OK so.
So I'd like to tell you a little bit about
some of the work that we've been doing
trying to exploit disorder and
the idea here is to design function
into mechanical networks and so this has
been a project that has been going on for
a few years of spearheaded
by Andrew you and
collaborators at University of
Pennsylvania and some people at University
of Chicago my student kneedeep Machina and
postdoc and our bishop Berger and
then some people from the engineering
department had Chicago Daniel Reed in one
to Pablo and I'll tell you about
their work as we go along.
So what I want to ask is
does disorder matter to the material
properties of the system and
so what I want to compare for
example is what happens for
jam glasses in comparison to
what you get in a crystal and so
up why I'm interested in this is that you
know we study glasses all the time but
if you know when I was a kid and
I was told by a glass was one to
the window and knocked on the glass and
it was solid just like everything else but
they told me this is very different and
so when well why is it so different
I mean and so well if you go to an X.
ray machine and you take X.
rays of it you'll see that
the structure is different and
somehow this was supposed to be so
terribly deep and
I never understood why I should care
if that was the subtle thing that
is the difference is just that
the particles weren't in organized places.
Organized in a Christian way then
why is the glass interesting and so
the idea here has been to try and
find out other ways in which glasses or
jam systems are more physis tems are
actually different from what you get in
the crystals and so there are a couple
of things that we know are somewhat
different between crystals and
glasses and so one thing is.
That the exit patients that you get
in crystals you know those very well
they're the by law for
the specific heat at low temperatures T.Q.
case it's just saying that the low
frequency modes are normal modes of
a long wavelength plane waves and
what we know is that for.
A game systems or
glasses that you have many more X.
A Taishan particularly at low frequencies
and so this is one thing that
makes things different between glasses and
crystals but are there other things that
are special and so what I'm interested
in asking Is Ken thought or.
Provide some unique functionality that you
wouldn't easily be able to
get in a crystal and so
what are some of the things that we know
about well we know that if you have a.
You know a glass or amorphous material
that there are certain things that
it has that is because
it doesn't have this.
Perfect structure we
don't really know what a.
Defect is and we have grain boundaries and
so this provides certain kind weaknesses
for the crystal that you don't necessarily
get in the glass and so this is.
You know.
This water is more forgiving
in that way that is you don't
have failure due to those kind of modes.
Another thing that we know about glasses
and jam systems is that your many many
quibbling ground states many more than you
get in a crystal and so this can lead to.
Adaptable features of the system
that there are multiple paths for
creating a certain kind of function this
is the kind of thing I want to explore
today and then there's one thing that
I just want to mention because it
came up in a lot of things that we've
heard about during this this conference is
that there's a common feature that
seems to appear in all sorts of.
Disordered systems which is this
idea of constraint counter.
And so we've seen that in case of
the origami that we've heard about we also
hear back in faint counting when you talk
about jamming or networks or glasses that
cetera and so this kind of idea is
throughout a lot of disordered systems and
so one question is well is this
something that you can grab ahold of and
therefore make some kind of
functionality out of this aspect and so
what I want to ask here is can this kind
of flexibility that you have in the glass
be harnessed in certain ways and
so there are two things on
the talk about global features and
also local features and so these
are the two things I want to mention.
And so here I want to focus
specifically on networks and so
I think networks I'm going to talk about
are derived from jampacked things and
so I don't think this is
actually necessary for the.
Conclusions I'm going to reach but they
are but that is actually what the studies
were and so that's what I can tell you
about concretely and I think a lot of
these things can probably be generalized
to more general kinds of disordered
materials so how do we make these
networks so we take a system of box
filled with frictionless shears that
are soft that you can squeeze together and
so you put these particles in a box
randomly and then you let them.
Go to their push each other apart because
they have repulsive interactions between
them until they no longer interact if
they can or if the vine is too small
then they will end up in some kind
of mechanical equilibrium and
this mechanical equilibrium will be
disordered in so this is what we have on
the left hand side here is that
these are the spheres that for
two dimensional jam packing and then what
we do to make this into a network is we.
The centers of those fears and
we put nodes at those points and
wherever there's an overlap that means
an interaction between two particles we
replace that overlap with an unstretched
spring between the two parties and
so this is what we get on the right so
we are racist for
years as fears are no longer there and
we have just this network
of unstretched Springs between nodes and
so that's the system that I want to talk
about today and see what we can what kind
of functionality we can have in that case.
