper and fold it
into an origami so that the crease
pattern has only one vertex.
2.Take the flat origami with the vertex
pointing towards the ceiling and fold it
about 1½ inches below the vertex.
3.What type of shape is formed when the
“altered” origami is opened?
4. As the “altered” origami is closed, what
happens to the interior angles of the
5. Some get smaller – Mountain Creases
Some get larger – Valley Creases
When the “altered” origami is folded
up, we have formed a FLAT POLYGON
whose interior angles are either:
0° – Mountain
360° – Valley Creases
Recap – Viewing our flat origami we have
an n-sided polygon which has interior angles
0° – M of these
360° – V of these
Thus, the sum of all of the interior
angles would be:
0M + 360V
180(5) – 360
What is the sum of the interior angles of any
(180n – 360)°
180(n – 2)°
So, we have that the sum of all of the interior angles of any polygon with n sides is:
180(n – 2)
But, we discovered that the sum of the interior angles of each of our FLAT
0M + 360V
where M is the number of mountain creases and V is the number of valley
creases at a vertex x.
Equating both of these expressions we get:
180(n – 2) = 0M + 360V
Recall that n = M + V.
So, we have:
180(M + V – 2) = 0M + 360V
180M + 180V – 360 = 360V
180M – 180V = 360
Thus, we have shown that given an arbitrary vertex x with
M mountain creases and V valley creases, either:
This completes our proof!
constructed by joining the edges
of many different polygons.
(Think 3-Dimensional polygon.)
0 SONOBE – A flat origami which when
pieced together with identical SONOBE
units can be used to modularly
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