This audience because for years.
The work I've been doing with my
colleagues to be a more child is more on
the mathematical side
than on actual science.
And we want to reach
audiences outside our own so.
When we talk about dogs had it been
paid here you probably have seen.
An object by bizarre hunger
months here which is
as structured then if yours if you pull
in one direction from some of its.
Bars it will widen it will
expand in every other
possible direction so this is
a really throng kind of property but
what I want to extend from this example
is that it's made of rigid bars
connected by joints at
the lower creation and
this one when you state in any direction
in the expense in any other direction but
it's looks like to
the stronger conditions or
maybe we have invested in something else
but what is important in this case is that
when you put it in that direction it has
only one possible motion it will not go or
pain to go in some way so
intimate you know science and
we have not met your scientists but
we live in the very
heavy material into one defined to
have a negative points on this ratio.
This is not this they can if the be there
if you stretch it in some direction
it will be the same in
some other direction so
for instance a non a positive force
on racial material of the something
like this when you find pool
it becomes thinner Yeah but.
Material like this will in fact expand.
OK.
OK.
So these are two videos made by.
Respected researchers in the field
that demonstrate both at their
own level in a model that might go
underneath which looks very much
like the baron joined model what
a non city behavior might look like
and what they're not sitting behavior
might look like so this one is not getting
thinner is getting slightly
thicker when you pull and
the underlying structure is this
infinite in your big structure which
if you look at the bars and the joints
are connected in exactly the same way just
the way displaced in space they often
refer to the one on the right is being
the entrant honey come this is
the classical honey company so.
Well so we are interested in
modeling such behavior where we look
only at geometry but because well where
is that there is matter that is geometry
sitting behavior can be also
study that the economic level.
It has been observed in a number
of crystal in matter in particular
this is a simulation of what.
Christoper let my call lapse and
you see it when it expensive expense
in all directions when it collapse
it collapses everywhere so.
Here our current approach is to study
such problems well if we want to play big
city behavior in chrysalis of course
we can do our level simulation but
this is computationally intensive
We have heard about simulations
some people said OK maybe take a day or
two more for the work week or so
on I'm a computer scientist I'm really
attracted by problems where I can make
a contribution themselves speeding them
up getting more into missions about the.
Underlying geometry and
getting better methods and
you want to note is that we are not
putting any kind of physics in
in this picture we are not talking
about other forces interactions and so
on we are essentially just talking about
the cinematics of these structures so
in terms of the zines So there
are two things it's we want to predict
the behavior in the you want to design
similar structures and in terms of
design there have been yeah there is
a problem with this OK so there have been.
A number of synthetic materials but
most of them are based on essentially
these two underlying designs
give you miles through thinking Square
says I have almost everybody design
something generalising this structure and
this is there the end from come
that we have seen there are variations
on this Monte eaves and various other.
Approaches that are do not fall
into the buy in green framework and
they have to be repeated in some way to
give three the opposite effect Our So
here are our girls first of all.
We have I'm going to have to call
the fugitive I'm going to describe in
the science that we have done with them
under the title of geometry sceptics just
so that it's clear that we are not
addressing all the physical properties
that may be delivered for
a complete study of such materials so
our goal is to develop a human think
theory that can explain in print the dog
sitting behavior and help design of
sediment I might T.V.'s so it's even
if it has these properties that you Kim
of came structures perhaps similar to.
The interim government others but
not just those two is just that
they are just a few of them so let's see
if we can get hopefully an infinite.
And an abundant.
Sit on the sides and to get in there
I'll give you a brief introduction
to Germany cut out civics
definitions in the main results and
then I want to address this problem
of the thinking in designing and
do both of them with the concrete
example so rather than putting equations
you would not see an equation
in my in my presentation and
a bunch of few items that are hard to
grasp to me that the I'm going to walk you
through an example to get all
these results in and in addition.
So how a very few want to read about this
these are some of the papers that starting
from two thousand and ten when we
attended a meeting with some physicists
we got interested in these problems and
we have a bunch of papers some
of them are very recent and
all of them are available on my web
page also together with a group
of students I'm developing a software or
digital it was meant for
this for the taking flexibility
in motion of proteins but
now it's extending the words crystals
in the goal would be to try to predict
among others if we make the algorithms
become as efficient as the hope
to detect the property is based on this
he already of some of the crystals
in particular it can be applied to
your life and get there in a moment so
now there are papers that I refer to and
all this theory it's.
