this is a a ppt to show how maths is used in origami

2. 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
Origami)
VI. Other ways how maths is used in
origami

3. •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
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.
Early origami was only performed
during ceremonial occasions (i.e.
weddings, funerals, etc.).

4. 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
unfolded.
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
intersect.

5. The difference between the
number of mountain creases and
the number of valley creases
intersecting at a particular vertex
is always…

6. • The all dashed lines represent
mountain creases while the
dashed/dotted lines represent
valley creases.
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

7. 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

8. 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?
polygon
4. As the “altered” origami is closed, what
happens to the interior angles of the
polygon?
5. Some get smaller – Mountain Creases
Some get larger – Valley Creases
polygon

9. When the “altered” origami is folded
up, we have formed a FLAT POLYGON
whose interior angles are either:
0° – Mountain
or
360° – Valley Creases
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

10. s
side
Shape
Angle sum
180
3
5
180(5) – 360
or
540

11. What is the sum of the interior angles of any
polygon?
SIDES
n
SHAPE
ANGLE SUM
(180n – 360)°
or
180(n – 2)°

12. 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
POLYGONS is:
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
M–V=2

13. Thus, we have shown that given an arbitrary vertex x with
M mountain creases and V valley creases, either:
M–V=2
or
V–M=2
This completes our proof!

15. POLYHEDRON
–
A
solid
constructed by joining the edges
of many different polygons.
(Think 3-Dimensional polygon.)

16. 0 SONOBE – A flat origami which when
pieced together with identical SONOBE
units can be used to modularly
construct polyhedra.