13 November 2014
100 Essential Things you didn’t know
about Maths and the Arts
Professor John D Barrow
Thank you very much. It is good to be back here at Gresham again, at Barnard’s Inn, to give a talk, which I have done on many, many occasions, although probably more often at the Museum of London. I happened to talk about the two previous books in this trilogy, which were also about applications of mathematics, first, to everyday life, and then to sport, in 2012, in the Olympic year. So, you might ask yourself, how come mathematics seems to be so useful and applicable in the world around us – why is mathematics so ubiquitous and so useful in the everyday world? I think the simple way to understand that is to have as your definition of mathematics that it is simply the collection of all possible patterns. Some of those patterns might be on the floor or the ceiling, it might even be on the shirt that you are wearing, but others might be patterns in sequences of events in time. They might be patterns in links and networks between disparate places or different packets of information. So, mathematics, at its root, is the catalogue of all the patterns that there can be. Some of those patterns may be rather boring and rather uninteresting and fruitless, but others have proved to be extraordinarily fruitful and powerful in understanding the universe.
So, some philosophers regard it as a mystery that mathematics works, that it describes the way the world works. On this type of definition, it is not a mystery. It is a triviality really: the world has to contain some pattern or we could not exist – there would be nothing to contemplate and no one to contemplate the universe. But the mystery is then why such simple patterns are so far-reaching and so powerful in understanding the universe. And so, we see immediately why mathematics is involved in art, in architecture and design.
Someone is probably wondering: what is the difference between art and design? I am not really sure, but David Hockney said that art has to move you - design does not, unless it is the design for a bus.
But if we just look at some snapshots of rather obvious applications of symmetry and geometry, spirals are both ubiquitous in this Fibonacci sequence to maximise the exposure to sunlight in this flower, and then in great pieces of architectural design - Leonardo Da Vinci was greatly fascinated by spirals and attempted to classify all the types of spiral patterns and offshoots that there could be, and he was particularly interested because he wanted to make designs of chapels and spawn these little side chapels off the side of the design.
This rather beautiful Leonardo drawing from Paris, in the Bibliotheque Nationale, this looks like a remarkable piece of technical drawing – it is actually a freehand drawing.
If we look elsewhere, in the Islamic world, the world of Escher or even here, in artificial images to represent information in science, one of the things that we soon start to appreciate in the images that we like is that the mind and the eye likes complexity in a type of intermediate category, so not overwhelmingly complicated so you cannot make head nor tail of it, or not so simple that there is really nothing challenging. So, we like complexity to be challenging but challengeable. We see that in the games that we devise. You do not try and play with a 20-dimensional Rubik cube. The success of certain types of games is their beautiful match to just being soluble and yet being a real challenge.
People have wondered what role reductionism plays in art and science. When we see complexity in the world and we try to reduce it, we try to make it intelligible in simpler forms, we are doing science. So, complexity with reductionism is science. Complexity without reductionism is generally art.
Art, of course, is rather individualistic. Someone said, “Art is “I”; science is “we”. Science is a shared communal knowledge – it must be possible for other people to verify and check it.
When you look at the first printed books, the first Vesalius’ “Anatomy” for example, or the first books of flowers, on the frontispiece of those books is usually something that looks rather mundane or even macabre. In the Vesalius book, there is a great table of gruesome saws and knives and instruments, dripping with blood. These were the instruments that were used to derive the knowledge in the book – that is the message, and you, the reader, can do the same. You can verify the information in this book by experiment and by observation. You do not have to read the works of the Ancient Greeks; you can actually check it for yourself.
Well, if we take this rather prosaic view of art, it is interesting to convince yourself that we have tried everything that, at first glance, we ought to have tried. So, you can think of artistic creations as having some dimension in space and some dimension in time. So, when there is no time dimension, a one-dimensional spatial creation would be a frieze; in two dimensions, it would look like a painting or an engraving, so it does not move, it sits in two dimensions; and if you go to three dimensions, you have something like a work of sculpture. If you start introducing time, you can make things move. So, in one dimension, you would be looking at music, a sequence of sounds, in one dimension; in two dimensions, you are looking at the pictures moving in time, you are creating a film; in three dimensions, the sculptures or the living sculptures are moving as well. So, you have created all of these possibilities with rather well-established forms of art.
People can then try and do clever things – we will see later on – of trying to work in between these dimensions, or linking together these media, theatre with film embedded, of course theatre with music embedded is opera and so forth, so there is a whole collection of combinations of these activities.
