What I'm going to tell you
about is really the story about

cause of course all of the discovery and
why it makes a destroyer and

a little bit about this
structure of Kweisi Crystal and

if I have time to work and I want to touch
upon a few properties of the system but

really the take home message is why did
you make so much noise that you have a.

Diffraction pattern as like this one.

And why did the dentures man
got the Nobel Prize for that.

So just to start would
like just to let you read

the statement from Swen Linden member
of the Noble committee that introduced

the identification on to the to the King
of Sweden when he received another prize.

Can you hear me on the back.

Yes OK.

So for three million now we have known
that five hundred symmetry is incompatible

always pray at a city this is well known
and we show you in a few slides why and

for almost three centuries we believe that
prayer basically was a prerequisite for

Crystal unity.

The electron diffraction pattern of
ten by Dan chessmen on a Paul eight

nine hundred eighty two shows that at
least one of the statement is flawed and

it has led to a revision of our we
view of the concept of symmetry and

crest in Christianity alike.

The object he discovered
are a page Arctic.

Ordered searcher that the exotic
symmetries that today are known as

quasi Crystal.

So I think of do do the interesting
part also of the story is not only

the discovery but
how much he had to fight for it.

And this is what linen is referring to.

Now having the courage to believe in his
observation and in himself than chest man

has changed our view of what order is and
has reminded us of the importance

of balance between preservation and
renewal even for the most.

Well established paradigms.

Science is a surgical construction
on an empirical fundament.

Observation made or break series as
an experimental I like this very much to.

All right so let's see this.

What was discovered.

But so this is a copy of done churchmen X.
book I dated from April one thousand and

eighty two he was doing electron
microscopy on a lorry of aluminum and

manganese So
this is per se also an interesting story.

What is the minimal or interesting.

There are interesting because you make
aircraft with them so you want an aluminum

system which is very light so you want
to have a lot of aluminum in it but

you want to be strength strong you know
dat aluminum is a very soft material.

So the idea in the community.

Was to put additions of
elements in a limb in him but

only a tiny fraction a few percent.

Done was working on doubts adding a few
percent of Meghan is an Ironman and

in aluminum but
instead of say staying was a few percent.

He went on ten percent eleven and
up to fourteen at fourteen percent

with this is the weight percent
fourteen percent at the make per cent.

Hugh knocked at the transmission
electron Mike micrograph and

he saw inserted area diffraction says
a Sadie's not a sad day a ten for

two mystery there were ten forcemeat.

So it was extremely surprising so
only is only

accounts so that there it is this is
what it's all so you have a turn for

symmetry you have shopped pics and
you have these five four to me tree here.

We showed was pentagrams and the Pens are
going to go in a hierarchical structure.

So here.

If you no longer loop.

Look along this line.

It is not periodic at all.

It goes at one and
towel which is a golden mean.

And then Tahrir Square and tell it to
the sewed and so there is an expansion

of the position of the peaks which
follow a pentagonal symmetry and

an inflation which is an inflation
in the golden mean ratio.

Well this is totally new it has never
been seen before and reported before.

So according to use own account.

He went he was so
you know it's a dock room so

he went out in the hallway and
looked around

asking people then when he was there you
can back the five hundred factions D.S.O..

Well this is really a fight for fashion
we have to understand what's going on.

So in fact this fight for diffraction is
only for one one type of an orientation

and he by tilting the sample you
can look at are the orientations.

And he found out that there was a two for
diffraction and

a three fold diffraction and
put everything together was right and

all these correspond to the destruction
of an I guess I hadn't.

So here see the five for diffraction
of the I could say hidden by there.

The two fall to Miter which correspond to
meet point between these nine here and

the three fold which is at
the midpoint of the triangle.

So we really have in this
electronic microscope the picture.

Of the symmetry of an accident.

So here is the need discussing with the
two co-authors of the paper laid on John

can and do any question us and
this paper was published the first report

of the paper was published in Nineteen
Eighty-Four nine hundred eighty four.

This is a major discovery it took
two years two years to be published

Don't try to publish it.

It was rejected one time two times.

It took for him to bring on.

Board these two people to
any cost to us and John can.

So John can was a director of
the NIST at the time it was

the national blue rule of standard.

And he was a very well known is still
a very well known mentalities I mean.

And he put all his weight and his name and
his reputation behind his discovery.

And then in house yes.

Whereas on the sabbatical from
C.N.N.'s In the US at the same

the same place and
he has a crystallographer Here's a series.

He was doing series of crystallography and

he is with John Kerry and
with John Candy out there.

The one who said well it is possible
to have an order treasure which is

non-periodic.

So let's go back.

So this is well known but
I just want to do for

the one in the room who might not be
very familiar with US go back to that.

