1 00:00:05,940 --> 00:00:08,910 Thank you all for coming to this this first talk of the evening. 2 00:00:09,840 --> 00:00:18,570 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. 3 00:00:20,100 --> 00:00:26,100 So as you look at the exhibition, you see various things relating to geometry over many centuries. 4 00:00:26,340 --> 00:00:34,160 In the coming up to the 19th century where you'll see reference to Ben Freeman with a, um, 5 00:00:34,830 --> 00:00:40,140 what ended up becoming a very much more abstract approach to geometry in the 19th century. 6 00:00:40,170 --> 00:00:45,810 So for example, the, the move to working in more dimensions than three. 7 00:00:45,820 --> 00:00:47,310 So the idea that you can, 8 00:00:48,270 --> 00:00:56,280 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. 9 00:00:57,030 --> 00:01:05,520 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. 10 00:01:06,540 --> 00:01:10,919 And so what I thought I'd do was, was pick up a different aspect of this abstraction. 11 00:01:10,920 --> 00:01:18,230 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. 12 00:01:18,240 --> 00:01:31,110 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. 13 00:01:31,140 --> 00:01:34,650 Now, some of you may immediately spot that this is not Alexandria. 14 00:01:34,660 --> 00:01:43,290 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. 15 00:01:44,550 --> 00:01:50,910 The reason it's here is that it contains representations of various scholars from antiquity. 16 00:01:51,270 --> 00:01:54,960 And so at the bottom right hand corner here we have Euclid. 17 00:01:55,580 --> 00:01:58,649 Probably some people argue that it's actually Archimedes. 18 00:01:58,650 --> 00:02:05,820 But again, we'll gloss over that detail. And so I suspect Euclid is a name that is known to to most people here. 19 00:02:06,750 --> 00:02:17,250 So around 250 B.C., he collated a collection of of geometrical knowledge, such as it was at the time, 20 00:02:18,300 --> 00:02:23,190 probably gathering together material that was known, possibly adding some new material of his own. 21 00:02:24,030 --> 00:02:27,540 And this is concerned mostly with the very much the sort of day to day geometry, 22 00:02:27,540 --> 00:02:33,870 a sort of plain flat geometry line circles and so on, the kind of thing that he's probably doing on the slate there. 23 00:02:35,790 --> 00:02:43,739 And so this this text in various versions became the basis for the teaching of geometry for 2000 years, essentially. 24 00:02:43,740 --> 00:02:49,260 And it's a very interesting story. And I should perhaps mention that a colleague of mine, Benjamin Walter, 25 00:02:49,260 --> 00:02:57,720 is will shortly be bringing out a book, traces this, uh, this descent because the elements has been. 26 00:02:58,170 --> 00:03:01,260 Sorry, I'm getting ahead of myself. So Euclid's elements, 27 00:03:01,260 --> 00:03:09,990 the text that I'm talking about has been transmitted and translated and corrupted and edited and generally abused down the centuries. 28 00:03:10,770 --> 00:03:20,250 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. 29 00:03:22,860 --> 00:03:29,100 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. 30 00:03:29,880 --> 00:03:34,430 There are also other versions of Euclid in the exhibition, for example, there is an Arabic one then. 31 00:03:36,750 --> 00:03:41,130 And so I've said that Euclid's elements was the basis for 2000 years of geometrical teaching. 32 00:03:41,460 --> 00:03:48,730 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. 33 00:03:48,730 --> 00:03:52,560 So he defines what he means by a point and by a line and by a circle and so on. 34 00:03:53,550 --> 00:03:59,580 And then he sets up his basic postulates, the things that you can do with these concepts that he's defined. 35 00:03:59,730 --> 00:04:07,500 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, 36 00:04:07,500 --> 00:04:15,840 building up the geometry through the various books of the elements, building up to very much more complicated geometrical constructions. 37 00:04:16,710 --> 00:04:21,960 And so the page that I have on the slide here, you may just be able to make out the diagram here. 38 00:04:22,230 --> 00:04:28,170 Oops. Which you might recognise as being a representation of Pythagoras theorem. 39 00:04:28,590 --> 00:04:31,860 And so this is the 47th proposition of the first book of Euclid. 40 00:04:32,160 --> 00:04:37,800 So he's worked systematically through a whole series of of earlier results and got to this one at number 47. 41 00:04:38,010 --> 00:04:40,140 And this depends on many of the things that come before. 42 00:04:40,150 --> 00:04:48,600 There's a very logical and rigorous structure to to the book, which is then replicated in other branches of mathematics down the centuries. 43 00:04:50,100 --> 00:04:57,479 And the story that I want to tell very briefly is the story of a certain amount of disquiet that was felt 44 00:04:57,480 --> 00:05:04,980 about one of Euclid's postulates that the postulates were supposed to be basic assumptions about geometry. 