I want to show you, just first of all, three or four different single examples that use some rather simple mathematics to tell you something artistically interesting.
The first is a story about Charles Hinton. Charles Hinton was originally a schoolteacher at Cheltenham Ladies College in about 1900 or so. His father was a rather unusual Victorian gentleman. He was a church minister, but of a special religious sect that he had created himself that preached free love and polygamy, so, in Victorian England, this was not really a winner. His son became, at first, more interested in polygons than polygamy, but later that was changed because he ended up on trial at the Old Bailey for bigamy. He married the daughter of the set theorist George Boole, but he was married to somebody else at the same time. He was sent to jail for a few days, but then he left the country and did the same sort of thing that people in that embarrassing situation always do: he went to Princeton University in America, where he invented the automatic baseball bowling machine, among other things, and then went to work in the US Patent Office, and he was working there at exactly the same time that Albert Einstein was working in the Patent Office in Europe.
But what Hinton was fascinated by, in his mathematics teaching and lots of his other activity, was: how could you visualise the fourth dimension? He realised that this was not a non-starter because, if you take a three-dimensional object, like this thing here, then, when you look at it, it looks like a two-dimensional object, and if we were to cast a shadow on the wall, the shadow would be two-dimensional. If we were to undo it into a flat plan, then that would also be two-dimensional. And so, he realised that there must be a simple recipe for what four-dimensional things would look like if you looked straight at them.
So, imagine that you have one of these little cubic boxes that you used to make at school. When I did craft lessons, this is all you used to do, was to make these little boxes – you would mark them out very carefully on card, and then you would fold along the lines and fold them up into a cube. But you can see that, if you lay them flat, what the pattern is…this cross, straight, straight, and there is a number of squares…there are six squares, and you can count how many corners there are and how many sides. Then you think, suppose we had a four-dimensional cube and we unwrapped it in the same way, then instead of having squares, we would have cubes, and we would have a central overlap. So, this is this famous hypercube that you would see if you were to look at a four-dimensional cube… So Hinton was the first person to apply these recipes and persuade you that you could visualise higher dimensional things.
In the 1950s, Salvador Dali talked to a number of mathematicians in the US about this type of geometry, and one of the results was this famous Corpus Hypercubus work by Dali, where you see the figure of Christ suspended in fact in front of this structure behind, and this is nothing less than the hypercube here, with a slightly reduced front, and the chessboard at the bottom and the figure representing his wife. So this is a representation of a higher dimensional geometry in his picture that obviously has some transcendental religious significance for the subject of the picture.
Here is a more mundane example. If you wander around London looking at things that are high up in the air, like this figure on the top of the column in Trafalgar Square, Nelson’s Column, something that you might wonder about is: where is the best place to look at this object? If you go up very close and you stand here and you look upwards, you will not see very much of it. If you go a long way away, the farther away you go, again, the less will be the size of the statue subtended at your eye. So, somewhere in between there must be a best place, where the size of the statue appears at a maximum for your point of vision…
So, with a simple piece of school geometry that I hope some of our students here have encountered, suppose this is the column, okay, and on the top is the statue, of height S, here is the tower, T, and you are at a distance X away, which you can change. There are two angles here: this angle b up to the statue, and a up to the top, so a + b is this total angle. With simple geometry, one of the things you learn at school is the tangent of the angle a +b is the tangent of a plus the tangent of b, divided by 1 minus tan a, tan b… So, if you are at school, you will know this, but you will have never had any cause to use this formula, until now…!
So, the tangent of b is just T over X, and the tangent of a + b is S+T over x, so by using that formula, you can work out the tangent of a here, if you wish, in terms of x, and you get this simple formula: the tan of a is Sx divided by x squared plus T times S + T.
Now, if you were to change this angle a here, and by changing the distance x, then where’s the maximum? Well, you differentiate the tangent of a with respect to x, you get sec squared of a, da/dx, and on the other side, you get this messy combination of xs and Ts… Where is it a maximum? Where do you maximise the apparent angular side of the statue? Well, it’s where da/dx is zero. And you can see that’s just where the square brackets here vanishes, where x squared is equal to T times S+T. So, the square root of the height of the tower, multiplied by the height to the top of the statue, is where you should stand to maximise the apparent size of the statue.
So, in the case of Nelson’s Column, Wikipedia tells me the tower is 50m, the statue is 5.5m, so you should stand about 52.6m away to get the best view of the statue on the top.