So what Danish man has seen is in is
sans a bin of electrons which has

a wave night which is about the wavelength
of the separation between planes.

And so what you normally
have in abrupt affection for

prior planes as it's very well known to
brag diffraction condition the the angle.

If there is a relation between the angle
and the distance between the two

planes the outcoming
waves can be in in phase.

So there is a constructive pattern.

They can be anything else but
they can also be out of phase and

you have no light at all.

So if you put a sample in it in
a microscope or if you use an X. ray

tube for that matter it's has a wavelength
about of the separation of the planes.

This is what you have you send a beam you
hit you hit a sample here of the world.

There was a lot of electronics
just to focus to be.

And at the certain angle you
have a constructive pattern and

you have a a spot in and it all the
position you do pattern is not construct.

And you don't have light.

So what this shows is that
you need to have order.

You need to have a certain distance
a relation between the plane

the separation between plane and the angle
in order to have a shabby function.

So this is very well known to one example.

This is an example structure.

If you don't have if you have a whole
distribution of distances and

planes between the To Do
list in a structure you and

up with random random light so
this is a hollow pitch diffraction pattern

like an intimate for space so this is
an amorphous faces of a human in Ireland.

This is interesting because you also
find quite a crystal in anime an IRA.

So you see there is a sharp
distinction between the two so

there is no doubt absolutely no doubt
that if there is a diffraction pattern

with chop eeks then
the structure is ordered.

Well this is so how do we know
what do we know or did they know

in one thousand eighty two about order of
structure whether this is the real Chris

or graffiti and it started in a
publication in seven hundred eighty four.

So it's very well established feel for
many many years and

this was in the treated amenable regime.

Henri.
There is still you have so

he was looking at the shape of crystals
and determined that it because of

the shape of Crystal you have
to had order in crystals and

this order was later on though
well understood by brother.

This is it if a sort of graphic by brother
and he just went on all the possible

possibility and came out for
is the fact that you can have these only

these order structure
in a periodic system.

So let me explain in a few words.

So what are the symmetries which
are allowed by periodicity So

you have a periodic set of planes.

While there are only a few.

To buy or to over.

Three or four over six and that's it.

You cannot have to Miter you can
have eight full ten full twelve or

whatever you want.

It's that and only that while it's
very easy to demonstrate but just for

the purpose of this talk I show you if
you have a square like this you turn it.

It's strictly identical to itself right.

So you can do the same thing for
a three fold fraction strictly that

tickle to itself when you turn
around it while you can build nice

nice periodic systems and
drawing this is by Max issue one And

so two for three for
four fold and Max if I do.

Maybe because I couldn't five or
six forty three.

But trust me you can have it.

One hour what what happens
with the Pentagon.

So if you try to use that to find
a series of planes wipes and

now if you make a five for
Taishan you define a new set of claims.

So this is not invariant by would
say you cannot have this city and

and five for symmetry.

You don't want to create more planes in
other words if you look at it at to make

distances for instance here this is an
atom of distance between these two points

here and here you create a distance which
is too short to this distance cannot be so

it can be deduced rated very very
easily but just you have a feel for it.

So this is a paradigm Crystal graph
in one nine hundred eighty two.

This is from a book of extra diffraction.

Solid composed a solid composed
of atoms arranged in a pattern.

It's three dimensional This
is what is a crystal or

if you go back to textbook tech
you tell introduction to sell it.

State physics then we are here
we have here who's in the right.

We cannot find a light is that goes
into itself under other were to then we

can make a crystal for molecules we
individually have an axis but we should.

Not expect the ladies to have
a far forward Taishan axis.

Well well known.

So no five four but
still man had seen a five or so.

What could that be.

You see here this isn't
a micrograph of of aluminum

island from Italy and
you have five crisps.

And if you look at the diffraction
picture of these five crystals together.

You end up with something like that.

So the are sure this is a five for
diffraction feature you have.

Five different orientation.

But what happens is that if you
select if you look very carefully and

if you look at all these these
dots that has been have been.

Outlined here by Red Wings.

They form a crystal lattice
they form a periodic system.

You see here you have peer
declines they repeat periodic.

This is not what minute seen you
know the first diffraction picture I

showed you have a progression which

is in the irrational golden mean number
one towel Tahrir Square and so on.

So this is irrational.

So this cannot be periodic And here you
have a five fold diffraction pattern which

has a sublight this which is pretty early
when you look very closely in fact this is

a pseudo five for symmetry it does not
exactly fight for very very very details.

Also something important
is that how do they observe

doubt is the this crystal leads.

This is just Despite here needs to this
diffraction pattern of the red dots

are right.

