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Thank you all for coming to this this first talk of the evening.
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It falls to me to address the mathematical aspects of the things that are in the exhibition, which I hope you get to have a look at afterwards.
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So as you look at the exhibition, you see various things relating to geometry over many centuries.
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In the coming up to the 19th century where you'll see reference to Ben Freeman with a, um,
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what ended up becoming a very much more abstract approach to geometry in the 19th century.
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So for example, the, the move to working in more dimensions than three.
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So the idea that you can,
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you can do geometry in this way that way you don't quite have the same sort of concrete way of visualising things as you do in lower dimensions.
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And in fact, this this moved to a more abstract geometry is part of a wider abstraction that happened in mathematics in the 19th century.
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And so what I thought I'd do was, was pick up a different aspect of this abstraction.
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So not necessarily to to do with the issue of dimension, but it is something that links to several things that are in the exhibition.
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So I thought it might be a nice thing to do. So in order to introduce this topic, I have to go back 2000 years to look at Euclid of Alexandria.
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Now, some of you may immediately spot that this is not Alexandria.
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This is Plato's Academy in Athens. Picture on the on the wall of one of the Raphael rooms in the Vatican, but we'll glossed over that detail.
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The reason it's here is that it contains representations of various scholars from antiquity.
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And so at the bottom right hand corner here we have Euclid.
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Probably some people argue that it's actually Archimedes.
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But again, we'll gloss over that detail. And so I suspect Euclid is a name that is known to to most people here.
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So around 250 B.C., he collated a collection of of geometrical knowledge, such as it was at the time,
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probably gathering together material that was known, possibly adding some new material of his own.
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And this is concerned mostly with the very much the sort of day to day geometry,
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a sort of plain flat geometry line circles and so on, the kind of thing that he's probably doing on the slate there.
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And so this this text in various versions became the basis for the teaching of geometry for 2000 years, essentially.
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And it's a very interesting story. And I should perhaps mention that a colleague of mine, Benjamin Walter,
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is will shortly be bringing out a book, traces this, uh, this descent because the elements has been.
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Sorry, I'm getting ahead of myself. So Euclid's elements,
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the text that I'm talking about has been transmitted and translated and corrupted and edited and generally abused down the centuries.
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It exists in in umpteen different versions. The one I have on the slide here is the oldest surviving complete version, which dates from 1888.
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And it's in the copy. And so this should be in the the exhibition, so you'll be able to have a look at it afterwards.
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There are also other versions of Euclid in the exhibition, for example, there is an Arabic one then.
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And so I've said that Euclid's elements was the basis for 2000 years of geometrical teaching.
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It's also the basis for a lot of the way that we do mathematics in terms of structure, because it begins with basic definitions.
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So he defines what he means by a point and by a line and by a circle and so on.
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And then he sets up his basic postulates, the things that you can do with these concepts that he's defined.
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I'll come to those in a moment. And then once he's got this basic setup established, he then works very systematically, step by step,
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building up the geometry through the various books of the elements, building up to very much more complicated geometrical constructions.
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And so the page that I have on the slide here, you may just be able to make out the diagram here.
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Oops. Which you might recognise as being a representation of Pythagoras theorem.
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And so this is the 47th proposition of the first book of Euclid.
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So he's worked systematically through a whole series of of earlier results and got to this one at number 47.
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And this depends on many of the things that come before.
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There's a very logical and rigorous structure to to the book, which is then replicated in other branches of mathematics down the centuries.
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And the story that I want to tell very briefly is the story of a certain amount of disquiet that was felt
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about one of Euclid's postulates that the postulates were supposed to be basic assumptions about geometry.
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And there's one that is somehow less basic, less self-evident than the others.
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And it's gets cause some worry for mathematicians down the centuries.
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So in order to to access the postulates, I'm going to do it through this, which is also in the exhibition.
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This is Henry Billings, these English translation of Euclid, 1570.
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This was the first English translation, as he proudly tells us on the the title page there.
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And so I've just pulled the postulates out of Billings, these versions.
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So we've got a sort of something reasonably authentic to have a look at.
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So the first postulate there are five in total. So the first one from any points to any points to draw a right line, meaning straight line.
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So if we got two points marked by crosses, we can draw a straight line between them.
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Simple as that. Very, very, very basic, very constructive.
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He's just telling you what you can do geometrically. The second postulate to produce a right line finite, straight, fourth, continually.
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Well, we have to get past some Elizabethan English here.
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But basically he says that if you've got the straight line, then you can extend it as far as you like in both directions.
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So, again, very straightforward, very constructive. Similarly, the third one upon any centre and at any distance to describe a circle.
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So there's a point. There's a distance from the circle as far as distance from the point.
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We can draw a circle centred at that point with that distance as its radius.
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The fourth postulate is slightly odd compared to the other ones.
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It basically says that all right angles are equal, which you would I suppose you would want it to be true.
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So this is a right angle and this is also a right angle.
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And this is the right angle. This is the right angle. They're all the same no matter where we are.
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A right angle means is the same thing. And he's put that in there explicitly, I think,
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because a lot of what he goes on to do depends on comparing one angle to another, on comparing an angle to a right angle.