You can apply the same principle if you are looking at the Sphinx. In fact, if you are kicking a rugby goal as well, you might like to think about this calculation as to where the best place to kick from is.
Here is another example I talk about in the book. I had encountered an exhibition in America a few years ago, and it was a photographic exhibition by Catherine Opie. Her exhibition was called Twelve Miles to the Horizon. So, what could this be about? She had gone on a very long boat trip from Korea to America, on a relatively small cruise-boat, and she took many, many photographs on this ship of the sea and the sunrise and the sunset and so forth, so of course every picture has got the horizon in it. And it is an interesting question to ask: how far away is the horizon on the Earth? Is it 12 miles? You might have thought this was just a guess.
Well, let us suppose this blue circle is the Earth, and R is the radius of the Earth, and, slightly exaggerated, let us suppose you are standing at a height H above the Earth. So, if you are standing on the seashore, H is your height. If you are on this boat, it is a little bit bigger than that. And where is the horizon? Well, you draw a line from here that is just tangent to the Earth’s curvature there, and so it is a right-angle down to the centre, and that is the horizon – that is the farthest you can see. So all we need to work out distance to the horizon is Pythagoras’ Theorem because R + H squared is equal to D squared + R squared, round that right-angled triangle, and I think that sort of sits along here… And one of the things you know is that your height is a lot, lot less than the radius of the Earth, and so you can forget about H times R compared to R squared – absolutely negligible. And so, the square of the distance to the horizon is roughly R squared + 2HR…along here… So, your final estimate of how far it is to the horizon is just given by this little formula here.
If we put some numbers in for the radius of the Earth and leave your height to be H, in metres, it looks like 1600 x the square root of 5H in metres or the square root of 5H in miles. So, if you were standing on the ground and you are 1.8 metres tall, say, then the horizon is about 3 miles away, 4800 metres. So, next time you go and stand on the seashore or down by the Thames, that is how far you can see.
Catherine Opie I think is not doing too badly. She is on the upper deck of her ship. She would be dead-right if she was 29 metres above the sea-level, and that is perfectly plausible if you are standing upstairs on a cruise-boat. So, a little bit of analysis of very simple geometry tells you this rather useful fact about how far you can see. You cannot see forever, but you can see three miles.
Another piece of simple geometry, but a very, long, long time ago, about 530BC. If you have ever been to the Isle of Samos, you will know there is one of the ancient engineering wonders of the world on this island. The island of Samos is quite mountainous. On one side is the capital, which is called Pythagoreion, after Pythagoras, who supposedly lived there, and Aristarchus lived there as well, the first person to have a heliocentric theory of the motions in the solar system. The monument to him in the square says that his ideas were stolen by Copernicus centuries later.
But the trouble with living on Samos was all the rain seems to fall on one side of the island and everyone lives on the other side because of the nice port, and so the challenge for the dictator of the day was to get the water from one side to the other in a way that could not be poisoned or stopped or messed about with or stolen by his enemies. So, what you have got to do, he thought, was to build a tunnel under the mountain. It is about a thousand metres of tunnelling to do in 530BC – this is a non-trivial engineering project. And he thought, to halve the time, we are going to have one set of engineers starting in this direction and another set starting in the other direction and they would meet in the middle.
So, the first question he is going to be asked is what if they do not meet? Or the question that he might ask the engineers may be what if you do not meet? So, they might say, so you will have two tunnels, you know, excellent!
But in fact, they are rather cleverer than that. They worked out a scheme whereby, eventually, they only missed one another by a few metres. They did much better than engineers under Mont Blanc, with laser-guided cutting equipment, in the last 30 years or so. So, what did they do?
Well, if you are starting here on this side and here on the other side, you have got two problems. The first is, you want to meet at the right level, so the people on this side need to know something about the height over here. Now, that you can do by having long extents of guttering, in effect, that you put round the side of the mountain, in long stretches, and you operate them like spirit levels, by putting water along this channel, and then along this channel, and making sure everything is level, and by the time you get round here, if you have kept the water level, a couple of miles away maybe, you will know that you are at the same level as the people starting on the other side. But from this picture, you can also work out this vertical distance, by adding up this length and this length and this length, and if you had to come back, you would subtract the vertical distance going in the other direction. So, you can work out this vertical distance, and by adding this to this, taking away this and this and this, you can work out this horizontal distance, so you know the vertical distance and you know the horizontal distance, so you know the angle in the corner by using our tangent formula we mentioned just now. The tangent of this angle is this distance divided by that distance. So, both tunnellers know that, if they put some stakes in the ground, separated, they can measure this angle to be right, and then they just keep siting more stakes under the ground and make sure they keep heading at the same angle every afterwards, and the people up here do exactly the same thing. They know what this angle must be – it is 90 degrees minus this angle, and this is what they did, over a number of years, and eventually, they converge at the same point. At the last moment, the strategy is that one of them actually changes direction, so that you will run into the other tunnel and you will not just pass a metre or two parallel to it.
