I also want to thank all

my officemates who of.

Voluntarily helped me put this
presentation together to start off.

With a thing about one thing about folding
we should think about functionality

in a way folding allows you to transform
systems from one state to another so

natural examples that occur in the world
are protein folding where you have some

are in a strand it's
some sequence of R.N.A.

and a protein strand and the biochemistry
that tells you how it interacts

determines what kind of final structures
it can form and this is what provides you

with all the biochemistry surfer life
you can also have a leaf budding

where the leaf starts up the bud and
up as a leaf as we all know but

in between there is this unfolding
process which So scientists.

Say that the actual pattern of the.

Veins on the leaves determine
the rate at which unfolds in these

five evolutionary purposes so

there's all these things in nature where
folding seems to provide mechanisms.

Necessary for life but you can also
design things by coming up with

nice folding patterns and so if you
want to put a solar cell in space it's

very difficult to fly something with
a lot of surface area through atmosphere

you want to fold it up and when you
get into space finally unfold it and

have an access to the Sun So
here is a model depicted by people J.P.L..

But what I'm trying to show here is that

by folding you can do you can
create crease pattern and

in some way actually things to go from
a single state into a final state.

And it doesn't really matter what
the system is here you have something

on the microscopic scale something and

some things that are very macroscopic
all that requires is you're thin enough.

To fold so
let's narrow down thing about origami.

Because of something we should be familiar
with everyone has a piece of paper is

the idea is you take a thin cheap shot
crease patter on it like this and

once you have your crease pattern you
are allowed to fall about increases but

your faces that are mapped out by these
creases aren't allowed to bend structure

intersect and
through some process which run

Turman you want to turn this crease
pattern into something like a body.

Now on the first side I showed
you this image of a satellite.

Which has a start and some pull it up
position and it has a single degree of

freedom so the only way can move is
to open into its unfurled position.

It has one single.

Continuous falling mechanism but how do
you get these things usually you have to

use a very spec carefully to geometry
the classical example is the mirror or

a crease pattern and so this is something
where all your creases are parallelograms

And if you have it properly designed
then the only motions are allowed

to fold are these gold motions where every
cell behaves the same and you go from

something awful that up into something
on for old and one continuous motion.

But like I said this requires very
carefully to geometry so what about not

fine tuned geometry what I want to
talk about is an entire class of.

Periodically triangulator origami so
what I'm saying is that we decide some

units cell where every
face of the triangle and

then we have two last directions
which you use to test laid sheet.

And so D other than these two requirements
the geometry is not very important so

what I mean by geometry is the lengths and
angles between all the edges.

What I'm going to show you is that this

trying elation puts out the edge of
mechanical stability and because of that.

We're able to determine just
by knowing all the faces or

trying the US that we have
some folding mechanisms.

So how do we characterize the state
you're on well let's talk about

the two single faces you can think about
the dihedral Engel measured between them.

And so here you can measure
the angle theta and then I call the.

I want to character is by.

Tract of a factor pie that way when you're
folded flat you say you have an angle of

zero and what I'm looking for
are mechanisms that take you through

some set of these that he
joined goals without it.

Costing energy and
you can think of your organ machine

as being in a configuration space
with that he will measure to every.

One of these creases but since we're
periodic we just have to think about them

and one unit cell so
my configuration space is.

About each one of these edges

now what happens when you
fold is if you change you.

Know by so now I'm going to cough I
think this is a very small change

then the orientation of this face is going
to change by some out five times the.

Unit vector of the edge.

What that means is that every face
is going to change some orientation

change its orientation so let's say we're
folding everything about the space here

then if we're going to be consistent and
say that our.

Is rigid you're not allowed to stretch
your bun your faces then as you go before

this face the space the space the space
you have to get back at the same place and

this is how we end up thinking about
the what's called the debt he jewel for

Angle vertex condition which
says if you some of the changes

in your day he's going all multiplied
by the edges they're measured about

around a vertex you must get to zero.

Now generally this is something that you
can you can find these guys using specific

geometry so we're not going to do
that we're instead going think about

changing from a mapping from this origami
problem into the spring that works so

now we're taking the exact same system but
instead

of thinking about the edges you creases
you're going to think about Springs and

so the thing about various issues
are going to think about some sites and so

analysis of your configuration
being decided by the chairman by.

Goals is determined by the site positions
so how do you fold this well we want to

move our sights through some configuration
with zero changing without causing

energy so we're looking for a zero modes
which don't extend these bottoms the.

Period this year versus them the fact that
we have these lattice directions which

test later sell means that if we translate
our Along these if we perform these last

operations then we automatically
get the zero energy linear

modes where we can displace our sites that
amount without causing energy but the.

Don't correspond the full So
can we turn on the folding modes

Well let's think about the other thing
if you were this morning here for

this morning for stuff you know stresses
in if you weren't I'll explain really

quickly every If this is some sight and
these are all springs attaching to it you

can associate the tension with each spring
you try to force by the tension with

applied by the unit vector of the edge and
dresses maintain equilibrium so

you get the tax condition which says the
sum of the forces around the site is zero

but this is exactly the same condition as
the day he joined overtax condition and

so if we can find some sort of tensions
which are stresses that we get these

days he's going to fall so how can we get
them while the fact that we're trying

you are means we have an equal number
of constraints in degrees of freedom or

at the Maxwell condition so we have for
example here we have four sites and

a total of twelve versus four times
three is twelve we will have to move and

three mentioned so.

That means that every.

Stress we have should have a zero Motor
Company with it but we already said that

we have some zero modes due to our
last symmetries and so here's a.

Diagram of basically the logic as we
have since we're periodic we have

some lettuce in the trees we can perform
some operation and everything is unchanged

from there we get your modes from the
Maxwell condition we get just tensions and

from the equivalence of the vertex
conditions we can use these two for

their origami so
I'm running a little late on time

I'm going to jump through things so
I can get to the end and answer questions

the basic idea is you when you fold
some sheets you're going to if you're

looking at origami that have some global
curvature and there it is a question of

what kind of purchase you can have it
turns out which I will have time to show.

So that you have this.

You form these screwy crystals where
you're related your last operator

sions are just translations with
the translations and rotations and

so if you skip through this basically
the rotations along the last

directions have to commute and
also the way you translate has to.

You can only have your screw crystal
can only have one reserve curvature.

And.

The can the summary of this
trying that origami is that

from the logic of having some
global motions relating to Folds

is we get the family of we get to folding.

Modes for our.

Are trying that origami and you can extend
the still there systems the same logic

if you have continuous sheets or

if you have some black hole karega me or
spin Kagami lattices so.

Thank you very much and
happy answer questions.