So as I said I want to talk about
two things global response and
the local response and so the global
response I want to just talk about
the module lie of the system so
the bulk modulus and
the sheer margin so want to remind
you briefly about what those are so
the bulk modulus is if I take
a system of the bulk system
the bulk modulus is I squeeze
it in all directions equally and
that cost a certain amount of energy
to do that the energy cost to squeeze
is a measure of the book market should
book on just as a measure of the energy
cost to squeeze this thing
compressed in all directions so
that's one of the modulus the other
modulus is the sheer modules that
I want to talk about and so
this is I take this system and I.
Find shear it in this case
it's just simple sheer and
again moving it in this direction
is going to cost some energy and
that energy one way of characterizing
the energy is that your Majesty math
the energy it takes to give it
a unit here in that direction so
these are the two kinds of modular life
that we're interested in in this case and
one of the things that you should.
Think is well how big
are these two modules I.
And the two module I are typically in
crystals comparable to one another and
why is that well simply that I have to
if I look at the microscopic I have
two atoms and I'm pushing them together
this way well they have to come up against
potentials of their nearest neighbors and
so that one energy scale the other thing
is if I do it your mind just will maybe
I'm coming at this direction instead and
while still going up against that same
potential between the two particles so
it's going to be the same order of
magnitude it may be off by a cosigned
theta I don't care about or
close on data here the direction which I'm
moving these things that's of order
unity so that your module and
bulk of crystals are typically comparable
to one another and so this is the.
What we typically get for
Crystal and behavior and then
I've been talking about these two things
because the one thing that we heard about.
In the first day was the side
DIA of the possums ratio until
I just want to remind you what the possums
ratio is possums ratio is a measure of
you know if I take a material and
an ordinary material and I
pull it in one direction it gets narrower
in the direction just as you showed us on.
Wednesday and so
this is the normal material and so
this has a positive possums ratio I
pulled in this direction and it squeezes
down in the other two directions and in
the ideal case it would be incompressible
that is the volume of this thing would
not change in that particular case now.
That is what you have for
normal materials and
then the word I learned
from Martin was that
the other end of this is weird right
that was your characterization of.
A negative plus on ratio of materials and
so this is the case of negative possums
racial material I call it in one direction
and it expands equally in all these other
directions so OK and so
material that does that is not common and
we found I mean this is so it's a weird
material and one of the things is could we
make materials like that and it would that
be something generic that you could find
in if you could make this in
three dimensions this could be a.
Something that you could say well we
can actually begin to make materials
that are not normal.
And so
typically if you think about it what
materials that you do you have that are.
Negative plus one ratio and.
There are probably only two that
you're commonly familiar with one is.
It's only barely negative which
would be cork that you have for
your wine bottles and so
you don't want something that has a.
Positive ratio which puts a cork in and
it breaks the neck of the bottle and
you don't want with native POS on
ratio because then as you know if you.
You know push in from the bottom it
squeeze in from the sides and it will go
out too easily so you you want something
that has essentially zero Possum's ratio
and so cork is very good it has zero near
zero percent ratio the other thing that
you may know that has negative parts
of ratio if you complicate a paper and
you couple of paper and this as you have
in Him Paul you pull in one direction and
somehow expands in all these
other directions and so
this is the two materials that
you typically see there you go
if you raise it up high so we can
all see that look at that brilliant.
Experimentalist.
Par Excellence OK.
Now I raise this regard.
To the well of the module I of
the system because in elasticities.
There's a one to one relationship
between the Possum's ratio and this.
Ratio of the bloke and it's your module so
if you have a very very low value of shear
compared to the bulk of the shear of
essentially goes to zero compared to the
bulk then you have a normal material and
so it's easy to shear the same hard
to compress it and so this is.
Something which would have a normal
positive Possum's ratio and
if you go to the other regime where this
your module is much much much much greater
than the bulk modulus then
you go into this organic
case the negative passant ratio you did
and so that's what I want to explore.
It so what I want to ask now
if one of these networks
is I've measured the bulk monitors
I've measured the shear modulus and
now what I want to ask you suppose
I go in to this network and
I grab a bond at random and
I pull it out of the system.
What's going to happen to the module well.
It will change by a small amount
because there's one Bond missing now so
there will be a delta be.
Labeled I for
that bond there we moved a delta G.
