From Euclid to modern geometry: Do the angles of a triangle really add up to 180˚? (28 Feb 2012)

From Euclid to modern geometry: Do the angles of a triangle really add up to 180˚? (28 Feb 2012)


>>Mark Ronan: Well thank
you very much Patricia. That’s very kind of you. The last time I gave this
talk, I’m going to have to move fairly fast
because we want to leave a little
time for questions. But a journalist;
well a journalist in the audience said he learned
more about ancient history than he did about mathematics,
because I’m going to start with Alexander the Great. And I don’t know whether
you know about him, I mean. When I went to school we never
learned that sort of history. But a remarkable guy who was
very keen on scholarly work. I mean he had Aristotle
as his tutor and when he was 20 years old
his father was assassinated; his father the King. And he had to put down
revolts and things, and yet within 2 years he
had got together an army of different Greek states
and crossed Asia Minor; over into Asia Minor,
to beat the Persians. Remember the Persians had
earlier sort of attacked Greece. The battle of Thermopylae
is a famous one which will be 2,500 years
anniversary in 2021. And that’s the place where 300
Spartans died to the last man, to defend Europe from the
invading armies of the Persians. The Persians had
a fantastic empire and Alexander conquered
the whole empire. Now you’ll see why I’m
saying this in a minute. See first of all he was;
he became King of Macedon. But very soon he became King
of Egypt because Egypt was part of the Persian Empire. And even before he conquered
the whole Persian Empire, he took Egypt and there are
many interesting stories involved there. But eventually a
couple years later, he took the whole Persian Empire and became the Great
King of Persia. And he died a young man. I mean, I think what
was his age? 32 when he died. After his death, his generals
divided the empire between them and the person who took over
Egypt was his general Ptolemy. Later known as Ptolemy Soter. The reason he is known as
Ptolemy Soter; he’s also known as Ptolemy the 1st, because
there were lots of Ptolemys; 1, 2, 3, 4, 5, 6, you know. The Rosetta Stone is Ptolemy
the 5th and those who have been to the opera of Giulio Cesare
will meet Ptolemy the 13th. You know, there are
lots of his Ptolemys. And eventually the dynasty
collapsed with Cleopatra, but that’s another story. Ptolemy Soter was called
Savior, Ptolemy Savior, because he saved the island
of Rhodes from an invasion by the King of Cypress and
they gave him the title Savior. And they built a huge monument
there called the Colossus of Rhodes, which exists no more. One of the great 7
Wonders of the World even after it had collapsed,
which happened fairly shortly after it was built actually. It was still one of the
great Wonders of the World. Ptolemy; we think
it’s Ptolemy the 1st, who founded the great Library of
Alexandria where Euclid worked. We don’t know; we
know about Ptolemy. We know about Alexander. We know about all these sort of famous generals
and kings and so on. We know absolutely
nothing about Euclid. I mean we don’t know
where he was born. We don’t know when he was born. We don’t know the
circumstances of his death. We don’t know when he died. There’s a famous quote here that
I quote; when the King asked him if he could you know, give
a sort of neat explanation of his geometry, he said there
are no royal roads to geometry. And you know, the idea of royal
roads existed at that time. In Persia there were royal
roads that were only available to the King and his
entourage, so on. We have no idea who
this guy was. He may have been
quite a young man and he may simply
have disappeared or maybe he didn’t exist at all. I think he probably did exist. Anyway, he wrote this
thing called Elements, Euclid’s Elements,
and they were; let us move on to those;
composed in 13 books. He; this is really
extraordinary stuff because it’s the greatest
textbook of all time. What he did was he sort
of layed out everything. He layed out definitions. He layed out postulates, 5 important axioms
for plane geometry. And it was written using;
if you’ve every done math; any sort of serious mathematics
and people always used to do this in school
at one time. Doesn’t happen anymore;
they all used to. They used to do Euclidian
geometry. And they would prove Lemmas
and Theorems, you know? Wonderful; wonderful for
the brain and development of our brains to do that. I’m very sad it doesn’t;
isn’t done as much as it is; as it was, today. But what were these
5 postulates? We’ll come to that in a minute but the Elements
