Maths with Pictures

Gresham Lecture, 5 October 2010

Maths with Pictures

John D Barrow FRS,

Gresham Professor of Geometry

Welcome to the new Gresham Geometry series.  If you come from north of London, like me, I also congratulate you on actually getting here today!

Today, we are going to have a gentle beginning to the lecture series by looking at the role of pictures in mathematics, but with a distinctly historical flavour.  If you browse through books of mathematics or you remember mathematics you studied at school or at university, pictures seem to be an integral part. Diagrams and figures, as they are called, seem to be essential for visualising and seeing what is going on but this has not always been the case. 

During the 19th Century and in the early 20th Century, there was quite a violent anti-picture reaction in mathematics, and there were philosophers of mathematics and mathematicians who were quite against the idea of using pictures.  They felt that pictures could deceive, that they were a sort of crutch to formalist logic, and that they should be avoided at all costs.  Even when I was an undergraduate, sometimes a lecturer would draw a picture and illustrate something, but say “You must beware – this is not a proof; you cannot have proof by picture.” 

So, in the 19th Century, there were books by de Lambert, in the French tradition, about mechanics, that rather proudly boasted there were no pictures at all. 

This was a subject in the tradition of Newton and British applied mathematics that was very intuitive and very visual. Then, in the early 20th Century, formalists like Brower and others regarded mathematics as being a logical process and Bachi, the French collective which tried to formalise mathematics, used no diagrams at all.  That movement led to the so-called “new maths” of the 1960s and then the 1970s, which had a huge and, in many people’s view, pernicious influence upon education and the teaching of mathematics in different countries around Europe.  Thus there was great emphasis upon mathematical structures, but not upon individual problems and diagrams and developing intuition.

There is something of a rather contentious history to this subject but I will start by going much further backwards, to think back and have a look at what mathematics books were like, long before there was printing and moving block printing.  The first printed book in science was in 1542, on the “History of Plants” – it was a botany book.  Just one year later, there was Copernicus’ “Revolutionibus” and Vesalius’ “Human Anatomy”.  1542 is when people began to start mass producing books with illustrations.  Before that, having pictures in hand-scribed books was a nuisance.  The pictures had to be drawn in every copy.  So, what tended to happen, is that a sponsored scholar working for some wealthy patron would have a beautifully illuminated manuscript copy, or a few of these, which would have all the diagrams.  He would present one to his sponsor, and others to wealthy, influential friends.  The students would have copies that probably did not have the diagrams, or they might just have a few of them. They would have lots of blank spaces to fill in, rather like those engineering lecture notes of modern times.

Many examples of such books, even in the 12th century, tend to be of Euclid’s “Elements” which was still a mainstay of the teaching of geometry all over the world 1500 years after Euclid. This was a key text for teaching purposes, not because they wanted to teach the students about geometry, but because Euclid epitomised logical argument, theorems, proofs and conclusions, which was what they wanted to teach.

Many famous philosophical works by Aquinas or Spinoza followed Euclid’s example in setting out their works as theorems, propositions and proofs. They were laid out in a similar fashion, despite not being mathematical books. Euclid was a touchstone of truth about how things should be investigated.  

The other thing to remember is that Euclid’s “Geometry” was not just a series of descriptions of lines and angles and curves.  It was not a model that was used to describe bits of the world.  It was believed to actually represent the whole world. Euclidian geometry was part of absolute truth.

One of Adelard’s books on Euclid is a good example.  This book would have been used by students in lectures, and the theorems are present in this version of the book, but there are no proofs so the students would have had to fill in the proofs. 

There are other versions in which the proofs are given, but the statements of the theorems are not so the students have to reconstruct them.  Usually, the diagrams are missing.  In this example from Adelard the diagrams are present, but they are not lettered.  There is a little stroke on each diagram, which shows which of these statements of the theorems this diagram is meant for.  However, the diagrams are very higgledy-piggledy. They are all written down the margin and not beautifully laid out as intimate parts of the text.  This is a typical situation because diagrams were expensive to produce and even harder to reproduce.  They were secondary so people tried to do without them if they could.

There is another example from Adelard dating from 1480.  Adelard was an 11th Century scholar who lived from about 1080 to 1152, and this example is a later edition of his translation from Arabic into Latin of parts of Euclid.  Again, it has beautiful script which would have been prepared by French scribes and there is something to guide the reader through the manuscript.  The starting letters of each new section have a colour – they are usually in gold-leaf. There is pale yellow, which just about comes out still, which marked the separate sentences, and then there are different colours which give you different sizes of letters following them that show the reader the statement of the proposition. Then, in smaller text, the proof comes afterwards – this is all rather hard to read because paper was expensive, so space was not wasted.  There is no index to a book like this, and there is no list of contents.  So this colour-coding and changes of typeface and illumination exist to act as a guide through the text – highlighting the statements of the theorems and the proofs.