The change in the bulk money
in the sheer amount just
due to that one bomb being removed so
there's a delta G.
and there's a Delta B.
for that one bond and so
I want to measure what those that is for
that bond but
now want to put those that bond back and
I want to now take another bond at random
out of this network pull that one out and
I do the same thing I measure what is
Delta be for that bond and Delta G.
for that bond and now I do this for
all the bonds in the system and
I now am going to get a distribution of.
How many bonds.
How many.
Bonds will give a contribution Delta B I
how many bonds will give a contribution
to the G.I. and so to see how this works
I want to start off by looking at what
you would get if you had a crystal.
And so a crystal is really a very boring
system that is I have a unit cell
which I repeat in term an ability over and
over and over again in all directions and
if I think of the very simplest case of
a crystal so I'm going to have one and
impune itself then it's a very simple
thing that is what does the distribution
of these B I's and geodes look like well
it'll just be a delta function that is
if I take this bond out it'll have
a value of Dell to be I I put it back and
because his next Bond that I choose is
identical to the first one I take it away
it will also have that same contribution
to the book munchers for this year modules
so all module I own the all bonds
contributes to the bulk modulus and
sure modules in an identical way and so
what I have is this is distribution of B.
eyes here still to be eyes will just
be a delta function everything will do
the same thing Bill nothing will
do more nothing will do less.
OK And so
what I want to ask is Well what happens to
this property of a material
when I make it disordered and
so typically what I would
have thought is that if I.
Go and I make a make the same disorder
I put disorder into mine a material
that I would get a blurring out of
that delta function that it would just
spread out a little bit and you know some
bonds will create a little more time with
create a little bit different less change
to the July but it will be basically.
Centered around that average value.
And so what I want to ask now is
well what happens if I do this for.
Or a jam system so
something that's fully disordered and.
As disorders as I can make it OK And
so what you're going to see
here is now I'm going to show you
this distribution function for
the bulk of my just if your mind to
light of this thing for these this.
These distribution function for Delta
beyond but the first thing you're going
to see is this dirty work at the foot
because the axes are going to change
from linear axes to log axes which means
something funny is happening OK and
so themselves this is not
what disorder does and
it this is the distribution
functions that we get and so
what you see about this is that first of
all the actually says changelog axes so
that means that I have an incredibly
broad distribution of how
a bond will affect either the book
knowledge of the few modules and
that's one thing it goes all the way
it's very broad it goes all the way it's
continuous It goes all the way down
to as low a value as we can find in
our simulations and so this goes all the
way down to zero as close as we can get.
The other thing about this is that it's
basically universal that it's the same
distribution here for
Delta B I N for Delta G.I.
both have the same behavior so it's.
Still something is you know kind
of universal about this and.
What of.
Then you'll have post-doc was able to show
is that this distribution function which
is sweeping something under the rug here
about how universal it is it's not exactly
universal to begin with but after a few
pruning things it becomes universal for
the bookmark to so once you've done that
then this distribution function has this
simple form of a power law cut
off by this exponential and
so this is something
we can show where the.
This country and so we have these
distributions that are broad and
continuous and universal OK So
now having gotten this far what I want
to ask you is well you know if you have
a bond over at the right hand side so.
You don't have a point there but.
Of Days of well of two I take
the side way over here OK and.
Well I thought you can give me your arm
is going to use your arm to the point.
So if I take this pond here and
that was a bond that made
a very big difference to the both modulus
I would have guessed that that would have
had a very big effect on the shear margin
says Well that is if it's an important
bond to the system is that
important bond to the system but
what we find is that that's
not true at all that is these.
Distributions are nearly completely
uncorrelated with one another so
the distribution of what you get for
the shear module and for the bulk on July
are less than one percent correlated so
they're completely uncorrelated So
if I took this pond away for
the bulk month list.
Its contribution to this year much is
could have come from anywhere in this
distribution it doesn't matter what
its value was to the book modules and
so this is.
These two properties are somehow very
different from what I would have
expected from a crystal and
this is what I want to use now in.
In order to do things and
so what I want to do is first go
back to something that Martin did.
Way back in two thousand and
nine which was if you took a.
Network of the kind that I mention and
I reach in and
pull a bond out at random I can ask well
what does it do to the bulk margins and
what does it do this year modulus and.