If you take the diffraction
pattern of just the one next

to it it's going to lead to
another diffraction pattern and

so the fact that the five crystals
are brought together in that way.

Here we have ten Crystal brought
together each one is going to contribute

to a periodic system and

the combination of them the addition
of the five or ten crystals.

Is going to make a five foot defection.

Right.
So each individually is a crystal.

But the addition of all of them makes
it as if it was a five foot symmetry.

But it is absurd to fight for symmetry.

So what was done legs which is OK so
if we have micro crystals.

We just have to make the approach or
of the electorate being very very small.

So we're just going to look
at something extremely small.

So this is what is done with a micro
diffraction pattern and this is it.

So Michael defection pattern with
a spot which is forty nine meters.

So if that my quote Chris or

twins that the combination of all
between specify for diffraction picture

then each crystal has to have a side
which is smaller than fourteen animators.

So it is really really Mike which way.

In that case we have a problem because
even if we have really a tiny tiny crystal

like that you wouldn't expect to
have the peaks to be that sharp but

that's that's another another problem.

What came up a few years later and
this is just one example there's

a beautiful picture taking in the other
lawyer which is aluminum copper iron.

I chose this image because it
was very representative but

there are many different honors
which come back to that.

So this is high resolution
electron transmission microscopy.

Which means that you can even usually
these dots corresponds to a column and

at home or interpretation is a bit
more complicated than that but

just for now just.

Let's let's take this assumption.

So we have just one dot and
it is surrounded by ten points so

you see there is no training there was
no Chris told there was no grand Bang

theory there is nothing is told or
uniform.

And if you look really again at a higher
magnification you see it again.

All those tiny Pentagons So

the structure is intrinsically was
a five fold five four structure so

we have to understand if we have an
ordered structure which if I fail to meet.

So it cannot be periodic.

And this is another example of this is
not only a detraction by electrons for

X. rays.

So this is an extra
diffraction on a single grain.

This is in menus in
your single crystal and

you see very clearly the five four
three four and two fold symmetries.

All right so
now we have we have a material

that has an I can say it or
symmetry which is ordered it.

So how can we reconcile the long
range order and the five Force miter.

Question is of course immediately
where are the atoms are this some

specific properties and also what makes
that the system is and can be realized.

Can can can be at least made a stable and
we see there are also stable phases.

So I will go very briefly
to these different points.

For the first model that
Danish man proposed.

Just Right right.

Have to Do The discovery
was it OK You know what's.

What we really observe is it
would take order it is not so

much of a predicate because it kind of the
periodic So he sought that you've taken.

I could say he didn't and you just put
it can I say he did together it can be

random fashion but
you keep the orientation order.

This is what you have.

If you do that.

What happens is you have very short
distances here between between the plane

and they don't show up in
a diffraction though this model is

not the correct model so we have to do to
get something a little bit more elaborate.

So the new paradigm is in order structure
is not necessary pre-op exam going to try

to give you an egg with just one example
of that was if you'd been achieved.

Sequence.

So here Sabrina she's become so
we have two rabbits.

A male and a female we need a male and
a few minutes.

So what do they.

Do I have to short time when
we have to baby rabbits.

But.

If we wait a little bit of time what
happens is my baby my two baby rabbits

are going to be adults whether it's.

And meanwhile do parent rabbits being
parents rabbits CONTINUE TO ME BABY.

So we have after one month we have our
adult baby rabbits and baby wrists so

if we go along a dead baby rabbits become
grown up ready it said while the brothers

are becoming of age also so
we're seeing a series of rabbits so

what do you have after one month to
more three months four months of how

many rabbits we have yes it is
a very relevant question for to see.

All right so because we are physicist
we're going to make an assumption for

the calculation Saudi read
bits have an infinite life or

eight knots or so
rabbits have an infinite life so

this arrangement of adults and
babies rabbits is very simple.

So we start with an adult couple and

these adult couples makes an adult and
a baby at the Next Generation Y.

the baby rabbits become adults
at this next generation.

So if you consent to construct
a sequence we have a and then a B.

A B. A and so forth.

So this sequence is
deterministic we have a rule

to build the sick once it
is told to be orderly.

You know after the end generation
how many rabbits babies and

adults you have it's totally known.

But it is not merely Can we show
you in a few strides how you can

determine that it's not a demonstration
it is just to have a feel of it.

So what is open ratio of adults
of a baby's Well this is also

a very relevant question may
you want to babies to be for

social with security
in the long run right.

So what is the it's a series of X.
is a number of adults of a baby.

At the end if the series
were pretty Arctic so

bear with me one sequence of final size.

So this is my sequence this sequence
really repeat itself beyond Bickley right.

So that the numbers of A's of the B.S.
would be at the for the infinite sequence

would simply be the one that you have
with the products sequence right.