The diagrams are rather incidental and squeezed into the margins.  Most copies would not have included them.  This might have been the lecturer’s copy or some well-to-do person’s copy.  Again, they are not terribly well labelled or linked particularly to some collection of statements of the theorem.

In the actual book, there is usually a flyleaf at the back and the front maybe as well, which will be packed with notes and other diagrams, to make use of the space.

We jump forward to another example from 1570 which was not long after the first moving block printed book was published in 1542.  This book, of Euclid again, was printed in London, by Day, the greatest printer of the day. It is remarkable because it is a pop-up book.  It was the first pop-up book.  In the section of the book in which Euclid talks about three-dimensional geometry, there are little flaps which sort of stick down on the page, and then they can be picked up and inclined to make sense of what Euclid is saying. They are lettered in the corners.  It is a beautiful little creation.  If anyone comes across one of these in your second-hand bookshop, do let me know!

One of the most important and impressive things in the development of science, like algebra, arithmetic and geometry, is the drawing of graphs. If the average scientist on the street was asked who invented graphs or how old the idea was, they would probably say that it would have been either the Babylonians or the Greeks, and that the idea went back to the beginning of time, like algebra and arithmetic. However, graphs are quite a recent invention in the history of scientific perspective.  They do not go back to early Greek representations of mathematics.  The earliest graph that has been found – it is not known who drew it – is this one. =

Historically, it is something of an outlier. It comes from the 10th Century, and was found in the papers of a monk who taught literature, astronomy, mathematics and the whole quadrivium of knowledge in France in the 10th Century. 

It is clear what the graph is. The labels on the y axis are planets, plus the sun, in the solar system.  Along the x axis, there are 30 divisions which are days of the month.  This is therefore a representation of the relative motions of planets over a monthly period.  It is not as simple and as elegant as that, because the scale is actually different for each planet, and there are bits of the picture that have been rubbed out and put back.

Medieval scholars later revealed that this picture was not quite what it seemed.  You probably imagine that this was a chap looking out of the window – there were no telescopes of course in the 10th Century - making observations and recording planetary positions.  Actually, he was giving a shorthand summary of a passage from Pliny, which discussed the motions of the planet, and so, instead of giving his students pages of text to learn and remember, he created this little diagram to represent what the text was saying about the motions of those planets. 

So, as far as we know, this did not really lead anywhere; it did not lead to a great explosion of people drawing little diagrams like this and mathematicians saying “What a wonderful idea!”

To find something looking more similar to a useful graph we go to 1350. The person responsible was Nicole Oresme, the great French polymath, a counsellor to Charles V.  He was a bishop, a musicologist, a mathematician, a geographer, a philosopher and a student of motion.

In some of his columns of his writings about motion he has these little diagrams that he called latitudes which are graphs.  They do not have numbers on the axes, but they all plot speed on the y axis and time along the x axis so he was using them as shorthand.  Instead of saying “let us now consider a motion where the speed increases steadily with time and then eventually remains constant”, he just drew this little picture. Sometimes, they are quite large, in the margins, but on other occasions, they are really quite small and just sit in the sentence.

When I first looked at them, I thought it was quite a handy way to say things in a shorthand fashion. However, when I looked further into the developments in philosophy and mathematics at this period, I realised that there was probably a little more to it. As I said, Euclid’s geometry was regarded as part of the absolute truth of things, and if anyone wanted to convince people that what he was doing was important and that it was part of the ultimate nature of the universe, it was a good idea to show that the subject being investigated was geometrical or had a geometrical flavour.  Motion sounded a bit boring.  It is about chucking stones, dropping things, moving carts around the road – it had a mundane feel to it.  Euclid did not write about it.  However, if the mathematician represented his study of motion geometrically, it suddenly seemed more important. I think that this idea of turning motion into geometry to enhance status was prevalent in these Oresme’s work.

On another page of Oresme’s notes there are little diagrams showing motion. On these, he started to use scales on the pictures, so he could measure the speed at different times.