Basically if I keep doing this
taking bombs out over and
over again I'm just doing
rigidity percolation And
at that point I know that both
the two mothers and the both
modules are going to drop in tandem
with one another and so the ratio of G.
over B.
remains constant and so
nothing much happens when I take the.
Things at random but notice that what
I'm plotting here is the ratio of
this year to the book modules and I just
told you that I have a broad distribution
of both your module and
contributions so I can reach in and
pull out a bond that has a very big effect
on one modules but not on the other or
vice versa and so
this is what I want to do now and
I'm going to take out the bond
which has the largest
contribution to the vault much less so
I know what all the bonds are I can ask
the computer to tell me which Bond if I
take it away has the largest effect
on the book modulus I go in and
I pulled up on the way what that's
going to do is it's going to make B.
drop as fast as it can because I've taken
the way the bond that created it was most
important for that whereas modulus is just
going to drop by an average amount and
as I do this I do this over and over and
over again so I'm starting up here at.
The disease just measuring
how many bonds extra over
rigidity threshold that I have so
out here I have a lot of extra
bonds in the system and now I take them
away one time and that means that Delta Z.
is dropping so I start here and
I take away one bond and
move up one little bit here I keep doing
it over and over and over again and
this thing just continues to increase G.
over be increases dramatically and this
is now again a log scale and so this Geo
would be is up to ten to the ten I mean so
ten orders of magnitude in that case.
Now I can do.
Something else I could have gone and
asked So let's prune the bonds
that has a minimum effect on
the book modulus Now if I do that.
Going to.
Be will not change at all but Delta G.
is going to change by its
average amount each time and so
that means this ratio of G.
over B.
is going to drop so
I start from the same point for
the same network and
I just keep pruning in that way and now.
Will be dropped and
this goes down to about ten minus four and
so just by pruning.
A few percent of the bonds in the system I
can tune this shear to the bulk modulus by
something like fourteen orders of
magnitude and so that's a large change and
what that's telling me is I can go from
something that's completely or exact that
is with a possible ratio
of negative one all the way
to one is completely incompressible which
is Possum's ratio in well in three D.
would be cause for issue of plus
a half in to do you be a plus one.
OK And so this is what we could
do fight pruning the Delta B.S.
I could have also done the same
thing by pruning on the shear
Montreux I instead of the book my choice
and here again I get another set of.
Curves that I can go I can make something
or that it by putting a minimum value
of Delta G.I. or make it incompatible
by tuning the maximum value Delta G.
Od so I have all these different ways of
getting to the end point that I want.
OK So this is.
These different algorithms produce
different kind of pruning.
Behavior and so I get different
lattices I can get the same.
Possums ratio by tuning in.
Riding of ways as a many
many ways of achieving this
the desired property of the system
the Possum's ratio by pruning.
In many different ways I
can get the same thing.
OK And so then what we try to do
with we want to make these things in
the laboratory and so we go in and so we
can do this either in two dimensions and
so in two dimensions we
take a sheet of rubber and
we take it to a laser cutter and we cut
out everything that doesn't look like
a network and
that's what we have on the left and
on the right we can do the see in three
dimensions by building this up by three D.
printer and so
we print these things in three D.
in two you can make.
Systems that are three D.
or two D..
And so this is.
One way that we test whether
we're actually doing something
correct in the system now.
What I'm not going to show
you the behavior for the.
Organic networks here are five
shows something in the.
For the local thing in a minute so
having said that with making these things
in the laboratory of now I want to ask
what else can we do with these networks
and so the question is can we make.
Other things that are on
a more local scale so
I told you about the bulk module the bulk
module I were the only things that
happened I would be kind of interesting
but it would be somewhat limited and so
what we want to do is take a lesson
from biology and ask about what do
what does biology do in biology makes
use of proteins in a way that is very.
Very important to the biology in this
is what's called protein Alistair E.
and so here this is.
Pulled from the biology website and
so you take.
A protein and
what this idea of Alistair does is
that if you have one side
on the protein where you.
Add your bind to something here then.
At a site way over on the other side of
the protein it now has the ability to bind
this substrate over here whereas
if you didn't have this.
Bond here it would lose the ability
to find something far away and so
this is an important way of the proteins
control their behavior their dynamics and
so what we wanted to ask is well can
we do the same thing in the case of.
Of our networks or
mechanical networks and so here is what
we were doing in so we take so
the idea is I ask you
please come up to this network that
we've just made this network and.