So which means in this
particular example for

the whole sequence who would have one two
three four five adults over three babies.

So this is a rational number.

It has to be irrational number.

No other way.

Now for the Fibonacci series
the one I just showed you before.

What you have is a step.

N. plus one the number of adults of
a baby is is simply the number of

adults of a babies at
the previous step right.

Divided And
here the number of babies comes from

the adults come from the middle
of babies at the step before.

So this series is nothing but
one of a one of an X..

At the previous step and this is this is a
series one two three five isn't of course.

If you take the limit of that
series infinitely many you find

it is the governing one plus five over
to say it is an irrational number.

So this notion if there
is a take home message

is intrinsically Kweisi periodicity
is linked to irrationality.

If you have something which is
rational you going to end up with some

with apparent exposure.

Quasiperiodic structure means
irrationality somewhere along the line.

So this is dishtowel square and so for

that I should India indeed the pattern so
here is here it is again.

So if this limit is irrational.

It shows and I wrote it.

Specifically it shows that
no subset of the series.

So there is no segment in
the series that has the say.

Adults' of the baby ratio
as a whole series right.

So because of that the series cannot be
a repeating sequence the series has to

be non-periodic OK So if you've been
out you see really deterministic series

which is nonpaying Arctic manana
the less it is in order to Sirius.

And now if you look at what you would.

So let's go through that.

Our first.

There was a very simple
representation of this ratio A over B.

which is simply a slope right.

So if I take a periodic lattice.

Two dimensional periodic lattice and
on this.

I built just a staircase which
is it's from a have an adult and

a a go up each time I have a B. I go.

Original because the series converge to a
finite slope is not the divergent series.

It is confined these slope is
going to be the rational number

tower that we have just
defined before right.

So and I construct a B.
a a B. and so forth.

OK.

By construction the slope is
irrational because I believe like this.

And no if I want to read
out my cereal is very easy.

I can just read out by projecting all
this points that I have constructed.

I read that ab ab I can read that.

So what you see his Here you what you
start from a two dimensional space.

OK And you can end up with a which is per
unit and you can end up with the a set of

a system which is a periodic on
the on the SO perpendicular So let

me explain a bit better in that how to be
of the one dimensional Fibonacci sequence.

So you take the opposite approach that
we want to just explain you take it to

the periodic area of Point is called
a lattice my dog's point here.

You define a strip.

With a slope which is a slope to Russia.

No with the directions of the lady's OK so
this slope is stout It says dear.

Golden Mean.

All right.

This is Number of a lot of a baby's right.

So you define this slope you select
a window around the slope and

you just project the points
within this window.

Defined by the slope.

On one line and then you have you have
a question periodic system right.

So you start from it predicts system
that you can very well define

if you just cut through it and
you get an irrational.

Periodic system Question Period existence.

Sorry I'm confusing you.

You can do so
you do you can do the same thing.

Well.

You can do the same thing.

So with a slope which is quite a bionic or
was a slope which is preallocate So

if the slope is Priya Dick
you have a staircase but

the sequence is going to reproduce itself.

The longer this sequence to close.

So the the system from
a quality products system.

All right so
you can have longer and longer and

longer periods at if we approach more and
more the question prerecorded so

you can imagine a series of crystals which
are called Crystal approximate crystals

which are closer and closer to
the question to question Crystal.

Now the point here is that if you take a
fully transform of such such a structure.

It is the conclusion of the four year

transform of the projection of the lattice
through my two dimensional like this.

So we have a set of.

Delta points shopped at a point.

With these acceptance window which is
this the strip that I have selected.

All right.

So if you take the product of the two
which is a Delta two rocket here and

with this funny.

There.

If you end up with shop diffraction
peaks you just get to speak and

realty the president dealt a peak.

So you have a simple way of seeing
that people actually sequence actually

gives you a sharp diffraction peek you
have an audit structure which has a shabby

fraction P. But you when you can relax
the constraints of the symmetry.

So for question Chris thoughts.

In a sense of course all the definition
of the city is a function.

Which for your transform
consists of a series of peaks.

Of position.

Missing one word on integer position so
each system.

Which can be described as the sum of.

Of that nature where

the base is vectors to in my previous
case is a two dimensional Rector's.

Is is called quasiperiodic So
we immediately see that.

If Ellen.

Is larger than the physical space my
physical space before was one dimensional

and was two dimensional then we have a
series which is a periodic On the contrary

if any is exactly the size of the physical
space you have a simple Crystal.

So you can go if the sun goes over
more if you need more integer to

describe your system than the physical
space then you have you can just

as I described previously between the two
dimensional to one dimensional system.

I think you lost everything with me.

OK Again we do it again.