Oresme was a student of many things, and I think it is rather likely that his thoughts about representing motion graphically were closely linked to his work in musicology.  In truth, the earliest graph was drawn about the same time, the 10th Century, as the musical score was drawn by Verezo in Northern Italy.  The musical score is a graph of pitch versus time. Oresme was interested in the representation of music, and forged the link between mathematics, music and geometry that the Pythagorean tradition started.

However, these graphs do not seem similar to the type of graphs that we read in the News of the World or the Independent financial section, telling us that things are going down and down and down.  Graphs like this came much later. 

The first graph of a continuous mathematical function that I came across was not published in any open literature.  It was a letter of the great Dutch scientist, Christian Huygens, to his brother, who was interested in life expectancy and the insurance business, the world of Mr Gresham.  What was odd about the life insurance industry in 1600 or so was that people were insured at a level regardless of what their life expectancy was thought to be, regardless of their age.  So Huygens’ brother wrote to him and asked, “Well, surely there is information about how long people are likely to live?” and whether there was a pattern to it, and Huygens revealed, of course, that there was a pattern.  So, he drew a graph with current age along the x axis and life expectancy along the y axis with a negative exponential function. If someone was 86 at the bottom right of the graph, it looks like they have virtually 0 years left. This was the graph of a mathematical function used to predict something and used as a substitute for great tables of data.

There is a graph of the sine function and this was the first time that somebody had tried to create a graph of a known mathematical function.  Again, it was drawn towards the end of the 18th Century, by the great mathematician Johann Lambert, who also developed the equal area projection of mapping that is sometimes seen today. 

So this is happened very late in the day.  Newton never saw a graph and never drew a graph.

However, the other type of representation of information in mathematics and science that is very familiar to us today is the whole business of charts, bar-charts, pie-charts and so forth.  This developed at the end of the 1700s and in the first quarter of the 19th century. 

One of the people involved was James Watt, the pioneer of steam engines, and much else. Working in Glasgow, he invented the pen recorder.  He wanted graphs that would tell him how pressure was changing in his boilers over a period of time, so he decided to automate the measuring and have the pen automatically create the graph. Watt was rather paranoid about letting out information of that sort to his competitors, so he kept his graphs a secret until 1822. 

However, Watt had a young man working for him called William Playfair, who was from Edinburgh. Playfair worked for Watt for two or three years until he was about 21, and must have learnt quite a lot about the mathematical representation of information from Watt because he created two remarkable books. In one go, in many dozens of new types of picture and diagram, these books conceived all the charts which we are used to seeing represent information, particularly in economics.  His book was called a “Commercial and Political Atlas”; it was called an atlas but there were no maps in it, just about 45 of these really elegant pictures.

This was the first time that there were economic graphs. One particular graph charts the change in the national debt of England against time, marking in various catastrophic and other events.  The general trend is much the same as today – increasing steadily and confidently.

A slightly later, rather beautiful graph of Playfair’s charted the price of a quarter of wheat as a bar chart and the wage of a good mechanic as a line graph. He was a genius at representing many types of information in the same picture. There are variations caused by the seasons and other socio-economic events. Along the top, he highlighted the centuries and the monarchs. The obvious observation to draw from the graph, although Playfair did not, was the growing gap between prices and wages.  This type of presentation of information developed in the early 1800s. 

Around the same time there was a growth of the Social Sciences, primarily in Belgium because of Adolph Quetelet who was a pioneer in this subject. He specialised in the creation of what he called a “social physics”, and he coined many phrases such as “the average man”. He argued that, statistically, if he examined a reasonable-sized city, with a population of a million, then a certain number of those individuals were going to become axe murderers, drunkards, lunatics or extremely good citizens, and that these things could be statistically predictable. One of his devices for doing this was the so-called normal distribution of statistics.  If there is a sequence of independent events – independence is the key – then, as the number of those events becomes large, then their distribution in frequency about their mean value follows this so-called bell-shaped or Gaussian distribution.

This distribution had been worked out mathematically by de Moirvre. He was a French mathematician who had fled to London back in 1685, but people did not start drawing this distribution until much later. 

The histories of statistics all say Quetelet was the first man to draw the normal distribution in 1846, but that is not true. De Morgan drew it first, in 1838, and he understood how the distribution could be tweaked so that it was either narrow or rather broad.

One of the points about this part of history of representation of information is that statistics was highly controversial at this time.  Writers like Dickens saw statistics as a great evil to be opposed in every way. Dickens saw concepts like “the average man” that Quetelet introduced as a device by which governments could ignore the poorest in society by making statements of the sort that “the average man is doing much better than ever before”. Dickens saw statistics as a way of hiding away destitution, as it were, in the tale of the normal distribution. 