45 00:05:05,420 --> 00:05:09,920 And there's one that is somehow less basic, less self-evident than the others. 46 00:05:09,920 --> 00:05:13,430 And it's gets cause some worry for mathematicians down the centuries. 47 00:05:14,660 --> 00:05:21,860 So in order to to access the postulates, I'm going to do it through this, which is also in the exhibition. 48 00:05:21,860 --> 00:05:25,950 This is Henry Billings, these English translation of Euclid, 1570. 49 00:05:26,010 --> 00:05:31,130 This was the first English translation, as he proudly tells us on the the title page there. 50 00:05:33,350 --> 00:05:36,469 And so I've just pulled the postulates out of Billings, these versions. 51 00:05:36,470 --> 00:05:40,250 So we've got a sort of something reasonably authentic to have a look at. 52 00:05:41,510 --> 00:05:49,170 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. 53 00:05:49,190 --> 00:05:53,630 So if we got two points marked by crosses, we can draw a straight line between them. 54 00:05:53,960 --> 00:05:57,530 Simple as that. Very, very, very basic, very constructive. 55 00:05:58,010 --> 00:06:06,860 He's just telling you what you can do geometrically. The second postulate to produce a right line finite, straight, fourth, continually. 56 00:06:06,860 --> 00:06:09,820 Well, we have to get past some Elizabethan English here. 57 00:06:09,830 --> 00:06:16,880 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. 58 00:06:17,210 --> 00:06:25,160 So, again, very straightforward, very constructive. Similarly, the third one upon any centre and at any distance to describe a circle. 59 00:06:25,760 --> 00:06:29,750 So there's a point. There's a distance from the circle as far as distance from the point. 60 00:06:29,990 --> 00:06:34,040 We can draw a circle centred at that point with that distance as its radius. 61 00:06:34,910 --> 00:06:39,020 The fourth postulate is slightly odd compared to the other ones. 62 00:06:39,050 --> 00:06:44,170 It basically says that all right angles are equal, which you would I suppose you would want it to be true. 63 00:06:44,180 --> 00:06:48,229 So this is a right angle and this is also a right angle. 64 00:06:48,230 --> 00:06:51,590 And this is the right angle. This is the right angle. They're all the same no matter where we are. 65 00:06:51,620 --> 00:06:57,620 A right angle means is the same thing. And he's put that in there explicitly, I think, 66 00:06:57,620 --> 00:07:04,970 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. 67 00:07:05,270 --> 00:07:08,509 So in a sense, he wants these this to mean the same thing. 68 00:07:08,510 --> 00:07:13,110 Wherever you are in this space that he's he's creating. Otherwise it wouldn't make sense. 69 00:07:13,130 --> 00:07:21,440 So that's why that's in there. And then we get to the fifth postulate, also known as the parallel postulate. 70 00:07:22,340 --> 00:07:30,620 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. 71 00:07:32,150 --> 00:07:39,380 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, 72 00:07:39,920 --> 00:07:44,120 the two inward excuse me are two inward angles, less than two right angles. 73 00:07:44,480 --> 00:07:50,000 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. 74 00:07:51,890 --> 00:08:01,580 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, 75 00:08:02,060 --> 00:08:10,580 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. 76 00:08:10,670 --> 00:08:12,380 So together they're less than two right angles. 77 00:08:12,680 --> 00:08:20,900 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. 78 00:08:21,440 --> 00:08:26,180 Okay. And this is Billings, this version of the diagram. So his lines meet at the point date. 79 00:08:27,740 --> 00:08:30,890 And so something that we are sort of invited to think about. 80 00:08:31,190 --> 00:08:39,620 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. 81 00:08:40,010 --> 00:08:47,300 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. 82 00:08:48,240 --> 00:08:54,570 Until we get to the case where the first two lines are parallel, in which case the two lines meet metres infinity, 83 00:08:54,850 --> 00:08:59,430 which I suppose is a mathematician's way of saying they never meet. But we say that they metre infinity. 84 00:09:00,510 --> 00:09:05,310 And so this postulate is very different from the other ones. 85 00:09:05,440 --> 00:09:12,360 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. 86 00:09:12,810 --> 00:09:17,639 This one is hinting towards things that are going on at infinity. 87 00:09:17,640 --> 00:09:23,600 So this is quite a different thing and. You might ask, what is this a basic postulate? 88 00:09:23,600 --> 00:09:29,120 If you were to write down a list of basic facts about geometry, would you necessarily think of this one? 89 00:09:29,150 --> 00:09:32,630 It's not necessarily that obvious a thing to think about. 