So, all this is going on for ten years. Lots of people die of course. It is about 170 metres under the ground. So, it is all a consequence of the geometrical heritage on that island, of people understanding simple Pythagorean geometry and applying it. You can walk in part of this tunnel today, and I have done that, and here is an illuminated section. It is rather cunning in that the water channel is under your feet. There is a little narrow opening here, and if you walked along this passageway, you would never dream that there was a water channel underneath, so it is significantly hidden. Unless you know it is there, you would not be delving down here and looking underneath.
Let us now look at something a bit more extended and unusual. If you look around the world about, there are two interesting differences between things which are human constructions and things which appear in the natural world. Things which are human constructions are more often than not smooth. They have been made to some type of optimal design. In these cases, it is to minimise air resistance and so forth. But if you look in the natural world, you tend to see things which are jagged and not smooth because they are trying to optimise something else. They are usually trying to optimise their surface area, whilst minimising their volume or their weight. So, that is what a tree is up to: it wants to minimise the weight of wood and branching, but it wants to maximise the surface area that is presented by the leaves to the outside atmosphere from which it absorbs its nutrients and so forth. If you look inside the human body, the lung system for example, again, you will see a great convoluted, tortuous mess that is trying to maximise surface exposure whilst not at the same time increasing weight at the same rate.
Well, let us look first at this jagged world because there are interesting ways in which people have tried to swap the two over, and, in artistic creation, use both of them in surprising ways. So, the jagged construction, something that we call fractals today, began really in about 1904 with this Scandinavian gentleman, Helge von Koch, who realised that the way of constructing designs and shapes, by copying the same shape over and over again, on a smaller and smaller or a larger and larger scale.
Here is a simple example that is usually called the Koch Snowflake. So you start with a triangle, you take out the middle of each side and you replace by another triangle, equilateral triangle, and then you take out the middle third of that side and that side, and you carry on doing that, ad infinitum. So, you end up with something that has an unlimited perimeter in the end, gets bigger and bigger, even though the area enclosed is finite.
This type of recipe for creating things you tend to see in graphic design and computer art very extensively, and it began to be quite a fashionable and big business in the late-1980s, thanks to Benoit Mandelbrot, who invented the word “fractal” to describe what is going on. It is no accident that this type of graphical representation appeared then because this was the time when the personal computer started to emerge, and when that happened, single individuals could start to experiment with mathematics and graphics on their own, inexpensively, easily, with very simple programming recipe. Before that, you would have had to baffle your way through enormous research teams who dictated what large computers were used for, and they were used for blockbusting enormous projects, like simulating [bonds/bombs] or exploding stars or predicting the weather. So, the whole fascination with complexity and computer graphics comes about as a result of the IT revolution.
Mandelbrot’s most famous creation was a very simple mapping. Suppose that you have axes here and you have an origin here, and you pick a point with some coordinates x and y, and then you apply this rather simple formula for some choices of numbers a and b, and this will send your point into another point on the plane. Most points get sent very, very far away and disappear off the page, but Mandelbrot was interested in the boundary, as it were. So, all the points in the black region disappear far away, but the boundary separating that collection of starting points that, after many applications of this rule, disappear, from those that do not, it has this rather intricate, almost heart-shaped configuration. When you look at it more closely, the configuration is astonishing. So, if you look under a magnifying class at this piece here, what you find is this is just a smaller version of the entire picture… And if you look again at any minute section of the boundary, you always find the whole boundary recreated in that tiny image. So, looking at this under a magnifying glass, you just see always the same structures, and however small you went with your resolving power, you would just see more and more recreations of the same intricate structure.
This has proved rather fascinating to many artists, but a few years ago, Richard Taylor in Oregon, whom I know quite well, started a project, that became I think rather well-known and eventually somewhat controversial, to apply fractal methods to try and understand various pieces of modern art.