A novel like “Hard Times” is actually a diatribe against statistics.  Mr Grandgrind has a disastrous life because of his reliance upon statistics and measuring. He even tried to determine his daughter’s marriage and her future happiness by a mathematical calculation based upon percentages.

There is a nice wood engraving of Grandgrind meeting Louisa in the study, saying to her, “Louisa, my dear, you are the subject of a proposal of marriage that has been made to me,” and then he goes on to work out what is the most advantageous course for them all to take.

I said that Playfair’s atlas had no maps in it.  The word “atlas” had become part of geometry and the great work of the cartographers, in Holland particularly, in the 16th Century. This was epitomised by the famous cover picture of Gerardus Mercator’s “Atlas of Cosmographical Medications upon the Creation of the Universe”.  This is why books of maps were known after that as atlases.  On the front cover is Atlas carrying the world on his shoulders.  The book was printed in a rather unusual size of paper which became known as atlas.  Therefore, books of maps, following Mercator, became known as atlases, and that was why Playfair used the word.  It was supposed to be a way to guide the reader through this unusual territory of information. 

Mercator’s projection of the Earth’s surface onto a flat plane, as with any projection of a spherical surface onto a flat plane, is distorted.  For example, Greenland, a long way away from the equator, appears to be an enormous territory.  This cannot be avoided.  The map projection can be changed or Lambert’s idea of keeping the areas of land masses the same but distorting their shapes can be used but no solution is perfect.

Very quickly, people were interested not just in drawing maps to show how to get from A to B, but to add and overlay other types of information onto the map, just as Playfair did with his charts.

This can be done, for example, with a map of the Earth at night. It is a compendium of different photographs taken of the Earth from space. It is a map of the money. Where there are lights on, that is where the money and the electricity are; whereas, in some other places, where there are enormous population densities, there is very little light at all.  Such a map is very revealing, not perhaps in the way that NASA had originally imagined.

I should say that, in this quest to represent statistical information, people like Playfair and Quetelet played a big role.  Another unsung hero of that work was Florence Nightingale - a serious statistician and founding fellow of the Royal Statistical Society.  The reason she was successful as a nurse was not just because of tender loving care, but because she kept records of what were the consequences of doing certain things, such as moving the distance between beds in hospitals in the Crimea to test how effectively it would stop the spread of infection. By keeping statistical records of the outcomes, she could do more of the things that worked well and less of the things that did badly.  So she invented lots of the varieties of pie chart that are now used to represent information in the newspapers or on Microsoft Office.  She liked those ways of representing information because it really hit her at a glance what was going on; she did not have to look through tables of information.

In the first edition of Playfair’s atlases, he included tables of data, as well as charts.  By the third edition, he had given up on the tables of data.  He was so confident of the impact and importance of the pictures that he no longer bothered with the tables and just showed the pictures.

Nowadays, people like maps which do not just show places but show connections between things. A typical sort of contemporary example would show internet hubs in the USA, thus showing the connectedness of different users of the internet and the web in the USA.

The first weather map was pioneered in Britain by Francis Galton in 1875.  It appeared in the Times on April fool’s Day 1875 although there was no connection with that date.  It was the chart of the previous day’s weather.  At first, the idea was just to understand the weather situation and what had happened. Later on, it became more ambitious and predicted the next day’s weather.  The original plates from which this was printed are still at the University College London Library in London.  Galton, again a great polymath, invented fingerprinting, did infamous work on eugenics and other important work on statistics. However, the general appearance of the initial weather map looks rather familiar in many respects – not just the fact that it was rather dull and damp. Later on, Galton would develop the symbols for the warm fronts, the ecluded fronts and the cold fronts.  This was a map with additional information about how things were going to change in time.

A map I particularly like, and, which for many people, typifies London, is the London Underground map.  However, the London Underground map was not always like it is today.  Back in 1908, there was a London Underground map pullout which was given away in the Evening Standard and it was a disaster!  This map was a geographical map, and the routes of the lines reflected exactly where they went. That was not very helpful, because they went to an awful lot of places and large amounts of the map did not even fit on. 

If anyone wanted to come in from way down on the left, they had better ask a friend because the whole line was not shown! If someone wanted to go from one side of London to the other it was not entirely obvious how they would do it because the map did not show where to change. The map showed many old lines that do not exist anymore. For example, the Baker Street and Waterloo was eventually merged to become the Bakerloo.