90 00:09:33,650 --> 00:09:38,480 And so from very early on, there's a certain amount of disquiet about this postulate. 91 00:09:39,140 --> 00:09:45,620 Even from Euclid, it seems because he doesn't use this until the 29th proposition of book, one of the elements. 92 00:09:46,580 --> 00:09:53,630 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. 93 00:09:53,680 --> 00:09:58,159 And this happens at the 29th proposition and then it is used later on. 94 00:09:58,160 --> 00:10:02,710 So it's definitely needed within this geometrical scheme that he's constructing. 95 00:10:02,720 --> 00:10:06,200 But people began to say, Well, do we need it as a postulate? 96 00:10:06,650 --> 00:10:10,610 Does it actually is it actually a theorem? Does it follow from the other postulates? 97 00:10:11,060 --> 00:10:16,060 And so for many centuries, this became the problem that people tried to address. 98 00:10:16,070 --> 00:10:21,790 So a very early or early ish commentator on Euclid was propolis in the fifth century. 99 00:10:21,800 --> 00:10:30,830 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. 100 00:10:30,860 --> 00:10:32,450 There was a lot of attention given to this. 101 00:10:32,450 --> 00:10:41,000 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. 102 00:10:42,080 --> 00:10:47,030 And also in Oxford in the 17th century. So this is John Wallace, who was civilian professor of geometry. 103 00:10:47,360 --> 00:10:54,350 He was obliged to lecture on Euclidean geometry, and he also tried and couldn't get anywhere with this. 104 00:10:56,120 --> 00:11:01,759 And so by the time we get to the 18th century or so, certainly the later 18th century, people are beginning to think, 105 00:11:01,760 --> 00:11:06,170 well, okay, we've not managed to prove the parallel postulate on the basis of the other ones. 106 00:11:07,100 --> 00:11:13,850 So can we actually construct a different kind of geometry where we don't have the parallel postulate all going further? 107 00:11:13,850 --> 00:11:21,620 Can we assume the opposite of the parallel postulate, whatever we mean by that, and still construct a consistent geometry? 108 00:11:23,240 --> 00:11:28,010 And I'm certainly not going to attempt to go into the details of that because I'm likely to come unstuck. 109 00:11:28,010 --> 00:11:34,550 But certainly as the 19th century, it all comes together in sort of the 1820s, 1830s, 110 00:11:35,870 --> 00:11:40,620 where the Hungarian mathematician John Paul II and a Russian Nikolai Ivanovich, 111 00:11:40,620 --> 00:11:44,900 along with Chayefsky, working independently, despite what Tom Lehrer might have you believe. 112 00:11:45,920 --> 00:11:55,190 And they constructed this a consistent non-Euclidean geometry, specifically a geometry in which the parallel postulate is negated. 113 00:11:57,770 --> 00:12:05,659 And these geometries work in a very different way from Euclidean geometries are not going to attempt to, to, to say much about them. 114 00:12:05,660 --> 00:12:10,520 But that's quite a nice representation of one of these types of geometries, which is called the Poincaré disk. 115 00:12:11,390 --> 00:12:17,240 But this particular representation of the Poincaré disk, perhaps recognised by M.C. Escher. 116 00:12:18,320 --> 00:12:23,000 And I suppose just to bring things to a close, 117 00:12:23,000 --> 00:12:28,550 the thing that I would maybe try to point out about this is that there are straight lines in this picture, 118 00:12:29,180 --> 00:12:35,329 but you would not necessarily recognise them as straight lines because in non-Euclidean geometry, 119 00:12:35,330 --> 00:12:38,840 a straight line is not necessarily the same thing as in Euclidean geometry. 120 00:12:39,080 --> 00:12:42,620 That's something that it defines a slightly different shape. 121 00:12:44,180 --> 00:12:48,919 And so this happening in the early 19th century alongside other abstractions that were 122 00:12:48,920 --> 00:12:53,210 happening alongside the realisation that you can move up to more than three dimensions, 123 00:12:54,470 --> 00:12:59,860 we have people rethinking the fundamental notions upon which geometry is based. 124 00:12:59,870 --> 00:13:05,090 So keep thinking what a straight line should be, rethinking how geometry is put together. 125 00:13:05,420 --> 00:13:10,100 And certainly this then feeds into the later work that I mentioned at the beginning by Riemann, 126 00:13:10,100 --> 00:13:14,480 for example, the idea of curved space time, the idea of different, 127 00:13:15,110 --> 00:13:24,980 more exotic types of geometry, which eventually sort of find their, their realisation, if that's the right word in the theory of relativity. 128 00:13:25,310 --> 00:13:32,180 So here in the first half of the 19th century, people this move towards towards a more abstract geometry that frankly, 129 00:13:32,180 --> 00:13:39,110 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. 130 00:13:40,790 --> 00:13:41,870 And I think I will stop there.