Before I talk about projects, I'm going to spend
a lot of time talking about tryinggles and basic geometry.
So I can vince, glücklicher Dreuecker und Unglücklichkeit, Dreuecker.
Remember the problem is guined to be.
We have increplate distancing information. And worst on that,
but distancies are not excect, we know the two pro-pro-
tons are lesson for not half engstrongs from each other,
all lesson to in the half engstrongs,
plot.
We don't know exactly that
the distancies
First of all, consider how much exact information unate,
to define a structure exactly. True evans is a little bit board,
because we are in the nate one distance to define as system.
But if I've got three evans, I need three paces information.
But that's the way the world is. What a five good fore Adams,
you're going internet, six bits of information,
in three dementions, we're going internet,
the first to the fine these points,
the first 3 distancers and then a noble three,
to fix this going in the plane.
And then out of the plane. So basically your guys nate,
three times the number atoms distancers. And how biggest approaching,
well, you would say the number vatoms in a person is about,
it's a potents or noten order menue you bigger then the number residuce.
Ah, because they're a big residuce and little little projecties.
But that mains for 100 main our assets,
you would like to have a 1000 exact distancers.
And a spect froscoposed is probably not guying to recorde
that much information. And even if I didn't,
it's not going to bay exact information
So, what's a interest tours? Under the termined distancies
and under the termined sistems. So let's thinking terms
of trying goals and a few points in spice.
I have got here two too points in spice.
And I say to you the distant, these goals are pretent are fixed.
So to make to for the purposes a vagument,
as side that the coordinates of eye and ji are fixed in spice.
And then you come up to main say,
I have missioned, smore distancies,
for a point ki
A third point. And you say, point Kai,
is within 6 eng strongs of RTMI.
So DIKI is lesson 6 eng strongs. And you say,
you have missured some more information. So you know,
the point Kai is within 6 eng strongs of JI.
So, point Kai, has to be within this area of other lab.
But so we have missured and you know, two and always.
But all we have is a original egin of
possibilities here
If I'm as a more distances. So, I have,
I and Ky, then some other answer to here.
And then you say, my point, Ky,
has to be nears of this sky, nears of webcar.
Engineered with third point, that were stricts the possibility saven more.
So my point, Ky, would have to be with
in this rage live a here. If I have,
if I add, smore information, and I say,
point, Ky, Atomkai, has to be neard eye,
would in this circle joy within that circle,
within the purple circle and would in the arrange money,
then I restricted the number of the spice of possibility,
saven firther.
So I would say, my Atomkai has to be with
in the smooth rasion here.
So, each bit of distance information, you guear.
It's not going to absolutely to to term in the structure,
but the more information you have.
The more your restrict the set of possible answers.
So animal spect froscabusts are row ways charging trying to record
as much information is possible
Let's make the problem even more interesting.
As well as underdeternment, regions of spice. We are
also going to nate regions of spice that do not exist.
Imagine, I going have assumes I am joying.
I had might a measurement, which says that add on
Kai has to be within this circle here.
And has to be within the circle every here.
But that company, because I and Kai,
are a little bit to far a why from
each other
That mains that if I are try to solve this problem using classic,
geometrical mains,
quickly going to rich a point where I have to do in impossible operation,
usually this is something like take the square roots of a
negative number or get the inverse scenes or a coscious sainers
of a number that's other with bigger them one or small them miners one.
This this happen, so lootly,
because we are dealing with real elect experiment with experimental areas.
And there also physical raisens to expect this to happen
If I have I always have motions in sort of person,
inside projects. So I enjoy a wobbling around in space.
And that mains that sometimes ky sames to be need to why
and sometimes sames to be need to joy, simile tiny as way.
So much alemotions late to impossible distancers. Should you give
up? No. What it mains is one names are nethod,
which does not explored, when first to with impossible distancers?
And this is one of the raisens that classic geometry,
school geometry, will not work, when trying to regenerite coordnets.