were transmitted; well they got translated
into Arabic. Here’s how they came to Europe. They got translated into Arabic. A chap called Adelard
of Bath, went to Spain, got hold of a copy and then
translated it into Latin. And this Latin translation;
I’ve mentioned Roger Bacon who was one of the great minds of this country in
the 13th century. Not to be confused
with Francis Bacon. The first translation from Greek
directly into Latin was in 1505 and what I’ve pictured here is
the English edition in 1570, made by Henry Billingsley, who
became Lord Mayor of London. Died in 1606. It contains; it’s difficult
to read this writing actually. But it says that there
is a preference by M-I-D. M-I-D is John D., Dr. John D.,
who was pretty much I think, in charge of the Elizabethan
Secret Service and decoding of messages, which of course led
to the execution of Mary Queen of Scots, as many of
you who have studied that period of history
will know. And the English National
Opera; they were kind enough to mention the fact that
I’m interested in opera; are putting on an opera
called Dr. D. pretty soon. So I’m very much
looking forward to that. It’s a new opera. Now, let us go to
the five postulates because that’s really
what this talk is about. You see I’ve got the
first four pretty simple. There’s a line between
any 2 points. You can extend a line
in both directions. If you’ve got 2 points you
can use one as the center of a circle and the other
one to measure the radius, and then you can draw a circle with that center
and that radius. All right angles are equal. Right angle means 90 degrees in our usual terminology
acquired from the Babylonians. But just call them right angles. You know you take
a straight line. You put a perpendicular there. This angle is the same as that
angle and they’re 90 degrees. But then there’s
the Fifth axiom. You can read it here and you’ll
probably read it and say, I’m not quite sure
I’ve got that, so why don’t I give
you a picture. That’s the picture. So these lines L and M, are sort
of so to speak not parallel. You see the sum of these
2 angles, alpha and beta, is less than 180 degrees. And in that circumstance
the axiom is that these 2 lines
L and M will meet. That’s an axiom. And he; when Euclid wrote his
Elements, he tried to prove as much as he could
without using that axiom. But eventually he had to;
you know he went so far with this theorem, so
eventually he needed this axiom. And many people since then
wanted to get rid of this axiom. They felt it wasn’t
really necessary. I’m only going to mention
a few of these people. We’ll go first of all to
a chap called Al-Haytham. He’s sometimes pronounced
Al-Haytham, also known as Al-Basri
because he came from Basra. Died in Cairo. Famous work on optics
which again, you know, Roger Bacon quoted
Al-Haytham’s work on optics. It was brilliant stuff. At one time, and this is
complete digression, you know. But at one time, people used
to believe that the eye sort of sent out rays that; oh
it’s all very complicated, what they used to think. But this guy had the right idea
that it was just light coming in straight lines and
entering our eyes. You think, well that’s
pretty obvious but you know, it hadn’t been obvious
to an awful lot of people and he did marvelous work
on refraction and so on. Now he also worked on
the parallel postulate. And tried to prove it. But here’s another
famous man who said, well Al-Haytham just got this
wrong, and that was Omar Khyyam. Omar Khyyam is often
remembered as a poet. The Rubaiyat of Omar Khayyam. But in fact he was even better as a mathematician
and an astronomer. And he criticized
Al-Haytham’s proof. I’ll tell you what
Al-Haytham’s proof was. I’ll just do it. He took a straight line and he
took a little perpendicular line of a certain length and he
moved that perpendicular line, keeping it perpendicular to
the straight line, like that. And the point at the top of
that would draw another line. And this would obviously
be you know, so to speak, parallel to the first
line and you just put. But Omar Khayyam said
this is nonsense. This really is nonsense. And it is nonsense actually. The idea of having
points move is okay. I remember my son was doing a
plane geometry course in school. He explained a proof to me
where points moved along here and another point moved
along there, and so on and his teachers [inaudible]. But he insisted it was right. I examined this proof
in some detail and said if you just rephrase it, and he
did; went back to his teacher and his teacher said,
that’s good. Which shows he had
a good teacher. Some teachers wouldn’t do that. But, excellent teacher. Anyway, to jump forward