The it's based on the legacy
of geometry papers of Maxwell.
In particular there is
a famous proper a few of them
of called next to a month year and that
gives a characterization of the G.D.P.
of Byron joined three
months in Dimension two and
nothing is known in Dimension your higher
end it would be a major breakthrough
to Maddox in mathematics and so on so
I want to tell you that everything.
That we have developed would not have
been possible unless we did we did that
get the generalization of this in the
periodic setting which I'm going to define
the moment so the I will feel I'm
of Maxwell it's a few of them that.
The length that I insist of a bar in
joint planar framework again is going
from the same words in
the mission to put where
he calls lifting the bar here surfaces in
dimension three and again this year and
is strictly related to the finances
in to their dimension two and
what we did in order to the I some of the
properties about Automator we generalize
it to beyond existing So these are the
main underlying mathematical result.
It's amazing that they believe in
the end to implications in the material
science good to let me do a bit of
the actual definition and the main
results of the model I'm working with it's
a pity over the bar enjoin framework so
meaning that something that
generalize is hearing from hunger so
I have rigid bars rotatable
joints any dimension hopefully
it's purely me can be taking
a magic definition or no forces or
other physical considerations now this
is not unusual in their literature so
this is a paper for up from a little
Spaulding from nineteen thirty
you see there is thought talking about
the framework of the saw the need for
the light Crystal and he notices that
while it is strong it is not the region so
he notices flexibility properties you know
this is the sort of wide color lapses and
while collapses there edge of the Union
cell it's something here we are a bit
of this I don't know some something is
touching maybe maybe three or four.
So.
You need cell actually
also collapse it folks.
OK So listen so
now what you see here is the honey
come of the gut feelings really
I just want to point power that this
is not a legal book of this character.
Because.
Now I I cannot point to that sort of
in the way that some weird things
happened with it but it was.
Somehow she did the job
frankly I don't know.
There will be.
Yes.
But.
They acknowledge I'm
a computer scientist Yeah.
I know what it means OK so.
OK good so what they mean is that these
joints are not to rotate about yes
in fact this is a very is it's you
know this is a very strong structure.
OK so for some reason they disappeared
from my screen which is very bad but
I'll do my best to continue up to a point
so they'll figure the by the Indian
framework is a movie need to get up and
there is a problem with analyzing the G.D.
inflexibly in Kenya to get upset
just want to point to one difficulty
is they can go for the emotions I mean
the formations with finite measure of
computational methods so
here's an ism why for this thing this this
think of it this being in thing
is just a more basic creed so
you can be forming a column by column or
go by a rope but in the end you get up
get an infinity what they mean a countable
dimension of that the formation space.
What we want it's first of all
all of these definitions
not all of them preserve.
The city so what we would like to rather
forget about all the possible definitions
and focus only on those just as of served
in in the Harlings set up
those dead movie in such
a way that they preserve the periodicity
girl and that is the basic definition so
I want to do something like this so
I feel superior This is the group of my.
So in my in to new.
Hampshire and the most of the bailout
is just growing to preserve So
it's just like a constrained in four step.
So I did leave exam on purpose because
you know you see that this one actually
since you have so certain directions so
the ultimate goal is to try to understand.
I'm getting into a top political
problem here OK trying to understand.
The emotion and what makes this
doc said to God not talk CITIC so
I have a big problem now because I
cannot see on my screen OK good so.
Let's see if it works good.
Try again good much better.
OK.
Good so the reason I wanted to see
my screen is because I want to do
an a little animation of this.
Just to give you an intuition about what
this whole their formation thing means so
this is my we usually see this lead
this is in a very static fashion but
this one is before my book is The formable
with exactly one degree of freedom so
when I'm rotating this vertex
when I'm writing this bar
around this paint joint you see that there
picture this it is maintained yes of
these are these are still the orbits of
Representatives However the left is the
forms very much like in polling space per
in the forms in the forms
their overall volume
if you want of this infinit structure
of which I only could make a piece.
Changes it shrinks in some directions it
expends in others and that's the kind of
of behavior that we like to pinpoint
to understand mathematically.
And predict OK So
to summarize if we go back to my slides
to summarize the object of study is
a movement the graph on which we
fix their the letter the letter
is not translations.