And what to we have have much real digitant to we
have for a prochane of in risk residuce,
you might geometry five a 10 terms enerys distancers.
If you are a good spectroscopist and you have a friendly project.
Bractually we want to three times the number of assume, good distancers.
Which might be 30 to 60 times the number of residuce.
And this would engly bay, I'm not if
the distancers would really perfect. So what are mains is,
we are going to have to addin more distances information?
For changly, we can do that. We have lots
of distance information from camical to archa
Oh, talk about the use of chemical jarged in the moment,
but first of all comes the question can you
approached this kind of problem with school geometry.
And I'm fortunately school geometry is not guying to work.
So I think about something like the hostszene through,
so a hose sign role or the sign role,
but eat line, when you do when you do metsel, at school.
So I have trying goes here, I had some angles,
could are use these can'ts rules to work my why are
long from this distance to that distance
and sound
No, snock guide work. And there are two raisens. Firstly,
if we look at my little trying to here,
so I be say, you don't actually know the distance,
I all the distance base.
You just know a right just possible distancers. We would say that
this distance for example is between three and five engstrums.
But we don't know any more than that. So we're
going to be working with rigngers and not exact distancers.
The second things it turns out, if you worked to
put in distancers into these former up. And if you
are are a little bit ron, you get new merical
areas adding up as you, as you worked along a structure.
And there is new merical areas a kimi like really badly.
Gen you quickly rature situation, we are trying to do something like,
take the square root of a negative number.
Or take the inverse sign of one and half,
something as impossible
So, the mission that we are going to said,
I not on, he's. We're going together
all experimental data, all the energys and the Angles.
But then we come the trick, mixing us, with nown camustry.
And I will expoint all this in pineful data.
Then we're going to use a tech nake to make
the distancing information as touch as his possible.
And we're going to put are balloned on the distance between
everypair of points, or everypair of atoms with then approaching.
And then we're going to put a lower about on the
every distance with in a priority.
And then come true methods ware, how do you generate
quord nots? Before we get to generating quodnants,
I'm going to take a tent at 10 minute detor,
talking about toys and tryangles
So, let's start into the mentions. And let's consider are
friendly tryingle. So my friendly tryingle looks like this, just some
rather arbitory tryingle, but I know the three distances on this tryingles.
So living, 30,16, or living, 13 and 16.
In this situation, I know, everything about the system.
I could put, I and joyed down on the exaccess,
by strong this on this distance here, I could say,
that's Zero Zero into the mentions, that's 11 Zero.
And then I can use the like news standard triggonomatory to
work out quote not's für Kai.
And I can solve that analytically. So that is
a friendly tryinggle. What happens with under determined distancers?
This is really important. This is the bicyces of
all of the ideas of distance geometery.
So his my tryinggle, solve half. But of
got two distancers,
not three
Arly know, this distances all even. And this distance
is that time. But I haven't measure this distance here.
Also this is more typical of the situation with enema disia.
Where what trying goes are consistent with this starter?
My trying go could be spread out, like this.
Or could be rather squash that. The point is,
there is no unique solution. If I have
two distances, there's a range of possible solutions.
So let's form or is that. What is the range
a violable to the distance
Let's consider impossible data and the limits. If I have
my trying all with distancers eleven in 13,
I can stritch my trying growth.
And you can say that the maximum impossible distances all even plus 13.
So I find masure a distance of 25 here.
Something is wrong. It could be the 25,
could be the eleven or could be the thought time.
Let's go in the other direction and take my trying goal
and squash it down and make it long and thing
So that would be this trying all hill. If I'm massure addistance of one,
that cannot a car physically, because no matter half I'm move,
roll type as point around here,
the distance has to be as big as the difference,
between 13 and the weaven.
So, it's faily easy to tell,
when I look at triples and points, what's possible,
what's not possible.
And what will limits
going to bay