to the 17th century; well actually 18th century too. An Italian mathematician
called Girolamo Saccheri, thought he had proved
Euclid’s Fifth axiom. In other words, he
thought he could prove it from the other 4
more basic axioms. And he split into 3
cases where triangles; the angles of a triangle
added to less or more or exactly 180 degrees. And then he had to
eliminate the first 2 cases and he published this in
a book which came out; if you read the date on here. I think there is a date on
it somewhere, isn’t there? Yes, right down there
at the bottom, you see? In roman numerals 1733. That was the year he died. Bit of a shame really,
because it was wrong. But he; never the less these
people were very clever people. You know, even they
got it wrong. So I’m going to give
you a false proof; I’ll tell you it’s
false before we start. You know, you can try and figure out what’s wrong
with this if you can. First of all, the first step is
to show that any 3 points lie on a straight line or a circle. So let me just run
through that very quickly. I’m just taking 2 of the points
then I’m drawing a circle through those 2 points and
the red dot is the center of the circle. Okay? Now we want
to make the circle; we can make the circle
smaller, we can make it bigger. Here’s a case of
making it smaller. And here is a case
of making it bigger. It just needs to get
a little bit bigger and it will hit the third point. In this animation actually, it does hit the third
point curiously enough. When I actually do it in the
talk it doesn’t, but never mind. I think it’s rather
good actually. It doesn’t quite get there. All right. So if you’re holding onto your
seats, we’re going to really go into this proof that 2 lines
are going to intersect. Now let me just remind you
of Euclid’s Fifth axiom. You’ve got these lines L
and M. The angles between; the angles that they make with
the third line add up to less than 180 and the idea
is to prove that L and M must intersect, okay? So I’m going to do that. And do it in front of your
eyes and then we’ll move on to some more history. So if you get fed up with this
mathematics, it’s okay, really. We’ll move forward, don’t worry. Right. So let’s take
these lines L and M and what I’m drawing here
is 2 lines; a line cutting L at right angles and a line
cutting M at right angles, yeah? And since L and M don’t have a
third line right angles to both of them, these 2 line segments
must intersect somewhere, yeah? So you just let them intersect
at a point Q. And you take P to be the point opposite
Q, across the line L, and R is the point opposite
Q, across the Line M. And that length of QR is
not necessarily the same as the length of QP. I know it looks like
that in this picture, but it doesn’t have to be. That’s completely
irrelevant to the proof. Okay. Now, let us move on. We’re trying to get
a circle that goes through those 3 points
P,Q and R. First of all let’s think
of it this way. You take a circle through P
and Q, which has its center of course necessarily on the
line L. You take a circle through Q and R, which has
its center on the line M. And you make these
circles bigger until eventually you will get a
circle that’s bigger than either of them and it will go through
P, Q and R. So the center; as you make these circles
bigger, the center will move; the center of this
one, of the top one, will move down on the
line L. The center of the bottom one will
move along the line M. And when you’ve got
a single circle, it goes through the 3 points
P, Q and R. The center of the bottom one and the center
of the top one will coincide because you’ll have
the same circle. Now in this picture
they don’t coincide because I never quite get that
far, because I’m you know. But in principle you’ve got a
circle; you will get a circle through P, Q and R and
its center will lie on L and it will also lie
on M. And therefore L and M have a point in common. Proof. Proof of Euclid’s
Fifth axiom. Now you can wonder
what’s wrong with that and I’ll ask you
at the end, okay? But we’ll move on, because
a great deal more work went into this question and one
particular guy who was a very, very, very smart guy,
Johann Heinrich Lambert. He proved; anybody who has
done circles, you know the area of a circle is pi r
squared and so on. He showed that pi cannot be a;>>Okay are we back again? Yes. Sorry about that. I must have touched the button. He proved that pi can never
be written as a fraction. Many of you who went to school,
particularly in the old days when you know, we used to
deal with fractions a lot because we didn’t
have calculators; used 22/7 as a good