The ones he liked to start on were Jackson Pollock’s artworks. Pollock’s pictures are the most expensive in the world, so if you wanted to buy this, you would have to pay maybe a £100 million at the present auction rate. It is very large. It is about ten foot by four foot or something like that. When you think back, in the 1970s, there was an enormous fuss in Australia when they paid a million dollars for a large Pollock – best investment that the country ever made for their gallery, so it is probably worth close to 100 million Australian dollars, or US dollars actually, today.
So what Taylor was interested in was whether you could carry out an analysis of the surface of this picture using some of the mathematical methods that describe complexity of Mandelbrot’s sort, his fractal sort. First of all, you have to understand what the idea is here.
If we just have a square like a canvas and we draw a straight line, you might think that is a one-dimensional curve – it is very simple and it does not need much information to describe it. If you start wiggling the curve around, it occupies more and more space of this area, so in a sense, you need more information to describe what is going on. And if you make it busier and more and more wiggly, you need even more information, and eventually, if you make it as busy as you can, so it visits every point in the square, the line starts to behave like an area – it is coloured in the whole of the area. What Mandelbrot and others had realised was that you can define a dimension for what this line is doing in between dimension one and dimension two, which just reflects how complicated the line is.
How do you do this? Well, suppose you have got something that is a bit complicated, like the outline of this fern, and you put it on a grid of squares, and the squares have a particular size, and you ask: how many of the squares are cut by the pattern, the fern pattern? And, over here, you plot this… So, this is the number of squares that get cut, and then this is the size of the square. And now you make the squares smaller, and you carry out the same calculation: how many squares get cut? And then you make them smaller still, and count again. And you can see that, if there is lots of very, very small-scale wiggliness, that as you make the squares smaller, you pick up more and more and more intersections. So, for this example, the number of intersections versus the size, if you plot the logarithm against the logarithm, you get this straight line, and what you are interested in is the slope, so: how quickly does the number of intersections increase as the squares get smaller? That would be this formula here… So, the number of intersections when the square is of size r is proportional to the size to the power minus d. In this case, if it was just a straight line, that number would be one. In this case, it would be two. But in between, you can calculate what the effective dimension is for this wiggly pattern…
And that is what was done for lots of Pollock’s pictures, looking at different levels of the painting and so on. The first remarkable discovery was that these pictures were all fractal-like, so they did indeed have this power law dependence of the number of crossings on size. So, what this means is that they are self-similar. So, like the Mandelbrot Set, when you look at them under a magnifying glass or you make them bigger, you get the same statistical impression, so there is not a dominant scale of pattern in the picture.
If you look at artworks in books and then you go to galleries, I have often had this strange experience that something that looks quite nice on the large format page in the book is actually disappointing in the real gallery when it is very large, and no doubt there are opposites.
Pollock is not like this because he has this self-similar structure, so whatever scale you reproduce his pictures, it has the same impact.
Pollock, of course, knew absolutely nothing about fractals and fractal mathematics. But there is lots of detailed film of him painting these pictures, so what he does is that he has two operations: one, he throws paint – he puts a big canvas on the floor and throws paint, and then he drips paint, on smaller scales, and then, at the end, he cuts the canvas out, leaving the edge parts of the canvas behind. What he had mastered, clearly, just intuitively, was this idea of not having the picture dominated by particular scales of pattern and imprint, so when you look at it when it is reduced or it is enlarged, you get the same impact.
So, if somebody asked you or says, “Well, you know, my three year old can produce a picture that looks like that,” they cannot. They cannot produce a picture that is self-similar at this degree of analysis. This requires considerable skill and expertise.
So, what was then done was to look at Pollock’s pictures over a long period of time and to compute this dimension, this effective dimension of the pattern, and it is rather remarkable: he gets the same dimension, to within about 0.1, over many, many pictures at any given time, so it is almost like a constant intuitive perspective that he has on the picture. As he gets older, he starts to make the pictures busier, he works them over many more times, and you start to see the dimension increasing with time towards the end of his career.
There are various other signature aspects of his pictures. I told you he does two things, he throws and he drips, and you can determine the scale on which the crossover occurs between the two activities, and you have a different slope, a different value of d, for the throwing and a separate value of d for the small scale activity up here, the dripping, and this is the throwing, and this is the transition.
So, what Taylor had tried to argue was that you could use these defining characteristics to authenticate suspicious Pollocks or unsigned Pollocks, of which there are many, because he was very reluctant to finish work and so he would only sign when he was sure he had finished.