I ask you to please choose at random Your
choice of where you want to source and
so this was where you happen to choose
a source that I ask you to come back in
please choose another place on this
network that you want to be the target and
so you chose this thing far
away to be on the target so
this is totally random is your choice and
then I ask you Do you want this target
when I pull the source apart so
I want to target to pull apart or
do I want to go together so it's your
choice precisely where you source where
the target and what you want the active
action to be and so the question is can we
can we tune the same to that case and
so we using the same idea and
this is what I wanted to be able
to show you if we had had the.
Overhead projector of so
this is the idea so here we have a.
View so the This Is It OK so you can see
it and so what you actually see here is
are two separate networks
there's one on top of the other.
What if I pull them apart you'll see that
these two networks really were almost
identical and what I can show you here
because I don't have four hands but
is that if I pull on the source
I can make in one case
the target come apart and the case of how
you come together and so I will show that
to you if anyone's interested they
DID WE CAN I can show that you.
Directly but but
this actually works OK and so.
This kind of behavior so this is.
We didn't know you could do this and
now the fact that you can do this and
you can do this with nearly
one hundred percent.
Success rate is something that we
didn't expect to be so easy to do and
so this kind of work has also done by two
other groups who ran the same time so
I met you group and.
See TO SEE THE has also done some related
work in the same the same general ideas.
Can you design out a star response.
So what.
I want to ask now was well how
complicated can that task be that is.
What I told it beginning is we have
one source and one target could I'm
now ask you to have one source but
more target so the sources
over at the upper left and I have three
things which I've asked the system to.
Create So if I pull on the source
I want those three other targets
to move in the direction and with the
amplitude that I've put right there and
the question is can you do that and
the answer is yes and.
What we're showing here is at the on
the bottom left is how many targets can I.
Have.
The the function that I prescribe at
random how what's the probability
of being able to get all of those
to work and so what you see here.
Is.
So for a system the size of an In this
case with a size eight system what we
can do with you can have asked will see if
I can tune one yes I can tune one can or
two to yes I can tune to but
pretty soon I can tune that many because
I only have eight particles in the system
but if I make the system bigger and bigger
YOU SEE THAT CAN TOO MANY MANY MANY sites
here many targets with that one source so
the complexity of the task is increasing
so there's a number of targets I'm asking
system to tune and each different color is
for a system of a different size and so
if I make the system larger and larger and
larger I can tune many many many more
sites almost uniformly so
there is a probability of success and so
I can get nearly one hundred percent
success of with many many targets if if I
make the system large and the question is
Well how does a number of targets increase
with the number of nodes I have in
the system and at least the present
behavior is the algorithm that we have
is a little bit sub linear it still
says that if I have an infinite system I
can prune an infinite I can to an infinite
number of sites but it doesn't scale
with the size of the system so
it says that in the infinite
size limit I can
tune zero percentage of the sides but
still be an infinite number but.
But this is depends on the algorithm that
we're choosing to tune this thing in so
that we may be able to
do better than that.
OK but
it also tells us one other thing which is
that what how many ways
could I have two and one.
Site right that is and
well since I can tune any of them and T.
of them then the number of ways
of getting that one function.
Is going to be some exponential.
You know it's a.
Whatever that thing is called the.
The Benefactor Oriel thing with a factor
of me that's lots so it's not one and.
So.
What it's saying is that this is
really a very easy thing to to create
five minutes OK so.
And so this is the.
Kind of showing the power of the said is.
We started with asking what.
Was biology doing and could we mimic it
and then what we found out was very easy
to do it and part of the reason so easy to
do it is that there are so many different
ways in which you can do it and
that's what this is basically telling us.
Until now I just want to end with one
last thing which is talking about aging.
Behavior and so this is now a little
bit different that is what I'm
thinking about is suppose I have
a sand pile you know and so
it's a big heap of sand and
down in the middle of the sense
of it's feeling the weight from
everything on top of it and so
these particles are under a lot of
force between the grains of sand.
So what's going to happen here is well
the grains that are under the most force
they're going to do form plastically
over time not a lot but over time a long
enough time you will see that these grains
are no longer going to be nice little
fears they'll be little flat top guides
and so they're going to distort over time.
And so
the bonds that are the under the most
stress most force are going to form
faster than those that are under
less force and so this is the idea so
it's a little bit different from
the pruning that is this is a weaker form
of pruning I'm just allowing the system to
evolve under its own steam
it's measuring its own.