Therefore, this type of map was not very helpful and in the next 20 years or so, the London Underground struggled economically because people were not using this train service.  It was struggling to raise enough income to justify its existence. However, by talking to people and listening to rumours, it became clear that one of the clear problems was that it seemed to be such a challenge to actually use it.  It looked very complicated and the distances looked so long.

At the beginning of the 1930s a young man called Harry Beck came to work for the Underground Railway Company as a draughtsman.  He had trained as an electrician, working in electronics – which was no accident. He was given the latitude to think about maps in his spare time, and a bit of his work time, because he believed he could come up with a much better map for the underground train system.

The first map that he drew in his exercise book is important to mathematics because it was the first topological map.  It was a map which did not worry about the real positions of things in space, but rather about the links between those things. With a topological map, if it was drawn on a great trampoline or a rubber sheet, and then stretched in any way without being torn, the links from one station to its neighbour would stay the same, but the distances would be altered.  Beck tried to imagine that he was looking at the map through a lens, and he would expand out the inner region, where he needed space, and bring in the outer region, so it would fit on the page. The first Beck map of the underground is a great design classic, known the world over for its ingenuity, its elegance and its simplicity.  All the lines were at 90 degrees, horizontal, or at 45 degrees.  Remember, he worked in electronics – it did look like a circuit diagram which was the model.  He also introduced the clever way of representing the exchange stations, and the Thames came wandering in to give a bit of orientation. 

However, if someone wanted to use this map to walk around the streets of London, they would be better off with a Monopoly board!  I once saw some Japanese tourists pouring over an Underground map near Piccadilly Circus, and they asked me how to get somewhere. I said, “First step, put that away!”  It is strange that, if someone comes to London from out of London, even quite frequently, and they find the Underground is working, they may find that they do not actually know how to get from Piccadilly Circus to the Embankment or to King’s Cross.  They do not have any feel for where these places are.  Our entire orientation in London is built by this map.  Of course, if someone is underground, they do not need to know where things are; they just need to know what the next stop is and where to get off.

This topological feature of the map is obvious. Places like Morden, Cockfosters, Uxbridge and Ricksmanworth are at the world’s end!  People living there really think they are in London because of this map. It really looks as though it would not take any time at all to get from Morden to Piccadilly Circus, but they people ought to leave half a day. It’s quicker to get in from Cambridge than it is to get from the ends of these tube lines to the centre.  It is only 45 minutes from Cambridge to King’s Cross on a good day, but it could be a lot longer.  This is a beautiful example of a particular type of principle in action.

Mathematicians did not only guide the way maps were created, but sometimes something very visual lead to a great problem or conjecture in mathematics. The famous one to do with maps is the so-called Four-Colour Conjecture, which is now the Four-Colour Theorem. 

This was conceived by Arthur Kempe in 1878.  He called it the Geographical Problem, quite rightly, and the problem is very simple. If someone wanted to create a map, whether of the United States or of the counties of England, and they did not want any two neighbouring states to have the same colour, what was the smallest number of colours which would colour any map. Trial and error convinces people that they seem to need four as they cannot get away with less than four as there are always examples where three would fail.  However, no one could prove, for nearly 100 years, that four colours really suffice and that there were no possible counter-examples to that.

This was finally proved by Appel and Haken in 1976 in a remarkable, and in some ways controversial, fashion.  This was the first time that computers were used to prove a theorem.  The proof of this theorem is so long that it has not passed through any single individual’s mind and this was the first time that had happened. Appel and Haken were able to organise the problem so that they knew that there was some enormous number of possible types of map which could give a counter example, but it was a finite number. However, they needed to check them all one by one to make sure that they were no counter-examples.  Human life is too short to check them all one by one, so they created a computer programme that did that checking. 

When it was announced to the mathematical world that the Four-Colour Conjecture had been confirmed, people were very excited. However, one famous mathematician, Erdos, when he was told how the proof was done, was very depressed, and said it was not a good problem after all! 

Thus, it is still open for someone to find a beautiful, short proof, at some stage, but it has not been found so far.

We move on to something a little bit different, where pictures in mathematics have played a very interesting role and they have been bound up with the computer revolution in the late 20th Century. 

A Swedish mathematician, Helge von Koch, is the pioneer of what is now known as fractals or fractal geometry. That name was invented in 1972 by the French mathematician Benoit Mandelbrot, who made a great personal industry from the whole world of fractals.  Von Koch was interested in a very simple type of geometrical construction. If pictures are not used, it is hard to imagine and very laborious, so it is a good example of the power of simple pictures. He wanted to create a closed curve that would have infinite length but just enclosed a finite area.