approximation for pi. There are better approximations. 355/113. You try
it on a calculator. Pretty good. Okay, there’s a way of
getting these approximations, we won’t go into that. Anyway, Lambert; now I’ve said
here he showed that the area of a triangle in the
hyperbolic plane; he imagined that you had a plane that didn’t satisfy
Euclid’s Fifth axiom. And then he showed that the
angles of the triangle would add up to less than 180 degrees. We’ll come back to that later. So like Moses, he
saw the Promised Land but never entered it. He died in 1777. And then, I just want to mention
another chap, Schweikart, who was a friend of Gauss. Now we’ll talk about Gauss in
a minute because he was, well, arguably the greatest
mathematician since Newton, at any rate. And one of the 3 greatest
of all time, as I often say. Actually somebody once said
to me; this is a funny story. This is in America and this chap
was a lawyer and he said to me, tell me who are the 3 greatest
mathematicians of all time? You know, question
out of the blue. I said, well, I don’t know. I suppose, Archimedes,
Newton and Gauss. He said, why do you guys
always give the same answer to that question? And, so. Anyway. I think somebody wrote a book and he had quite
fairly strong opinions and that was his opinion. But Archimedes and Newton were
fantastic, they really were. Fantastic. But so was Gauss. Anyway, he wrote a letter to
Gauss saying this being assumed, or whatever, you know. He made some assumptions
about the existence of a plane that wasn’t Euclidian. The sum of the angles in a triangle is less
than 180 degrees. It becomes less as the area of the triangle becomes
greater and etcetera. Let me just talk
about Gauss now. There’s a nice, lovely
picture of him. And there is the statue
to Gauss in Braunschweig. I’ve been there. I’ve seen the statue. And I’ve been around the back of
the statue and if you go in back of the statue you find
there’s a 17 point star because when Gauss
was 19, he proved; well, he didn’t just prove. He showed how you could
construct a 17-gon, a regular 17-gon. A regular polygon with 17 sides. You’ve maybe; in school you
might have constructed a regular triangle. You can construct a regular
pentagon with 5 sides. You can’t construct a
regular heptagon with 7 sides. Let’s talk about prime numbers
there because; but anyway, but you can do a regular 17-gon. The reason is that 17
is a power of 2, plus 1. It’s a prime number;
power of 2 plus 1. There aren’t many of those. But anyway, I digress. This man who one,
he’s just amazed at what Gauss did in his life. But any rate, he discovered
the hyperbolic plane. But he didn’t publish
it because he knew that it wouldn’t really
be accepted, you know, and they’d be in an
awful lot of trouble, see, and everything else. So he intended that these papers of his should be
published after his death. So who did discover
the hyperbolic plane? Well let’s go to
Farkas Bolyai who was; this is a Hungarian
mathematician, Bolyai. He was a friend of Gauss’. They went to University
together. And he was, unfortunately,
he was very poorly paid and he wrote and published
dramas while he worked on his mathematical
masterpieces. He tried to dissuade his son
from ever going into the study of Euclid’s Fifth
axiom and he said, detest it as lewd intercourse. It can deprive you of all your
leisure, your health, your rest and the whole happiness
of your life. But Janos Bolyai persisted
in working on the Fifth; Farkas Bolyai; there are
lots of lovely quotes. I mean he was; this is a man
who wrote plays and so on. I mean he was perfectly
capable of writing and writing very elegantly. And I’m not sure I have
the particular quote here but he said, I’ve been
on this journey before and I’ve always come back with
a broken mast and torn sail. You know, so don’t go there. But Janos Bolyai;
this picture of him is on a Romanian postage stamp. We don’t have a picture
of Bolyai; Janos Bolyai. I know it seems incredible
but we don’t. We have a picture of Evariste
Galois, who died much younger, because a friend of his
sketched his portrait in school. But nobody ever gave; this is
perhaps an invented picture. He wasn’t Romanian,
he was Hungarian, but the town he was born
in is now part of Romania. And he said, I am resolved to
publish a work on parallels as soon as I can put it
in order, complete it and the opportunity arise. I have not yet made
good the discovery but the path I’ve followed
etcetera, etcetera. And his father said, well this
is brilliant stuff actually. We must publish it immediately. And since his father
was publishing a book, he could easily slot his