Here is a fake Pollock that people were claiming was a Pollock about fifteen years ago, and when you carried out this analysis, it was not self-similar at all and it did not have the transition between the two activities – it is not a Pollock. There is no doubt about that. But if you just looked at it in Shepherds Bush Market, if someone was trying to sell it to you for $50 million, you might well buy it, or you might buy it on eBay, because you think, well, it looks a mess, it must be a Pollock…but it is not.
And then, a couple of years ago, twenty or so prospective Pollocks were found in someone’s attic. It was an old friend of Pollock and his wife, a couple, who had been given them. They were rather smaller, and so suddenly there is a big challenge: are these real or not? So, things then started to become very controversial in that Taylor authenticated a number of these, and then other people came along claiming that they could make other patterns which were designed to pass his published tests and that this would pass the Pollock test – okay, you can add colours and so on later. And then Taylor said, “Ah yes, but I have other tests that I apply, other than the ones that I have told you about in my papers, because if I told you all the tests and you were a forger, then you would just make sure that the forgeries passed the test.” So, we are still at this stage, where Taylor claims that this cannot pass his secret test plus the public test. But you see how the use of the study of complex systems and complex patterns can be used in understanding art, and perhaps even in authenticating it.
Well, let us leave painting behind and go to not jagged fractal things but things that are smooth. In the 1920s and 1930s, there was a movement in sculpture, led by Henry Moore and Barbara Hepworth, which tried to introduce what were called stringed figures, so, they introduced spanning strings into works of sculpture. One of the fruits of this period, if you go to John Lewis’ store in Oxford Street, on the wall outside, there is a great winged figure by Barbara Hepworth which is on the side of the store, on the street, and this was created really as part of this movement.
So what happened was that Henry Moore was looking around the Science Museum and there are a lot of beautifully made models in the Science Museum, of which this is one – they are still there. They were made by a student of a famous French mathematician, Monge, to illustrate particular types of smooth curved surface. So what is going on here is, imagine that you have these two metal struts, and many strings are passing from one strut to another, and you then twist them – you have probably done this, even if you are only knitting or making some type of tapestry. You can imagine what happens: you create this curved surface here, an envelope of lots of straight lines that, in this case, has a parabolic curve.
This fascinated Moore, and he set about creating a whole series of sculptures, usually in wood, but sometimes in stone, which combined these stringed arrays, sometimes with a twist, to create a sort of beautiful transition from one straight surface to another. These are rather beautiful things because you are used to Henry Moore sculptures being enormous. This one, I think it would just fit on my hand, and there are some that would just fit in there – they are really tiny. They would still cost you half a million pounds. But this is made out of a beautiful piece I think of cherry-wood, which he had found at a shoemaker’s, as a shoe last, and he used to keep pieces of wood and stone for an opportunity in the future when he was going to do something with them, and this he did in 1939 and called it Bird Basket.
Well, this type of mathematical transition you find used in architecture in rather elegant ways as well. This is the Jerusalem Chords Suspension Bridge, and you can see the same basic idea, where you have a series of straight cables going upwards from the base to another straight support, but you introduce a twist, and when you do that, you create this beautiful smooth envelope. The idea here is that these envelopes give you the smoothest transition to pass between two straight lines.
So, here is the simple recipe in the previous two pictures. Suppose that you just have a graph like this, and you are drawing lots of straight lines, one starting here, down to here, one from here to here, and so on, so you make a whole family of straight lines, and you see how you get this envelope structure of the straight lines, that as you increase their density, it becomes an exact parabola.
You can then do other things. You can make the joins go between several lines, so here is a little illustration here… Again, you are in the process then of creating the smoothest path that you can follow in going from one line to another, with some other constraint.
These mathematical algorithms are really what are used to create all the fonts on your computer, or your typewriter if you are an antiquarian. So, these fonts are created by these mathematical envelopes of curves, and you can see one in action here, doing something a little more complicated.
The people that first understood these mathematically and systematically are two mathematical engineers called Bezier and du Casteljau, and they both worked for car companies in the 1950s. Their jobs were to produce beautiful, aerodynamic, aesthetically-pleasing car bodies for sports cars, and one of them worked for Renault and the other for Mercedes, and they set about using computers and these types of mathematical curves to produce the type of car bodies that are familiar in sports cars.
So, I have made here a more elaborate one. Here is just the equation of a straight line, okay… So, T is just the fraction of the line that you have covered up, so we start at the point P0, when T is 0, and we end up at the point P1, when T is 1.