This can be done, at least theoretically. The way to do it is to start with an equilateral triangle and then take out another little triangle, on the mid-side of the triangle, and then do the same again – take out triangles out from every side, and keep on doing that forever.  It can be proven, fairly straightforwardly, that if someone did go on forever, the length of the outline of that curve, around all the surfaces of the triangles, tends to infinity.  It becomes arbitrarily large as there are more and more triangles, but the total area that is enclosed by all those lines is finite.  It is rather curious.

This was the paradigm for the study of objects of that sort.  There is a three-dimensional version, called Menger’s Sponge. This is a surface where cubes are continually hollowed out of a large cube. If this is done ad infinitum, again there is a surface which has a finite volume, but an infinite surface area.  These sorts of structures are very interesting, not just to mathematicians, but for biologists as well. 

A tree, for example, is trying to maximise the surface area of foliage that it exposes to moisture and oxygen in the outside world, while at the same time not having the mass of wood increase at the same rate, which would cause it to collapse. The network of our lungs has vast surface area, to ingest oxygen, nutrients, while keeping the mass of the lung finite. All over the natural world, this recipe of wanting to maximise surface but keep weight or mass in check, is found.

In the 1970s and later, this type of study was resurrected in a big way by Mandelbrot, who then worked for IBM in Yorktown Heights in America. He had almost unlimited computer power at his disposal, and he discovered things that could not have been discovered without the use of computers and pictures.

He created another version, in effect, of von Koch’s snowflake surface.  Imagine there is a graph with an origin. Mandelbrot had a rather simple mathematical transformation, which says that if someone started with a point somewhere and applied this transformation many times, keeping a note of where it goes, then most of the points end up being sent far away and never come back, but some of them keep moving around at just a finite distance away from where they started. Those ones are interesting.  The shape encloses a black region which is the boundary of all the points that do not get thrown far away.

The boundary has different coloured shades which say something about the speed with which the points get thrown far away.  However, the boundary has extraordinary intricacy, in that, if it is studied under a magnifying glass, it is shown to be exactly the same structure mirrored all over again. 

A close-up of this infinitely long and infinitely complex boundary, any part of it, will show the whole Mandelbrot set, as it is called, the whole structure, recreated again. If part of its boundary in turn were to be looked at under the magnifying glass, it would show exactly the same structure created over and over again. It is an infinite sort of mine of intricacy. This boundary has this infinitely complicated structure, even though the original formula for generating it just involves three symbols.

This was a wonderful example showing that very simple equations, with completely predictable, non-random starting conditions, could produce behaviours which were almost completely bottomless and unpredictable in some sense.

The exploration of this type of structure is part of the subject which became known as complexity.  It grew up in the late-1970s and 1980s, hand-in-hand with the computer revolution. The availability of small, inexpensive personal computers meant that single individuals could study such structures completely and compellingly.  Previously, computers had been the preserve of huge research groups with multi-million dollar funding who were focused on blockbusting projects like making bombs, predicting the weather and understanding the formation of stars.  Now, anyone could use powerful computers with good graphics, for single individual research work. The development of experimental mathematics, in which mathematicians would explore, by examining graphics like this, interesting structures, and then subject them to further analysis, grew out of the personal computer revolution at the end of the 1970s.

The Mandelbrot set appears in all sorts of places such as in a rather nice piece of computer art or even in the advertising world. A bottle of vodka seems to be celebrating its chaotic structure all around the boundary.

We move on to things which can be visualised, but in some sense do not exist in our space.

At the end of the 19th Century, a rather unusual schoolteacher called Charles Hinton, who worked at Cheltenham Ladies College and then at Uppingham School, was forced to make a hasty exit from England.  He went off first to Japan and then on to the United States.   His problem was that he had married a lady called Maude Weldon, but he was already married to someone called Mary Boole, who was the widow of George Boole, of Boolean algebras and Boolean functions, a famous logician.  Bigamy was a serious offence in England at this time so he was tried at the Old Bailey and jailed for a short period. On his release he disappeared across the Atlantic.

He had started as a maths teacher, but his father was an unusual influence.  His father was a sort of freelance minister of religion who formed his own religion which preached free love and polygamy, which was not guaranteed success in Victorian England.  His son started working with polygons and then he moved on to the polygamy so he fled to America.  However, through all this period, he was interested in the Fourth Dimension.