As on it and
then responding with a memory of what it.
Learned about.
From all the weight that's
been on top of it so
that it has memory in it it's also
a greedy algorithm so in the sense that.
What it's doing is it's no longer asking
pulling the bond away asking which Bond
was the most important I'm just asking
which Bond has most stress on it and
that's something that
the system just knows and
it responds to what that stress is so
it's a very greedy algorithm it's not
having to do any complicated computations
and it just goes down hill and
so this is the results that we def
of this so the idea here is that.
The bonds are in the most stress
to form classically faster than
those with less stress and so what I mean
by that so the energy of this pile is
the sum of the energies of each of
the bombs separately minus a lot
a lot is the unstretched length and I'm
going to let the unstretched length relax
very slowly depending on how much force
there is on that actually and so L.
I zero changes slowly in time and
we feed that back in and this is
what the ageing of the system does and
what you see is well the bulk modest drops
this year modulus doesn't do very much and
so after time in two dimensions or
three dimensions you get that the ratio
of geo be it gets larger and larger and
you get some of this really it drives
itself towards inorganic behavior
automatically on its own and so this is
something called directed ageing that is
you usually think of ageing is just some
bad thing that happens you don't typically
think of aging into this directed in
one pathway to a particular place and
this is a ageing that you somehow knows
it wants to make a weird material items.
OK And so that's where I basically
want to end so I just want to.
Conclude that.
Designing function into
disordered networks there is
what I've tried to present here is a new
paradigm for designing punctual materials
based on the idea of using the disorder
in the material to make it
possible to design the functionality I've
shown that you can do this globally and
you can select the passant ratio you can.
Do locally and get action at a distance
Alistair effect in the system.
But as I've said many times before no
good talk is really finished until
you have the both feet line and so this is
a bullshit line so everything above this
line I hope you believe in I mean I
think that's all facts everything below
the bullshit line you have to believe it
all OK And so this is what I want to.
See them the picture if you will and
so what I've tried to show here is
that there's a new principle for
disordered matter and this principle
is that this independent of Bond
level response that you have these very
broad distributions for the Delta bialys
in the Delta G.I. they're very very broad
They go all the way down to zero and
these distributions are uncorrelated with
each other and this allows a new kind of
functionality to be tuned into disordered
materials that is not available for
their order can't counterpart the second
thing that I want to make is OK so
what does this have to do with real things
and so can you do this in situ and so
on in our lab we're working on doing this
in situ those we're trying to figure out
whether we can actually measure without
going to the computer which Panjandrum
most stress and then flip them and make
the systems do do stuff on their own so
that's one thing could we do this
microscopically so that you go down into.
Design interactions between particles at
the microscopic scale which you could then
if they're under stress a laser could.
Come in and zap them those bonds that
are in distress where as it would
leave the other ones alone that would be
a different kind of thing that you could
you could do and so this would be so that
we can dream of happening one thing is
biology which is you know we
started with the South stuff having
to do with biology but what does it
really part about biology that's kind of.
You know those great you bristle on our
part to think that what we have to say
has anything to do with biology but you
can at least say that well since it's so
easy to do in these mechanical
networks perhaps that's why biology
can make use of it so easily and so
that was one idea that of how we can
return to biology something that we took
from biology to begin with and finally you
know the idea is that you might be able
to use this stuff in architecture I'm so
buildings no longer have to have
rectilinear walls I mean we have you know
you look at a Frank Gehry building and
they are all doing all sorts of things and
what this allows you to do is OK so
you have this disordered structure and
so if my colleague across
the way pisses me off
what I want to be able to do is go to
my window push it a little bit and
have the window fall out OK And so
this would be how I could get even with.
The.
People in mind but my my last OK
with that I thank you very much
I should just say want to hear all
the people who did all the work and
so and really who was involved
in all of this and then
Goodrich started this work out on
the organic materials and they.
Took that over improve all
these various things about it
Jason rocks has done the work
on the Alice Terry and
then you'll read in one the Pablo
from engineering group.
Has applied this to actually show how you
can add on bendy constraints into these
materials and then the machine and
Berger did the making the.
In the lab and then Elaine it headed for
E N N Rick run Ellen Fitch
who did the work on many you know
how many different ways you can.
Create the same kind of how
complex your function can be
without Thank you very much thank you.