Now let us replace the point P0 by this line… So, if we substitute the line in there, okay, after 1-T, and we substitute it in there, for two more points after T, then we end up with this two line movement, the smoothest possible transition from this line down to this line.
If we do the same thing and we put this quadratic parabola into the formula and another one into here, we end up with a smooth transition which is now a cubic curve, and it is being generated by the motion of three straight lines.
If we put that in here, we will get next a quartic, with four straight lines generating this smooth curve.
So, within all these worlds of engineering design, the creation of fonts, nice ways to go smoothly as possible from one line or surface to another, this type of simple mathematics is what helps you.
Well, lastly, let us look at something about art galleries and mathematics. What do we know about art galleries? Well, they are full of things that people want to steal and things that you have to keep an eye on, a bit like this room really, and you might therefore say, well, suppose we have all sorts of pictures around the wall like this, and the gallery is divided up into little nooks and crannies and small rooms, what is the smallest number of cameras that we could get away with buying so that we could keep an eye on every wall? Or, if you like, what is the smallest number of guards that we need to pay, if the guard stays still and they do not walk around but they can look 360 degrees?
This is an interesting little problem. You can see that it is not entirely obvious what the solution would be. And we could imagine a very simple type of gallery that mathematicians would call a convex gallery. So, this is, say, a polygon, but all the corners are pointing outwards. The technical definition of being convex is that, if you draw a line between any two points in the gallery, the line stays completely inside the gallery. But if it is not convex, you can see, you could draw a line from here to here which goes outside the gallery. The convex gallery is very nice because one camera, placed anywhere, can see everywhere, so you need one camera or one guard and you can see everything. Over here, if you have got N sides or N corners, it is not at all obvious what the story might be, although, if you are a mathematician, it can quickly become obvious.
Let us suppose we have that same gallery, which has got 19 sides, then the right way to think about this is that, if you had a triangle, for example – that is one of our convex figures, it is the simplest one – you only need one camera in a triangle. So, if we were to divide up the gallery into a number of triangles, we would be in business with one camera each into seeing how many we might need. So you can always divide a polygon, no matter how wiggly its side, into a collection of triangles, and I have done that here. I have put the dash lines in here… So this has been divided up into a collection of triangles, and what we want to do now is to do something that is called a three-colouring.
We would imagine that we have got three colours, and we are going to start by putting one of the colours in this corner, say, and then we want to put down the other two colours in such a way that every triangle has the three colours appearing, and there is no line which joins one colour to the same colour. So, this is not as hard as it sounds, because if you put one colour here, let us suppose it is red, then the other two colours, you put one here and one there, and therefore that colour must also be the colour here, so you know what this colour is, so you know what this colour is, so you know what this one is, so you know what this one is, so it is completely determined once you start. If we make those three colours, there is how it goes. So, there is the pink one, and we chose these as yellow and green, so this one has to be pink and therefore this one has to be green, and therefore this one has to be yellow and so this one has to be green, and you see how it goes.
Now, what this three-colouring, as it is called, now gives you are possible places to put your cameras. So, one solution of the problem is to put the cameras at the pink places. If you put a camera at the pink places, that camera would see every place in the gallery. Similarly, if you put cameras at the yellow spots, it would also see all the places in the gallery, and likewise with the green places. But if you add them up, how many pinks do you need? One, two, three, four, five, six. How many yellows? One, two, three, four, five, six. How many greens? One, two, three, four, five, six, seven. So, you should go with the pinks or the yellows. That is two possible solutions to the problem – they are both equally good. And in general, you can quickly figure out what the answer is: the answer is you count the number of sides or corners, divide by three, to get the triangulation, and then you take the whole number part of the answer, and that is the most cameras that you might need to watch the gallery. So, with 19, divide by 3, 6 and a third, whole number part is 6, and that is exactly what you have got with the yellow or the pink solution.
You can carry on doing this. If all the gallery walls were at right angles to each other, you would not do a triangulation, you would do a quadrilateralisation – divide into four-sided figures, and the answer would be the whole number part of a quarter times the number of sides.
You can generalise this problem in all sorts of ways. You can allow the guards to move around. You can put obstacles in the middle. You can divide the gallery into separate rooms.
So, this became quite an interesting mathematical problem when it was first posed in 1972 because it was a new type of problem. Mathematicians had not seen a problem, they thought, of this type before, but very quickly, Chvatal at Montreal realised that you could transform it into this colouring problem of graph theory, where you are assigning people with a particular view of things.