In other words appear in the group and we
allow it to move only in such a way that
maintains their action so
I want to emphasize that
specifying the unit cells
in its language it's
important because you may get different
properties both from flexibility to get
them in from other things
then on which sail you.
You choose so depending how what is
their unit sell this one to whose X.
I'm going to pay good so let me keep all
these definitions of this is our model of
buying a player the buy in joint We have
a default measure theory we understand how
it should move and the adventure of
this different mission theory is.
That it can be addressed with finite with
finite methods meaning with taking from
one to break Joe Mitty that the measure of
the configuration space will be a finite
number and we can talk about one degree
of freedom two degrees of freedom and so
on which was impossible in the previous
it so now in this class of frame or
sell I can understand what is
an ox said the patriotic one and
now has that another little issue with
the concept of what an exotic material is
because the definition it's a little vague
Yes So the direction in which thinking
the material produces opposite is stench
and letting the not fall in a plane is
not well defined so some materials
have different behavior for for
different reactions some have several
directions in which you can observe this.
Behavior the stretching itself may not be
uniquely defined in other words there may
be degrees of freedom so what you
have seen before with the enter and
honey come Actually it has
two degrees of freedom so
you have to to do specific
information in order to for
it to be accepted so we like to understand
not so much what is an offsetting
material this is not well defined by
the rather what is an ox said the past and
once we have not said the past we say
that the material has or the pay or
the claim or has accepted behavior and we
call or not sitting material one that has
said it behavior so that in other words
they exist some the formation path that
is accepted so therefore mission past
means they are the trajectory and.
This is the main definition and what we
have to retain from it is essentially this
equivalent definition which says that
the one parameter the formation of a video
the frame or his ox say the wind
the current given by the name matrices
of the letters of basis of periods
has all the velocity vector or
the tensions in the positive
semi difficult so
I'm to emphasize the fact that you
can come up with the least six or
seven variations on these definitions so
you can look at increasing volume of
the units and if you look at one direction
giving them all the elections you can
look at all of them but so far this
seems two hundred is more than many but
that income taxes are the worst and
it's it has very beautiful properties and
I want to convince you that this is
a very definition to be investigated
in particular when just very quickly to
mention that this condition with the gram
interest is being in the same
positive definite cone has the other.
Analogy with causal trajectories
in special relativity in fact
this illustration is done with three
dimensional McCoskey space but
it is exactly the.
Corresponds to what happens for a set
this is the in the plane in dimension to
the engine from which the mentioned
two in dimensions the M.
higher it's not the same.
OK So in other words an existing
material we define as being one that has
an accepting pass starting from a given
state so let me give you an example yes so
this is a bar enjoying framework and
it's in this particular state and
I'm going to move it and
I know happy already we did all the Apply
the field Ensign did all the analysis
it has only one degree of freedom so
moved in exactly one determined
patient in opinion in the failure way.
This is some proof and you see there.
You think if any direction that you
focus on it goes in that direction but
it goes simultaneously in all
the other directions Yes And
we've got lots of devoted girls and
some point some some changes happen
remember that they're at the end
honey come have this been insisting
probably that this was yes and
that they we're all of the we're not
complex that we'll be flexing looks and
this is in fact the defining
property of this object and
when property stops being true so
where is this angle now.
On X..
Well things can break it's it
can stop being the opposite and
that's what we would
like to understand OK.
Good so let's talk about
predicting infant there's a lot of
possibilities of the definition that they
gave before was a global definition that
about a trajectory but then we look at the
infinitesimal level in the city and with
the Grammy trees gives us directly are
going to make way of very fine of tasting
that condition so there's some good in
the concept the consequence is that.
Of.
Existence of non-trivial infinitesimal
tipping the formation can be
decided using simulation in program.
It's known that in things dimension
this can be done only on the time but
this is a very complex algorithm in
practice there are implementations in
matlab that the reportedly work does
not do well I'm talking with a number
of specialists in this area and
we hope to be able to at least tested on
slightly the larger crystals soon so
with this.
I want to know how much time I have so
that they can.
See.
Yup I don't see a clock at all so that's
why yes if you can point to the I Have a.