All right so let's move on.

Another example.

So maybe two.

To show you something
a little bit more concrete.

This is the Penrose lattice.

So this plan was like these were
discovered where before the question.

You have to realize also that.

When Danny Schiffman made his discovery.

Everything was ready.

The notion of course the prayer this city
was were it was in the nature of for

the mess a magical mathematics
is was well known.

Well well well understood
what it Penrose in one nine

hundred seventy five had published this
this which is quite repaired it and

had demonstrated that it was quite
a periodic So everything was ready.

What was really new in what
man did this he saw it and

he recognized it and he fought for it.

OK so what you have here is
a timing of a thick here and thin.

Wrong value.

So you have the proliferative
oblate they call it.

This is a MS non-periodic.

It has as you can see
a five foot symmetry.

It has a lot of there was a lot of
symmetry in this in this system.

So for
instance here you have local orders.

You can see if I fall and
you have plenty of them everywhere and

this star is this separation between
these local orders is about the size of

of this of this local arrangement.

So for and as you take a bigger one.

You see you find them but the most credit.

And they are not P.R.
because obviously here even bigger ones.

Is that now more apart so
that property is very important for a lot

of physical properties like electronic
properties because the the way function I

want to have time to go through that too
I just want to to give you a hint here.

The way found function is can be
localised on these local at any order and

can hope from one place to another one and
still is because a deception

a hierarchical structure there will be
a very very peculiar propagation and

estates are not extended nor
localized they can be self.

Self assemble it would be they
call that critical states.

This is dispositive that you see
here on the simple pen was ladies.

Translate into very
special electronic states.

In a real system.

So other properties of this of this

prayer decoder is first of all you
want to make sure that the did it.

Diffraction gives you a five
fold diffraction it is it is

chopped a fraction.

So this paper was published just a few
a few weeks after she discovered so

it was already in a review when when
she published is that of his paper and

it is very clearly the same diffraction
pattern as watchman as is so

this make really a big big impact to
have the two papers next one next after

the after the other so describes
the crystals The paper was entirely quasi

Crystal a new class of order structure and
this is what gave the name to the field.

So coming back to see
a pen was right there.

So I showed you the local order.

There was something which
is interesting also is.

Self similarity.

So if you take a series of
tires you can cut them or

out and
reconstruct a new panels like this.

You can see here you have the flat.

Rhombus And this scene from verse you
can build a new pen was like this.

So a blue starting from the black one.

So this is blue and of course you can do
that at many different inflations what

is different from a crystal is that the
inflation is of course by the golden mean

so here again you have this of similarity
with the government as this is a next

generation and next year a ship
that you can go on for Phantom.

So I was telling you about
the question prior to city and

the way to approach course you periodicity
with a periodic structure so this isn't

an example that may be a little more clear
than what I showed in in one dimension.

So here this is what is called
an approximate face so you have a pen.

The timing looks very much
like a pen Restylane But

here the order can repeat itself.

So I outlined a blue box.

So if I copy just copy the blue box to
here you see it's just it's pretty Arctic.

So you can how this plan was tying in but
periodic.

Now if if if you dislike a change your
reorganization of the of the tiles.

Then you have a structure that is is you
cannot find a subset a subset of it.

So you can have was a same time as
something that has the same local order.

You see the there arrangement of time
in this quality product timing here and

D. in this in this one is just the same so
locally it's exactly the same but you

one case it repeats periodically and in
this case it never repeats periodically.

So with the same set of targets the same
local environment you can grow.

You can have something periodic and
something non-periodic.

OK let's go back to to something
a little bit more concrete.

This is the first view of the I was ahead
or face as reported by fire just man.

As he saw it in nineteen eighty two.

So it is for those who are interested
in this is it dendritic structure.

So actually it goes actually quite
a crystal it is black flower here and

it grows from the seed and
it grows in the entity structure and

expel the solution of of aluminum so far
you have a mattress of aluminum around us.

The diffraction pattern to fight for

diffraction pattern actually come
from the center here or any any on.

And this is an S.T.M. suspending electron
microscopy picture where you very

very nicely see the flowers shape.

So we had a two dimensional cut and now
this is a three dimensional aspect of it.

This is still rapid liquid.

Selenium agonist in this system
is met us to all it does

it is the rapid quench of the melt.

But.

Right.

There is a sort of a right after
the discovery of the published paper.

Within a year or two there will be
tons of papers in the literature.

People had seen also across the crystal.

But at the time.

Nobody recognized it so
there you are you five you find a five

four diffraction pattern and I must
be wrong and this was discovered so.

Where when the paper the paper
paper appeared in the literature so

people just went back to the doors and
I have seen it again and so

there were many many many Eloise that
I'm sure if I felt a fraction seven and

this is after many years
of research a kind of a.