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So in a sense, he wants these this to mean the same thing.
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Wherever you are in this space that he's he's creating. Otherwise it wouldn't make sense.
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So that's why that's in there. And then we get to the fifth postulate, also known as the parallel postulate.
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And perhaps without even reading it, alarm bells are starting to go off now because this looks a good deal more involved than the previous ones.
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Certainly, there's certainly a lot more going on. So when a right line falling upon two right lines doth make on one and the same selfsame side,
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the two inward excuse me are two inward angles, less than two right angles.
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Then shall these two right lines being produced at length concur on that part in which are the two angles, less than two right angles.
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So I think that needs a bit of unpicking. He's saying if you have two straight lines and a third line falls across these two lines,
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and then clearly on the right hand side of the line, there is an angle here and an angle here that are each less than a right angle.
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So together they're less than two right angles.
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And so on that side of the third line, if we were to extend the first two lines sufficiently far, they would meet at some point.
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Okay. And this is Billings, this version of the diagram. So his lines meet at the point date.
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And so something that we are sort of invited to think about.
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Euclid doesn't quite do this explicitly, but we're invited to think about what would happen if those lines were not angled quite so sharply.
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What if they were angled outwards a bit more? Well, the point at which these two lines meet would get further and further over to the right.
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Until we get to the case where the first two lines are parallel, in which case the two lines meet metres infinity,
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which I suppose is a mathematician's way of saying they never meet. But we say that they metre infinity.
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And so this postulate is very different from the other ones.
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The other ones are very much about things that you can do in front of you, that you can draw a line, you can draw a circle.
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This one is hinting towards things that are going on at infinity.
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So this is quite a different thing and. You might ask, what is this a basic postulate?
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If you were to write down a list of basic facts about geometry, would you necessarily think of this one?
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It's not necessarily that obvious a thing to think about.
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And so from very early on, there's a certain amount of disquiet about this postulate.
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Even from Euclid, it seems because he doesn't use this until the 29th proposition of book, one of the elements.
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He could have used it earlier to simplify certain things, but he seems to have put it off and put it off until he absolutely had to use it.
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And this happens at the 29th proposition and then it is used later on.
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So it's definitely needed within this geometrical scheme that he's constructing.
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But people began to say, Well, do we need it as a postulate?
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Does it actually is it actually a theorem? Does it follow from the other postulates?
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And so for many centuries, this became the problem that people tried to address.
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So a very early or early ish commentator on Euclid was propolis in the fifth century.
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I think he attempted to prove the parallel postulate on the basis of the other ones and couldn't do it in the medieval Islamic world.
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There was a lot of attention given to this.
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So we have al-Hasani, we have Omar Khayyam as all to see, all trying to prove the parallel postulate based on the other ones and failing.
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And also in Oxford in the 17th century. So this is John Wallace, who was civilian professor of geometry.
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He was obliged to lecture on Euclidean geometry, and he also tried and couldn't get anywhere with this.
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And so by the time we get to the 18th century or so, certainly the later 18th century, people are beginning to think,
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well, okay, we've not managed to prove the parallel postulate on the basis of the other ones.
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So can we actually construct a different kind of geometry where we don't have the parallel postulate all going further?
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Can we assume the opposite of the parallel postulate, whatever we mean by that, and still construct a consistent geometry?
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And I'm certainly not going to attempt to go into the details of that because I'm likely to come unstuck.
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But certainly as the 19th century, it all comes together in sort of the 1820s, 1830s,
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where the Hungarian mathematician John Paul II and a Russian Nikolai Ivanovich,
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along with Chayefsky, working independently, despite what Tom Lehrer might have you believe.
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And they constructed this a consistent non-Euclidean geometry, specifically a geometry in which the parallel postulate is negated.
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And these geometries work in a very different way from Euclidean geometries are not going to attempt to, to, to say much about them.
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But that's quite a nice representation of one of these types of geometries, which is called the PoincarĂ© disk.
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But this particular representation of the PoincarĂ© disk, perhaps recognised by M.C. Escher.
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And I suppose just to bring things to a close,
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the thing that I would maybe try to point out about this is that there are straight lines in this picture,
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but you would not necessarily recognise them as straight lines because in non-Euclidean geometry,
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a straight line is not necessarily the same thing as in Euclidean geometry.
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That's something that it defines a slightly different shape.
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And so this happening in the early 19th century alongside other abstractions that were
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happening alongside the realisation that you can move up to more than three dimensions,
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we have people rethinking the fundamental notions upon which geometry is based.
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So keep thinking what a straight line should be, rethinking how geometry is put together.
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And certainly this then feeds into the later work that I mentioned at the beginning by Riemann,
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for example, the idea of curved space time, the idea of different,
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more exotic types of geometry, which eventually sort of find their, their realisation, if that's the right word in the theory of relativity.
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So here in the first half of the 19th century, people this move towards towards a more abstract geometry that frankly,
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not a lot of people were interested in at this time. By the time you get to the end of the beginning of 20th century, it's finding a use.
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And I think I will stop there.