son’s work into an appendix of 24 pages at the
end of the book. Which he did, and one
historian of science; I can’t remember his name. Very famous historian
of science. Anyway, said he considered this as the 24 greatest
pages every written. It was, you know, it was
really incredible stuff and he never published again. Can you believe it? You see it’s really; it’s really
sad what happened with this. I mean you’d expect that this
man did this incredible work, he would be offered chairs
here and there and everywhere. His father sent the thing to
Gauss but Gauss was a slightly; well, he was not an
outgoing, enthusiastic, what a brilliant thing
this is, kind of chap. What he wrote back
was, and this was read to Janos, who took it badly. If I commenced by saying that I
am unable to praise this work, you would certainly be
surprised for a moment. This he wrote back
to Farkas Bolyai. Indeed the whole contents
of the work; the path taken by your son; the result
to which he’s led, coincide almost entirely
with my meditations which have occupied
my mind partly for the last 30 or 35 years. And so on and so forth. So Janos just felt; you
know, I did all this stuff and it’s already been done? I mean he was really despondent
and he just; he never recovered. Very, very sad. But you see all this
was in the air. I mentioned Schweikart, I
mentioned Lambert and so on. This was sort of in the air and
Nikolai Lobachevsky, a Russian, also discovered this hyperbolic
geometry independently. And he had no better fortune. I mean he became head
of his university and they got rid of him. I mean it is quite incredible. It’s almost as if there was
a curse on this you know? It’s like the assassination
of Kennedy’s. It’s a little cursed thing. You can’t win. You just sort of mustn’t
do anything here at all. Anyway, but I don’t
believe in curses, you know. I was in Egypt recently and
they tell you the big story of how Howard Carter
discovered Tutankhamun’s tomb and all the workmen
who had worked on it all disappeared and so on. I don’t really buy this. Anyway, let me go on, go
back to Saccheri’s Triangles. I remember once and I’m
saying this to people who may not know these
things, mathematics students. I didn’t know this stuff
when I was a student. Not even as a graduate student. A very famous mathematician
came to Chicago and gave a talk on hyperbolic plane
and said that the area of a triangle was proportional
to 180 degrees minus the sum of the angles of the triangle. I was absolutely mind boggled. I thought it was
mind boggling, yeah. Absolutely incredible. I mean even if you send
those corners of the triangle out to infinity, you’ll still
only get the triangle being a certain size. That’s why we tend to
draw them with curved, with concave lines like that. Okay. So what’s the difference
between the Euclidian plane and the hyperbolic plane? Well in the Euclidian
plane, if you have a ; I’ve sort of condensed
the whole Euclidian plane into a disc here. And I’ve taken a point on
the boundary of the disc; in other words a
point out at infinity. And I’ve taken all the
lines that go to that point at infinity so if
that’s the North Pole, then these are all the
lines that are going north. The direction is north and
they all end up at the sort of North Pole at infinity. And at the other end
they end up so to speak at the South Pole at infinity. Hyperbolic plane’s
quite different. There are lines emanating from
the same point and they end up; these are straight lines. I know they look curved on this. Straight lines and they
end up all over the place. Anywhere around the boundary. This is called the Poincare Disc
Model of the hyperbolic plane and I’m just sort of really
mentioning this perhaps for mathematic students. It’s a very interesting
model of the hyperbolic plane where the lines are;
oh, that’s funny. It didn’t; yeah. The lines, if they go
through the center, the straight lines look
like straight lines. If they don’t go
through the center, they look like curved lines
but they are curved lines which are arcs of circles that
meet the boundary at 90 degrees. Here are a few examples. This is the hyper; this is
a wonderful representation of the hyperbolic plane. Whenever lines cross, the angle at which they cross is
always exactly the same as the angle they should cross
at in the hyperbolic plane. It’s a very useful, very useful
thing, this Poincare Disc Model. But I must finish now. There’s another model called
the Upper Half Plane Model. I think I’ll just
skip that actually. What goes on in the
hyperbolic plane? The reason the lines don’t meet
is here are the 2 lines L and M, crossing a third line at
