台本（折り紙）：English version

Origami is an art form with roots in Asia (might be in Japan)
more than 1,000 years.

When you fold a sheet of paper in half, you create a straight-line,
called "crease".

There are two kinds of crese.
One is created from by "mountain fold", and
the other is created from "valley fold".

I think you know the Japanese expressions of each.

I will use a dashed(green) line ------ to indicate a valley fold,
and a dash-dot(red) line _._._._. to indicate a mountain fold.

Here is an example
I think you can fold the pattern.

It is easy, isn't it.

Ok I will give you some examples.
Please try by yourself.

Then I will give a formulation for more restricted
way of folding paper.
We say that the folding is a flat folding, if you can
make the result of the folding flat, that is, it is
flattened on a plane (or, in a table table).

Let me ask. Among those examples that you have,
which one is a flat folding?

Then I will pose a question. Let's think about it together.
Please take a paper of this pattern.
(Four mountain creases meeting in a vertex. )
The question is to introduce valley creases
(mountain crease is not allowed).
A typical solution is this.
Let me ask: can you find other patterns ?

You think you could find all of the patterns ?
(This a kind of exhaustion problem. )

Is there any one who can give an explanation
why these are all?


=================

In the next stage, I will add more restrictions on
flat folding.

That is during the process
from one paper to the final flat position　wa have to
keep each face bounded by creases flat.
Just imagine that each face is made of hard materials
like iron.
Let us call such folding a rigid folding.

Can there be a rigid folding ?
Let us think about this question.
Please take a strip in fron of you.
Can you fold it in this manner ?

The next is what is called Miura folding.
This is a folding pattern of Miura folding.
Before folding it by yourself, please try to imagine,
the outcome, I mean, the image of the folded paper.

Miura folding is known as an efficient way of
folding maps.

Let mee try to convince you that Miura folding is a
natural way of folding.

Let us think about folding patterns of a long thin
square, which looks like a paper bag for wood
chopsticks served in restaurants.
From the left side you have a mountain crease, and
from the right side you have a valley crease.
The question is:
Find out the simplest way for joining the creasese
in the middle of the square.
Please tell me if you can do it, and try to explain.

OK, for the explanation of the next stage,
please cut the folding pattern of Miura folding
along the middle line into two equal two pieces.

Next, please fold the piece.
Can you catch the idea that it is a sequence of
the previous simple folding above.

Then try to put the pieces side by side.
It will give the Miura folding.

I hope you have caught the idea, and
I am hoping you something more:
that is to expand the idea of Miura folding
for more general settings.
For example, does the quadrangle in the pattern
must be the same ?
Well, the truth is that the situation is more flexible.


Here is another example, something artistic.
It is called Roert Lang's Oval Tessellation

Please try.