Of a summary of what has been found so
a lot of ago is composed of aluminum but

there are also zinc bays and cadmium
based aluminum they have been compiled

here by side who is really someone who
found a lot of those of the system.

So the system can be made a stable with
a lot of stable quality Crystal and

which is really interesting
is not only that stable but

you can have not only
very complex phases but

some some some faces are extremely simple
in the composition just binary ellipse.

If the fairies are using.

Have of simple composition are to
leave the face a gram is a nightmare.

So just as the cut of the face there for
instance for aluminum copper iron So D.C.

is a get at seven hundred seventy or
so here you have a live in him.

This is a human corner.

This is the irony corner so we're here and
this is copper corner and lose lives

here corresponds to the phase the area
in composition when the phases stayed.

When you can find it.

So if you zoom on that area.

So here you have a better phase which is.

Cubic.

This is the only get face
which is very close to the.

To Z. I can see it or face but

when you zoom on that this
is what you have to have.

I don't know if you can see from the back
but the scale of his one percent here.

So the extension of the I can say he
will face is only a few percent in each

direction so it is a stable phase of
the equilibrium phase there Graham but

you really really have to tune
the composition extremely well to have

just one phase and not to have the room
the heat or phase they're made a phase

of the beta phase which are intimately
compound extremely close.

What is interesting in those compounds
this is why I spend a little bit of time

on that it has been proven that these
crystal and phase are actually approximate

in the sense of the Penrose that is that I
showed you before the approximate of the.

I could say heat or phase which means that
the local environment in this crystalline

phase mimic the local environment in
the Arkansas had or face very very close.

So it's interesting because you have this
almost the same composition and you can

check what use the role of the long range
order Kweisi periodic versus period it and

with different kinds of periodicity depair
the city being closer and closer and

closer to the to the quality of the city.

So this is an extremely
interesting system.

I spend a good deal of time on it.

And this is it.

This is a stable.

Chris also you can grow it in and
this is kind of nice to see.

This is a dictator and so you have.

Five five passes.

This is not every day that you can
see a five fold sitting system and

you can grow it in a small grains but
also a very big system.

And says System aluminum paid imagine
is I mean a little inside history is

we are very interesting looking in
the magnets because it is stable.

India has Meg Mangan is so good it is it
can become a little bit magnetic So what

is a magnetic odor in a quasi periodic
system this was a really big question.

So I was handed a single crystal
of aluminum pellet a magnet

that big it was it was a bar which was.

Five centimeters I said well I
cannot put that in a magnet or

meter it can and I had to cut it.

So it was a big deal.

Having to cut a single question Chris I'm

going one of the first into in
the world that was made to go anyway.

So you can really grow them quite
a crystallized long range order.

I mean in credibly good good quality.

And I heard very recently.

So in one thousand two thousand and
nine in that question Chris and

I have been found in nature day
downed natural consequence.

And this was finding this mixture of
whatever spin a lot she hadn't only

been and the stock material consists
predominantly of copper aluminum but there

is a light granules which are aluminum
copper iron which is the exact composition

that I showed you before the face when so
it was fine in the middle right.

So nature does it for
you where you have to look for it.

There are not only five
fold symmetries but

they're also octagonal dodecagon all and
deck again all symmetries eight twelve and

ten fold symmetries which
is also interesting.

This is there are called two
dimensional quality crystals.

Of because there are two dimensions.

There are quite equal in planes.

And the planes are stacked periodically.

So this is also something
really interesting because at

the same composition with this
in kind of local environment.

You can look at what happens and
you quite plain and

look at what happened in
the parodic peerless it is so

unravel which is a rule of parody
city versus versus periodicity.

Structural models which have to have
oil on them and go quickly on up.

I said all phases.

Well I come back to the question
where are the atoms.

OK I showed you before that in
principle it is possible to have

a quality product ordered structure
which is ordered by non-periodic.

But no where are the atoms so
because of this.

Proximity of of crystalline
phases to crystalline phases

people have looked at the local order and
that also was ready in the literature.

There was what was called Frank Casper
phases so it's a the Mattel or G. of this.

This is them was very well known in and
there was a lot of work

that has been done in particular
by polling just to mention that

they consist of a cluster of cluster
invaded another I close a little

cluster imbedded in a different kind of
of five ultimately clusters a dick and

he drew an oval When it because
he to decode Dudek and he didn't.

And so
these blocks have been used to build

the do do do quite a christening
structure so that two broad classes.

One is to take a pen was like this
just the one I showed you before but

generalize a three dimension and
to put a cluster at each corner or

inside and work with that and the other
one is a generalization of what I

tried to explain to you two dimension
to one dimension projection.