angles alpha and beta, which add up to less than 180. But you see they sort
of drift off and go off to infinity completely
different ways. So what was wrong with my proof? Anybody know? Yeah, go ahead.>>[Inaudible response]>>Mark: Yeah. Actually, well done. Congratulations. I think you’ve hit the
nail on the head there. It is exactly that point
that any three lines lie in a straight line or circle. And what I did was gave you a
very sort of hand wavy proof. Well you see make
the circle bigger and bigger and bigger, yeah? And there’s a limit you see. What happens if I take the model of the hyperbolic plane
given us by Poincare. The Poincare Disc Model. Here are the 3 points in blue. There’s a line going through
2 of them and you can; I haven’t got all
the circles on that because the picture
will just get too messy. I’ve got the biggest
circle you can get that goes through those 2 points, yeah? The 2 lower points there. The biggest circle goes
right out to infinity. The center is out in
infinity believe it or not. And the circle so to speak
goes through infinity as well. That’s the biggest, and it
never hits that third point. So that was a hand wavy proof but you know people
did this kind of thing in geometry at one time. And they would get to
some conclusion like that and then they would use that to
sort of leapfrog onto showing that the Fifth axiom was a
consequence of the other four. And the leapfrogging was fine. What was wrong was their,
their starting position that they managed to get to. But it was just nonsense. So I’m going to; I’m going
to stop there actually because I’ve got through
my talk with enough time for questions I think. So we have about 10
minutes for questions if anybody would like to.>>Right. Do we have
any questions? Yes, we have one.>>[Inaudible]>>Mark: Yes.>>[Inaudible]>>Mark: Yes.>>[Inaudible]. You said it’s only coincidence that that point is
in the middle. But it seems to me how can
it avoid being in the middle.>>Mark: Oh, well let’s just
run back to that very quickly. And we’ll see what you mean. You mean, you mean
this thing, yeah?>>Yes.>>Mark: That picture. Q; well, what I said
was that the length of the line PQ is not
necessarily the same as the length of the line QR. You’ve got one line. You’ve got another line. They will hit, they will meet. Okay? There’s no question. I mean that point Q; if
that point Q could be; if you shifted; if you
shifted this line over slightly to the right, then Q would be
closer to the line L than it is to the line M. But when the
chap who drew these pictures for me drew it, he did actually
make them I think pretty much the same length. But that’s just, you know, yeah. But I was very grateful to
him for drawing the picture because I can’t draw these
pictures on the computer so. Yeah?>>Yeah, I came in slightly late so I might have missed
something, but;>>Mark: Yeah. You missed Alexander the Great. That’s what you missed.>>All right, yeah.>>Mark: Yeah. I was almost going to go back
to that and get a question on it, but go ahead, yeah.>>Nothing to do with
Alexander the Great. I sort of get the feeling
that somehow this is sort of cheesing a bit because;>>Mark: Yeah.>>Surely when Euclid was doing
his geometry he would have assumed that you were talking
about what we call a plane which is a flat plane.>>Mark: Exactly.>>Is there nothing in his
assumptions that sort of relate to that flatness or
is it just sort of;>>Yes, it’s the Fifth axiom. It really is the Fifth axiom.>>Right.>>And it’s that Fifth
axiom which he, I mean, I think Euclid was
a brilliant guy. I mean he just you know got
it down to what he needed, nothing more than he needed. The first 4 axioms are
also things that you know, you’ve got to start somewhere. And the Fifth axiom, you
know, he really needed it. So that’s what makes it fact. And there’s nothing wrong
with the Fifth axiom. You know, it’s very
useful for architects and all sorts of people. And of course now days I don’t
know if you keep in touch with what goes on in physics. I mean I try to, but you know
they talk about dark matter and expanding universes
and all of that stuff. What would happen if you made
a triangle in the universe with the points millions
of light years apart? Would the angles add up
to about 180 degrees? Or wouldn’t they? And the latest thing I read,
probably in the Economist or something, so you
know, [inaudible]. But it said that
actually they would. That’s the latest. So I don’t know. We don’t know, really,
for the universe. But of course on the
earth it’s not true because the earth is curved. But in the ideal flat plane