Okay, just to alert you to something you can read about in the book. Another interesting area of mathematics is in regard to texts. So, people in the Humanities today are familiar with the idea of using statistical analysis on texts, whether it is Shakespeare or early editions of the Bible. Every so often, someone will come up with a new sonnet of Shakespeare and they will apply a type of mathematical analysis to see whether the number of new words in the text or the words being used in the text follow the same statistical pattern as the author’s other work.
The person who invented this line of inquiry was a famous and rather ill-tempered mathematical statistician called Andrei Markov, whose name is everywhere in the world of statistics nowadays – Markov processes, Markov chains. He worked around the year 1900/1905, when he was about 60 then, in St Petersburg, and he hated all the people in Moscow and he hated the Orthodox Church and all the politicians, so he was always arguing with somebody – he is the Victor Meldrew of the world of Russian statistics. He was interested in a particular type of statistical process that no one had studied carefully before. Previously, people were always interested in statistical processes where each subsequent event was independent of the others. So, if you toss a coin, it does not matter how many times you have already tossed it, if it is a fair coin, the chance of it coming up heads next time is a half. But the probability of the weather being rainy tomorrow is not independent of the weather today. It depends on the weather today, in some way, and there will be a probability that rainy days are followed by rainy days, and sunny days are followed by rainy days, and so on. Markov was the first person to develop the mathematical theory of these types of chains of dependent processes. He got involved in a big argument with mathematicians in Moscow, who, being very close to the Orthodox Church, who almost controlled the Mathematics Department, strangely, in Moscow University, they had written a lot of papers arguing that you could prove the existence of freewill from the fact that so many social variables -criminality, destitution, marriages, birth, and so forth – were normally distributed. So, because you get a normal distribution as the sum of independent random processes, they therefore inverted the argument to say, because we see all these normal distributions, they must ultimately have been the result of independent choices and therefore of freewill.
Well, Markov was having none of this nonsense and set about demonstrating that dependent sequences of events could also end up giving you normally distributed distributions of outcomes, and as a part of these investigations, he started to analyse texts – this famous prose poem of Onegin which every schoolchild in Russia studied then, and they probably still do today. He showed how, by hand, he could do an analysis of the way consonants followed vowels, or vowels followed consonants, or particular words followed other words in this author’s work, as a way of characterising that work and distinguishing it from other authors’ work or attempts to fabricate it. So, this was the beginning of that type of textual analysis. Poor old Markov did it all by hand, reading all 120,000 words in a novel, adding up all the vowels and the consonants – a dedicated chap.
Lastly, instead of looking at words, sometimes mathematical structures and networks can display information in rather beautiful and rather powerful ways. This picture was the winning entry in a competition about five or six years ago in America for students, for graphic design in the Humanities, I think, largely, so ways of using pictorial images very powerfully in their subject. They were students of religion and theology who won this competition, and what this picture is about is about cross-referencing in the Bible. So, along the bottom are the books of the Bible, in fact the chapters of the books of the Bible, and the length of the stalactite is the length of the chapter. You can…is it Psalm 119 is the longest, is it not? Right in the middle…you can probably see it better than I can, it will be that one… So, along there, they are plotting the chapters in the books from Genesis to Revelations, and if there is a cross-reference, then they draw an arc. So, if you refer to the Book of Life in Genesis and in Revelation, you have an arc joining the two. And the colours represent, in some sense, the significance or the depth or the length of the allusion and the cross-reference. So, what you are seeing in this picture is a representation of the interconnectedness of a multi-work text, and it succeeds beautifully in conveying information that would have required huge amounts of words, so you begin to see the level of interconnectedness, which parts of it are more connected than others, and what is, say, the degree of interconnectivity between Old and New Testaments, short books and long books. So, I have always found this a rather beautiful example of a very simple idea. You can only do it because you have computer assistance. You can do this type of analysis very, very quickly, in a few minutes. Because theologians are keen on concordances and cross-references, they have done all that cross-referencing information about the texts for you already. This is about representing it.
So, I hope that has given you a taste for a few of the topics of the 100 which I cover in this book, and I have tried to stress, in the examples I have talked about, it is not all about geometry and tiling and tessellation, so I have tried to be as imaginative as I can about what is meant by the arts and what sorts of things in engineering, in design, maybe cutting diamonds, the visual appearance of objects, stamps, coin design. So there are many, many applications of mathematics where there is some aesthetic quality, and also some optimisation problem which is novel or interesting.
Thank you very much.
© Professor John D Barrow, 2014