So describe in your high dimensional
list of periodic structure projected

in an irrational manner to get a quality
player take system on the physical space

so pulled out before we went from
two dimension to one dimension so

if we want to go to and
up at is three dimensional space

our physical space we have to
start with a six to mention space.

So this is sixty.

Angel crystallography maybe not.

OK So this is the kind of of a local
environment that there was found

here you have a beautiful beautiful images
that you recognize the dodecahedron and

each of these colors correspond
to add to make decorations so

because you have to enter
a low as is becoming.

Rapidly very complicated so
this is one approach to local order and

this is the second approach so
bear with me.

If you do six the crystallography.

You have a six to mention
critique like this right.

That you have to defer it locally
on its stead of putting one

atom like you would do normally for
a three dimensional Crystal

instead of putting an atom you
put a three dimensional space.

OK and that you're going to cut with
a physical so dimensional space so so

people have done that.

So this is this is the did
the decoration of each

point of the light is that you're
going to cut irrationally.

The beauty of that is
simply very complicated but

once you have once you have the sixty
mention or system when you have

the irrational cut you have everything you
can find all the rational numbers you can

find all the local environment and
then you also have all the different

approximate phases all the crystal and
phases you can you change the slope and

you find all the DA to decrease and
faces so it's complicated but not so much.

So it's fun.

In and
example of searchers people are going on.

All is well but
always was not well for everybody so

Lynn is piling that I had mentioned
before for having worked on the mythology

of these phases for a very long
time opposed quazi periodicity and

cosmic was a with all his might and
all his power and whole his reputation.

And his is objection was very strong and

he proposed a lot of different models and
I.

All these model days of gold
crystals were proven wrong.

So what.

What did you want to say what he
said is that in fact the analysis

of of a quasi crystal
could be Dawn was a simple

units which is a christening unit cell but

you have to put a lot of atoms in it and
to try to repeat it.

So the number of atoms you had to
put in this particular example

where twelve thousand.

So you have to describe the position
of twelve thousand atoms to be able to

mimic Kweisi periodicity and
only too many question per the city for

those who care on the power difference
of a powder diffraction pattern.

He couldn't mimic to predict the the which
is light is he couldn't understand

the actual resolved five full diffraction
it was just a powdered fraction so

well describing with twelve
thousand items per unit so

it's not very efficient so
if there is only one thing about Kweisi

periodicity it's very efficient we're
describing extremely complex structure.

So this is also take home message or
take a message irrationality irrationality

you have on one case
question purely city or

periodicity and also the efficient
way of describing the structures.

So what it all lead to is eventually
a new definition of crystals and

this is what the International Union
of crystallography in the executive

committee of one nine hundred ninety
one has decided by Crystal When mean

any solid have in an essentially
discrete diffraction diagram.

And by a periodic Crystal remain
any crystal in which a three

dimensional light is pay or
the city can be considered to be absent.

So you cannot you cannot find
a subset of a periodic subset.

In other words.

Question periodicity is totally included
in the description of crystals so

now Crystal can be periodic or
non-periodic.

In a one minute and
the specific Probert is.

Well.

Yes and no.

There is something which is
extremely specific to quasi crystal

which is called phase ins.

So I go back to this little
example of of a pen was like this.

So this is a Pen.

It was like this if you
put an atom here right.

This atoms may move to this
other position right there.

And by doing so.

Defined the same exact or no shape.

But with the displacement of the room so

we had the big one in the two and
the two C. in and is just shifted.

So all from the outside nothing changes.

So these two configuration
have the same entropy.

Very If they are totally equivalent.

So this is one of the of
the argument about why of course you

Crystal a stable is view have many many
equivalent configuration to system.

All right so in the two dimension
description for those who care.

It just corresponds to
instead of in straight lines

just having a fluctuation of the line.

So for this point it was taken before is

replaced by would be
replaced by a point here.

So would it correspond to
moving the point right.

And so if you look in higher dimension
you have fluctuations of there's

a whole of zoology of these
of this collective meld and

it seems that these collective modes of
displacement of atoms have been seen.

So this is kind of this kind of
exciting day in the community of more

direct properties more
close to what we do phones.

Phonemes don't have to really something
very special you would expect to have

a higher right because they
are the did unit sell equivalent

units that would be extremely extremely
large she would have a lot of gaps but.

There are there are not clearly seen and
there are very subtle differences but

also doubt differences the versatile
between approximate and quasi Crystal.

Awful depreciation of electric of
electrons I mentioned it briefly before.

In theory for quite a pair this city.

You would have critical states.

So there are states which
are a have a drawing.

Neither extended nor
localized So this is a Pen.

It was like this and
this is a calculation.