which Euclid was dealing with, yeah, it’s absolutely true. And yes, sorry. Does that answer your question? Yeah. Okay.>>Any more questions?>>Mark: We should
go back to Alexander.>>[Inaudible]>>Mark: Alexander the Great. Yes?>>So was this; was
the hyperbolic plane; was that the time, was that
the moment that people realized that you could have a
geometry that wasn’t;>>Mark: Yes.>>What we think
of as kind of our;>>Mark: Yes.>>Kind of normal
spatial geometry. So they didn’t have the idea
before they had the example of the;>>Mark: That’s right. That’s a very good point. You needed the example. I mean otherwise, the
whole thing is a bit sort of airy fairy. If this happens then that
happens and the angles of a triangle will be
less than 180 degrees and so on, but can this happen? Because if it can’t,
then forget it. So you needed an
example; this example. It had to be constructed
and that was the great work that Janos Bolyai did. Of course once that’s been done; once that the ice
has been broken, then people can start discussing
other sorts of geometry where things aren’t flat, yeah? They can do not just
2 dimensional geometry but 3 dimensional geometry and
a man called Bernhard Riemann in Germany devised sort of
multi dimensional geometry where you could have curvature that would be intrinsic
to the geometry. It wouldn’t be curved in
some bigger flat space. It would be intrinsic
in your geometry. And that of course would be
vitally important to Einstein when he did his Theory
of Relativity. Sorry, I’ve moved away from here
and I’m not supposed to, I know. When he did his Theory
of Relativity because in spatial
relativity, space-time is flat and there’s a sort of glitch
in spatial; not the right time to go into it but you know, you have to take
account of acceleration. You have to take
account of gravity. And once he did that, he
needed to curve space-time and it was almost Riemannian
geometry that got used; these ideas that
got used by Einstein in the general Theory
of Relativity. So this was absolutely vital
for the development not just of mathematics, but of science. Of physics.>>Thank you.>>Mark: Amazing.>>Any more questions? Yes, we have one over here.>>Mark: Yeah?>>At what point in this version
was replaced by the new one? [Inaudible]>>Mark: Which version?>>This version of Euclid’s.>>Mark: No, we still have
Euclidian geometry, okay?>>Yeah.>>Mark: We have
Euclidian geometry. It’s very, very useful. Students in school
learn it and if you go to a good school they prove
theorems and so on and so forth. Hyperbolic geometry people
don’t tend to learn about. It’s just a bit more
difficult to deal with. But it is extremely
useful in mathematics. And operating in the hyperbolic
plane has a completely different symmetry from operating
in the Euclidian plane. And it gets used in the
study of symmetry in a way that I won’t go into here but;>>Yeah.>>Mark: Both of
them are useful.>>I understand that but
usually the Fifth axiom is; has got a new version that there
is only one parallel outside;>>Mark: Well that’s Euclidian
geometry where the plane is flat and these lines don’t meet and you’ve just got a
unique parallel line. There’s another way
of stating that. It’s called Playfair’s axiom. If you’ve got a line,
it’s Playfair’s axiom. But it’s equivalent to
Euclid’s Fifth axiom.>>Yes.>>Mark: If you’ve got a line
and you’ve got a point not on that line, there is a unique
line through that other point that doesn’t meet this line.>>Yeah, that was my question.>>Mark: Yeah, yeah.>>And these 2 that prove
to be equivalent and that;>>Mark: Ah, and
that was Playfair. I guess Playfair proved that this was equivalent
to [inaudible]. There are other ways, I mean. That business that I said about
3 points lie on a straight line or a circle; if you took that
as an axiom, it’s equivalent to Euclid’s Fifth axiom. You can prove one; you know, you
can go both ways with the proof.>>Well, thank you very much. I think we haven’t any
more time for questions.>>Mark: Oh wait a minute. We’ve got one more here. Go on, yeah.>>If it’s very, very quick
and a very quick answer.>>[Inaudible question]>>Yes, I don’t know. Don’t ask me about that. That’s a very tricky one. I would have thought the
universe sort of curves around on itself
and just expands like a balloon expands
and I agree. But it could just be
sort of generally flat and expanding, so I don’t know. I think we don’t actually know
the answer to that question.>>Well, a nice end
to the lecture. A bit of uncertainty which
inhabits mathematics.>>Mark: Yes.>>And thank you very much Mark. [applause]>>Mark: Thank you.

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