Twenty minutes perfect because that
allows me to talk about these two
topics so
not just in the infinitesimal level
I would like to understand how on
the whole one dimensional trajectory or
of such a deformable object which part of
it is accepted because it can develop but
it cannot grow into infinity it
stops being upset at some point and
the box it is in the property it stops and
the.
Other words we want to go beyond
infinitesimal behavior we want to identify
what is going to accept the sick men or
segments Yes we want to understand
more of the general congregations
face on the formation project.
And the second topic that I want
to have this is how to design so
I pointed out earlier there
are essentially they're rotating squares.
With the intern honey come
in the few other designs but
actually couldn't design something
different couldn't design something that
is intrinsically three dimensional and
played it then say that once you build
it it's a basically the printing and
it has a bit of a joint is going
to expand or have the city property
as if we are talking about and
I'm going to show you that our theory
actually these two two techniques for
designing an abundance of in fact
the Infinite Family in Dimension two and
we have the criterion
that gives us place size.
In the lot of frameworks in dimension
three and the futility of higher
dimensions but no one nobody wants to
build framework mention more of them or
so once again we'll start with the large
the formations in other words not just
invented in Mali we want to characterize
your typical region in design so now
I'm going to do this with a sequence of
demonstrations rather than giving you the.
He already there are no
equations by the way but
there are behind the scenes
in there in the papers
let me give you a preview of what
you'll see and then I'm going to run.
The animation of actually their
interactive tools to help us again
about what is going to go well first
of all I want to emphasize it will only
work with one degree of freedom
because it's the trajectory is clear
it's one curve and we are looking
at segments on this trajectory
in of course the projector is an object
sitting in a high dimensional space but
it's still a one parameter and it's the
first thing that we have to understand and
then move on to more degrees of
freedom which will give surfaces and
hyper services and so
on but essentially once
you have an exit it through object three
on a higher dimensional surface it can be.
Built combinations of the ones from
the one dimension one degree of freedom.
Project body so
one degree of freedom is essential so
that's where we are focusing
on understanding.
OK so here's.
The theory.
We wanted in order to predict the vice
York City regions we have a theory at him
and then the women will look at.
Him and me wrong them for
example will all of this
is in those leave on and
leave so it and you know if it's also
said is find some properties we will
know that it's accepted so essentially
is remove the object then raise these
these incidents is not is the our lives so
it's we know in subset and
I'm going to show that in a moment when
that change happens if it's a high
parable in India mentioned it's a hypo
then it stops being accepted that's
what you see here you have so the premium
differently from it I mean with the.
Or when it's here it goes up to
a point but when does it stop and
that is what this criterion says and
it's very easy to test so
in order to the most of what I want
to do I'm going to actually design
a periodic framework to convince
you that what I'm doing is true so
I'm going to cover these two top
except with a single example.
OK And here's how I'm going
to design it and that is that
they can meet that allows us to design
things in arbitrary dimensions so
first of all you start and
then demonstrating it in Dimension two but
the principle is valid in higher dimension
and that's where we'll be applying
start with the one degree of freedom bar
in doing mechanised a finite one this
is not periodic a finite one in
the know everything about them or
something that we know about
them what what is needed OK.
Then.
This one we end our lives
this configuration space we.
Have on they are not their lives
is not fixed and we choose them
in such a way that they satisfy the
infinitesimal infinitesimal oxidization
criteria not only in there because
this is one bit of freedom structure.
In any home crowd the circle
here from what we can do with.
A case of this is they fill
in on this instance of this
one we go with the math and.
These points here will hold
on as a citizen in for
us to argue that each interval gives us.
The good property from the point
of view of the meeting and
this property carries over to the.
To the opinion for the more those two
arrows I'm going to choose them as
the generic interest of my ladies and I'm
going to build the framework from the back
and when proved that it inherits it
has the same configuration space so
that's where the design work comes.
In then.
So we generate all this it has
the same configuration space and I.
And the force and now.
They're in the least makes sense only in
the video saying they don't make sense in
that case and is only is the I live so
its I know that I am exotic
as they star being well if so
it's I know that is going to be.
The stopping of sceptic So
let me do the demonstration here because
of this is to see where they go and
I think this is the fun part of
my presentation I hope
it's more intuitive OK.
OK so here it is so.
I start with this
wonderfully the mechanism
this is the whole configuration space
there's a whole trajectory Yeah and
I can design it I mean I can change
the lengths I do whatever but
once I designed it to have what they
were properties I want it's fixed OK So
there's the first top ject what I want
to do is to turn this into this object.