If he she were and trembly.

So did the dark spot is the is
a weighted of the of the way functions.

So here you have higher density of
stated which are the dock area compared

to the white area.

So you see it looks kind of ordered but
not really ordered and

there are restaurants is of the of
the density of state from some some region

to some other regions and so in all this
system is very interesting but in reality.

Well I didn't find any difference
between a quality player decoder and

a quasi parity system but
there was something interesting

is that this system's at and
make it destroyed or not.

Yes OK This is them for

the aluminum reach faces
which are stable once again

the resistivity goes to
a middle into a transition.

Meaning that if you are really in a quasi
periodic order in this system and

even played in rhenium
then the resistivity.

Should stop and it's extremely
different from from aluminum So

this is this is well understood in
terms of local order in and local.

Of the wave function induced in
this group of atoms they were

talking about the I could say atoms and
the fact that there was no periodicity

prevents the a wave propagation on and
on the long range so.

Some of this is this is my story and
that may be a little bit too far off.

And I'm just going to stop here.

So in conclusion after more than

six thousand published article
the new subject what have we found.

It's a new civil order to structures
a series of new stable ordered structures.

Which lead to a read if
redefinition of Christophe's.

De atomic structure is
very well described.

I briefly mentioned it but
it's very well understood.

Now the new properties of quiet
Liberty City has been found but

in real systems of properties are pretty
much dominated by the local order which is

not too different from day as xmas faces
know the real question which is really

not answered at all but I'm not sure it
is answered who are intimately compound.

Chris or intermittently Come on either
is why our course a quest or stable.

How does an atom know that he has to
be there which is thousands of miles

from the other atom and what makes to do
a long long way and order correlation.

And besides it makes a nice
Sharp's night scarfs a nice tonic

in the nicer wallpaper and this is an
attempt to do a foot on the crystals also

with quasi crystal with a quasi
periodic array of of tubes.

So there is still a large press
of work on projects systems.

Thank you.

Thank you

thank you very much for
the great thought and the question.

Me a Magen is that if
you are near needs it.

When we transform into an phase which is
well known it is now it's made a stable.

So if you go back to an eye on
the contrary if you take a cup or Iran or

cadmium the Kenyan base systems they
are part of the it can be in phase their

gram so you can any of them until they
melt and it will stay quite a crystal.

So it

was no or

yes yes yes.

And so was small or

small if you are

only

there you go with all the work.

I think that most of these people have
published I mean they just published it

and they did their day went on I
mean that was kind of interesting.

Though the one who really pursued the
interesting indeed crystallography where.

People like mine and cyan and

others but not necessarily
the one who republished later.

On the way you're

Yes Yes I'm going.

Yes story.

Yes this is this is the end this is
actually the controversy was telling you

this is exactly the kind of so you can
you can you can always find a unit cell.

That is so large that experimentally
you wouldn't see the difference between

quality periodicity and
paid up that is absolutely true.

But then there was a question where do
you how do you describe a structure.

If you have to take a unit which is
pretty much the size of your sample it's

meaning yes.

So you have to find an efficient way
of describing they did the system and

you calculate the public
is that the system is well

yes.

You know what it would what can be said is
that the fact that you have a single grain

faceted signal Graeme's tells you
that at least the order goes.

To pretty much and to that to.

Centimeters big yes the yes
the biggest I had in my hands but

it was that which was it.

Is maintained over and over that it was
done it was done at Grantley great lengths

idea was No two Newton
deflections where you can really

look at big stars system in an X.
ray diffraction prism.

When that was that is the conclusion
that the people came up with.

Which diagram.

Yes yes.

So this this is this
is this is understood.

Pretty much what you have is that when
you have a middle existing THE So

you have you have a block wave propagating
into into your system when you have

a question periodicity you
don't have that you have two to

two to get rid of that in a real system.

What matters is is how there's
a way function propagates But

this is how does a word for probably gets
when you initiated and so what happen

when you have a local local order such as
such as the I can say he will order you

have a lot of of of of reflection within
this this this is local and Vironment.

And which leads to a low asleep
very slow diffusion of the of

the way Faget it because you
have a very slow diffusion

then you tend to localize
the system right.

And so when dissent becomes
localized then what happens if

if you want to increase when
you increase the temperature.

Then you how to do look is ation effect at
death comes into effect and so the does

the localization reduces the do of
the resistivity as you go to a higher and

higher temperature and

the more perfect the systemd the more
localized the system becomes.

We've had you will.

Well they didn't publish mashed
up much after his discovery

that is I mean Crystal he continued
to work on intimately compounds.

But he didn't pursue the.

The properties nor
did I know dear did actual refining

of destruction but
he's still very active in the field.

I mean on the metallurgy side.