In Victorian England, the Fourth Dimension was of great interest to all sorts of people.  Spiritism and the occult was a great Victorian interest. A Psychic society, which still exists, was formed at Trinity College Cambridge with famous members including Nobel Prize winning physicists, and people like Crooks were among the founders.

Hinton, though, was not interested in the occult side of the Fourth Dimension.  He was interested in how he could visualise it, and he had a very clever idea, which he wrote about in articles in the 1880s and then eventually published in a book, called “The Fourth Dimension”, in 1904, after the Scientific American magazine had run a competition for the best way to visualise the Fourth Dimension.

He recognised that in three dimensions, a three-dimensional hollow cube which is cut along the right sides can be unwrapped.  Remember those little lessons at school – craft - where pupils had to cut out these horrible little things, glue them together and take them home to Mummy and Daddy!  Hinton realised that there must be an analogous situation for fourth dimensional versions of the cube if it was cut down and laid flat.

He realised, likewise, that if we people look at a three-dimensional object in shadow on the wall, it is two-dimensional, and so we can work out what the shadow of a four-dimensional object would look like if it was cast in three dimensions. 

The famous hypercube, tesseract structure that Hinton first drew is the unwrapped four-dimensional cube.

Dali had a lot of interaction with Thomas Bankoff, a geometer, in the early 1950s and learnt about these representations of higher dimensional structures in three dimensions. He used it in his remarkable and famous Dali Corpus Hypercubus picture. The cross here that Christ is crucified on is one of these hypercube projections.

Representing things in higher dimensions is one thing. However, there were other people who were interested in representing things which could not exist at all in any dimension. 

Oscar Reutersvard was a Swedish artist who pioneered the representation of impossible figures.  Sometimes these are associated with Roger Penrose, who produced other versions of them about 20 years later, but Reutersvard was the first person to create these seemingly possible pictures which cannot exist in three-dimensions. Someone has shown me a version of this – they can be bought – which is a rather unusual three-dimensional object which is not as it appears and looks like this when seen. 

Escher made great use of this type of device to create physically impossible scenes, and then ones which are perhaps not physically impossible.

There is a great resonance between this type of geometric construction, in three dimensions and beyond, and extraordinarily skilful draughtsmanship.

Escher also represented what is known as the Mobius strip.  If we take a strip of paper, and joined one end to the other, we would make just a little loop. However, if we give it a twist before we join the ends, we get this unusual surface which has only one side.

If I was my predecessor, Robin Wilson – remember his penchant for absolutely terrible puns and jokes – I would say something along the lines of “Why did the chicken cross the Mobius Strip?” and everyone would say, “To get to the…”

The Mobius strip was first drawn by Augustus Mobius, who lived from about 1790 to 1868.  He was a mathematician, a geometer, and head of the observatory at the University of Leipzig, so he was an astronomer also.  In his notes, he spent some time thinking about this type of geometric structures, and he drew it in different ways.  It was a notebook entry in 1858, but it was only discovered a little bit later, after he died. Somebody else, Johann Listing, had had the same idea then about the same time, so they are regarded as having independently created this idea, although it has ended up with Mobius’ name attached to it.

It became greatly interesting to engineers and people in manufacturing in the early 20th Century.  If a conveyor belt that is moving things along, is given a twist, and made into a Mobius strip, then it will last twice as long, because there is only one side being worn.  If there was no twist, it would wear out the outer side, and then you would switch it over and you could use the other side.  There was a patent in the US for making tape drives and tape recorder drivers and so forth to maximise the storage and transfer of objects along conveyor belts. Thus, even this rather esoteric idea turned out to have considerable practical use.

The other place that people see the Mobius strip without even noticing is the standard recycling symbol that can be found all over the place, such as on milk cartons. It was invented by an undergraduate student at University of Southern California in 1970, when there was a design competition for a logo for such an idea, and he created that logo from the Mobius Strip, with arrows pointing around, and we have used it ever since.  I do not think he won a penny for it actually, except his sort of prize in the competition, which was probably a $5 book token.  It is not a trademark, so people can use it as and when they like.

I want to finish off just by looking at a few physics pictures that have a rather more unusual mathematical complexion.