Which is the petty or
the framework built on top of it so
starting from that simple object
in that simple object can be
not just the poor bar
mechanism can be anything and
we have a whole theory that allows us
to build there an infinite number of
of finite Birendra in mechanised with
the desired properties choose the.
Two three vectors and
generate periodic structure yeah so
that's their object that I want to be able
to so how do I build it so first of all.
Let me skip that one.
Yes and no.
OK I'm sorry it's.
Get.
No So.
Let me try another one.
It will not.
Now I know it's not working itself anyway.
OK found out that's why have
I had it on my slides just in case
it does anything like this but.
The idea is the following
if I do it's the.
Kind.
There is something with Java so
that's that's a major problem nowadays
is all the demos that they have built
over the years that are available
online it's hard to see them because of
the job our incompatibilities and
various other things but.
Let me still try to get this.
OK.
Try to get another one.
It's so frustrating I'm sorry.
OK so tried this way.
So.
OK but.
So let me skip a few things.
Right everything.
So the idea was that as this.
Finite.
Object.
The forms by violent drain framework
the forms I want you to keep
an eye on the two diagonals and notice
that in this segment so when I move from
here to here both of them increase in
length yes of this is an increase.
On this interval from here to here one
diagonal increases namely this one and
the other one decreases so
it's a quantitatively different part of
the computation space here again
they both have the same behavior
both decrease or if I go in the other
direction both increase Yes and
here it's again the non
expensive property so
this property where these two
things expand is called that
expensive interval of the configuration
space and that's other will be using.
Then I have to do it again this way so
then the next one is to
build the periodic framework
and I want you to notice that
it has a similar behavior so
here everything expense but
here it starts certain parts
start collapsing and it's reasonable
that when this will start again
everything decreases if I go in the other
direction everything increases and
then the sun is not even
a framework it's overlaps so.
Good so finally.
Let's show you Let me show you what
happens with the comics in this simulation
in this animation so
on when we start with this framework.
OK You see that reality expense and
you see that.
Ellipse here and when we cross this event
this becomes a hyperbola at which point
sides with the part of the finite frame or
and it's this parable our love the hype
it will light changes a little bit if it
changes until this point and from this
point on on it's again an ellipse and
from this point on it's a hyperbola So
our theory him which is valid in
any dimension says that as long
as you have a Hyperball lead for
these carefully defined vectors points
then it is ox said take as long as
is not an ellipse so if they have to they
have to say it is five more than that but
they don't have time to go all
the details then it's not box it and
this one allows us to have in fact the.
Full characterization of
the configuration space.
OK.
So.
Yeah so it's this is the.
Trajectories of the on
the trajectory I characterize
what is the opposite except meant in
fact there are two one is here and
you are other one is here and
this is the general principle and
that's what the three of them
that helped us and the questions.
Now now is just the property that
they have to be an analysis so it so
let me tell you what
the theory means because it's
this was just a demonstration of what with
what tools you have trying to capture.
The property OK So let me continue so
what I again can you please tell
me how much time I have because.
Four minutes OK So essentially these
are the three topics that I wanted to
cover and they can cover them very
quickly one is how to design.
The design by expensive frameworks
in The reason being that they have
a complete understanding of expensive
frame or seen dimension too but
not in mention three and some of the
recent results we have we are building.
You can call it a family and even a family
of exhibit things in dimension three that
with the same technique of
building of city frameworks we can
expand it to dimension three and
this way we get than an infinite set of
oak CITIC an expensive is an infinity and
probably this one is the most thing we
call is how to design decide if a frame or
let's say a zero light framework is
a mark of of let's have a mark of vertex
kissing that are here drop that one has
three degrees of freedom that's and
that's a more challenging than this
one and we have a now got in this
an algorithm that is based not on the same
definite problem which is expensive but
on the use of elliptic curves so these are
the two things that they can continue on.
And let me just walk you
all the way to the end and
then I will see how much time I have
to go into just one of these young so
that's summary So what I'm trying
to bring to the attention of
of this audience is that we have developed
a theory that can cause plane and
pretty talk sitting behavior and this.
Sign of city materials and
their basis of this theory is.
Part of a series over a long series of
papers that we have published in the last
eighty years and.