There is a cover of a magazine that I have at home.  The magazine, Life Magazine, from the USA, no longer exists. It is a 1950 edition which shows a US nuclear test. Inside, it shows several photographs, a few milliseconds apart, of this test.  Now, the energy yield of this atomic bomb was top-secret at that time in 1950. However, there was a rather ingenious gentleman called Geoffrey Taylor, known as G.I. Taylor who was Professor of Mathematics at Cambridge. His portrait sits in the tearoom I walk past every day. He was the inventor of all sorts of things to do with hydronamics, the standard anchor that is used in sailing boats and so forth. He had a remarkable mind and was able to intuitively see what was going to be true, by very simple arguments, before being able to prove it rather more rigorously.  During the War, he worked on the British bomb project, which never made a bomb.  It was called the Tube Alloys Project, and he was on that project because he was an expert on hydronamics and explosions and so forth.  When he saw the photographs in the magazine, it took him about 45 seconds to work out the energy yield of the bomb. He then made it known to people, that it was rather interesting that you could work out the yield of the bomb.  This caused a big international incident, crisis, as you can imagine.

How did he work it out? One of the things that physicists get used to doing is dimensional analysis. They have a feel for what some phenomena must depend on.  So if there is an explosion in air, and the interest is in how big the diameter or the radius of the explosion is, at any one time after the explosion has gone on, it must depend on certain variables. It is going to depend on the energy of the explosion and it is going to depend on the time – the longer you wait, the bigger it will be. The explosion is ploughing into air, so it will depend on the density of air.  If the explosion was in water, it would go more slowly than if it was in a vacuum.  Taylor realised that the energy of the bomb can depend only on the air density, the radius and the time. There is only one way because the dimensions of mass, length and time have to be right in the formula, in which those quantities can be related. The only way it can be is that the fifth power of the radius is proportional to the energy, multiplied by the square of the time since the bomb went off, divided by the air density.  Air density does not change – it is constant.  He could measure the radius with his ruler and the pictures helpfully had the times printed on them so he realised that he could divide one picture by another and that the energy was 25 kilotons of TNT, which was spot-on!  This was all very embarrassing for the powers that be, but it shows the power of pictures. There is sometimes proportionality and scaling in complicated events, and people gain information from the picture that it would not be easy to gain in some other way.

Feynman diagrams are famous for physicists.  Recall Oresme with his diagrams in the text, saying how speed changed with time and so forth. 

Feynman did physics in the late-1940s in a very different way to everyone else.  Other people were solving equations and doing very formidable calculations of how electrons and photons of light should interact with one another in the quantum picture of the world.    Feynman developed a sort of visual calculus to do these calculations, and those pictures were known as Feynman diagrams.  If there is an interaction between two particles, an electron can come in, exchange a photon with electron, two electrons go out and then there is another photon that is exchanged.  There is a whole hierarchy of these diagrams that appear like the terms that have to be calculated in a series.  If it is done correctly, the terms in the series get continually smaller, so you do not need to calculate too many of them before a very good answer is reached. 

This was a completely new way of looking at and doing physics.  It was extraordinarily accurate. The best predictions of Feynman’s quantum electrodynamics agree with experiment to 12 or even 16 decimal places.  We do not know anything about anything better than we know the way electrons interact with photons, according to quantum electrodynamics.

Here is a typical diagram, showing the exchange of a photon to ingoing and outgoing electrons.  The diagram has time going upwards and space going across, so it is a way of looking at the interaction between these two particles, and if there are two more going out, they exchange this quantum of energy.  The shapes of these diagrams give physicists the machinery to do the calculations to work out the numbers which govern the interaction strength and outcomes. 

Modern particle physics, finally, is very diagrammatic and pictorial.  There is an early bubble chamber picture, from the 1950s.  It shows particles going through liquid hydrogen at very low temperature which cause the hydrogen to boil and leave tracks in the chamber. It also shows the spirals of the charge particles, and by studying those trajectories, physicists can understand the nature of the particles, their masses and how their speed. 

Today, at CERN, there are simulations of what we would expect to see for a Higgs particle decay.  There is much more going on because they are at much higher energies.  It is an impact which is millions of times more energetic than the previous one, so there is much more debris. The challenge here is not just to create pictures but to somehow throw away 99.999% of the information that is irrelevant to find the one piece of information, the telltale signature, of the Higgs decay, and this means a battery of computers to reconstruct these events.  Therefore, particle physics is still very much linked to the representation of information in space and the analysis of pictures.

I hope I have given a quick overview and guide to the way pictures have been used at different times in the history of mathematics and mathematical science.  Next time, we are going to come very much back down to earth and look at continued fractions, a branch of mathematics that is rather simple but you may never have come across, unless you are a graduate mathematician.

©Professor John Barrow, Gresham College 2010