London Institute for Mathematical Sciences

itrus sinensis. Oranges. Not the most momentous start to a story, but this mathematical tale is going to be about humdrum things, so I'm not at all ashamed to be talking about fruit. In 1611, Johan Kepler - who is famous for his work on planetary orbits, and rather less famous as the first ever writer of (very bad) science fiction - was walking along the street outside his house, and noticed the orange sellers. In fact what he noticed was the way that they stacked up their oranges in neat pyramids, with a hexagonal layer of oranges at the bottom, then the next layer placed into the dimples underneath (actually there's two ways to do this, but only one of these gives a nice shape), and so on, until there is a complete tetrahedral pyramid of the things. It looks great, and it sells oranges, but what attracted Mr Kepler's attention was the thought that this was a very good way indeed of stacking oranges - in the sense that you could fit a lot of oranges into relatively little space.

More particularly (remember he was a mathematician and therefore prone to wild flights of fancy) if you filled up the entire universe with oranges stacked in this way, then most of space would be filled with orange, and only a small part left over because of the gaps between them. When he went home, it was the work of but a few moments to conclude that about 74.1% of the universe would be orange, and only 25.9% would be the left-over gaps. He then played around with some oranges, or maybe tennis balls (actually, this part of the story is not recorded, so I am making it up for dramatic effect), and eventually found that he couldn't do any better than the fruit sellers on the street outside. To a real mathematician, that's not much of a conclusion however; it's more of a challenge to action. For example, I can't find any prime numbers bigger than a million before my patience runs out - but that doesn't mean there aren't any. What Mr Kepler wanted to do therefore, was to prove that the orange sellers' stacking (technically known as face centred cubic) was the densest possible packing of (to be precise) perfect, equal sized, hard spheres.

Who would be interested in the answer to that? Certainly not the fruit merchants in the street outside, because they were only interested in the decorative effect of their displays, and the fact that they wouldn't be knocked over by the vibration of a passing horse and cart. In fact, there was probably no one alive who would find the answer of any practical use; not even the army, who might take an interest in stacking canon balls. Moreover, even if someone could make a few quid out of the answer, I'm pretty sure that wouldn't have added one jot to the effort that Mr Kepler put into the problem. Instead, I'm certain that he saw it as a beautiful problem for it's own sake: it's simple to state, it's very pictorial, and nobody knows the answer. To a cultured man of reason, that also raises a moral question for society: surely someone must know the answer to so simple a problem? No? - then shame on us!

Then, of course, there is the fact that everyone is hungry for honour and praise. Well, actually, most of us just want a bit of respect and empathy from our friends, politeness from strangers, and the undying admiration of our pet dog, but Kepler was (most probably - again I'm making all this up, because the sources are not very fulsome) a driven man, and would have crawled over broken glass (mathematically speaking) to win the astonishment and adulation of his fellow scientists, and so he worked hard at the problem to try to find the answer.

These, I guess, in varying proportions, are the motivations of all scientists: you might find a problem and work on it because it is beautiful, or just for the joy of exercising a talent, or to win reputation and respect from colleagues. To a certain extent, it's nice to think that the research will be "useful" - but in practice any real advance in science is going to take forty years before it gets translated into a widespread technology, if that ever happens at all (which is as rare as hen's teeth). Thus, for basic research, "usefulness" and usefulness are two completely different things - and the one with quotation marks is just a way to get money from the government. If you open the pages of the journal "Nature" you will find more "potential cures for cancer" than you can shake a stick at. Is this bad? As a scientist, everything you do will probably be of no use to anyone. As a taxpayer, your money (which you thought was going towards a cure for cancer) is actually being used to pay for personal hobbies and to propel academics up the greasy pole. OK, that's very scathing, but there is also an "and yet..." lurking under the surface: And yet, out of science, which is funded (partly) by this process, has come the foundations for technologies which have shaped the modern world, and the spiritual gift of understanding ... and even some good treatments for cancer.

So, society wants people to do useful science, but if the useable output of science is indeed so harshly delineated, with few winners and many losers, then how do you keep everyone motivated? Or do you just fund the winners? Ah! but the point about research is that it is about the unknown - therefore there is only a limited extent to which you can guess in advance what work will be useful (and history teaches us that many big advances come completely out of left field). Thus, given that society wants the new toys, and the present way of funding them looks pretty wasteful and corrupt ... could you keep ambitious people motivated in a better way? I don't know!: it's not a question unique to science, which in this respect is just a microcosm of human society. All commercial enterprises and even the church use ambition (the prospect of promotion), power (ditto) and money (also ditto) to motivate people and direct them towards a common purpose - and then hope to be judged by the usefulness of the outcome rather than the squalor of the process. So, if you have any better ideas for doing this, I'd be very interested!

For a scientist who just wants to contribute to society, there are a few comforts though: your work might be pretty (like a minor piece of art), and when it comes to usefulness of scientific work, the distinction between the elect and the damned (chosen at random of course, just as does Calvin's God) is not quite so bleak: there is a spectrum of research from pure to applied, and the applied kind is much more likely to have a (small) impact soon, while the pure researchers are gambling for higher stakes. Then there is the pleasure of handing on your hard-won experience to the next generation, so that maybe you contribute at second hand. And lastly of course, there is much peace to be gained from following a calling; and society benefits because on a Friday night you will be looking for zeros of the Riemann zeta function, rather than getting drunk and having a fight in Bedford town centre. In fine, science is not a profession which rewards humility, but logically, that's usually what's called for.

orry, that was quite a digression! - but what I was trying to get around to was the question of why should we be interested in filling up the universe with oranges? The answer is that (at the time), no one was: but it was a beautiful problem worthy of study for its own sake, just like a beautiful sculpture is worth carving for its own sake. Also, Kepler was a wealthy man, so didn't have to ask the taxpayers for a penny.

Nevertheless, as is usual with especially elegant or elemental problems, they tend to acquire a life of their own, and the attention which Kepler gave to it started a curious trickle of ideas, which, gathering strength from many other sources, has led to the production of artificial opals, computer simulations of the strength of soils (which you might want to build houses on), crystallography, the analysis of protein structure, molecular biology, and of course ... many potential cures for cancer.

Removing tongue from cheek (actually, a lot of that last sentence is true), how did Mr Kepler get on with his problem? Does the orange merchants' method give the densest possible packing of hard spheres? Well, after a great deal of hard work, Kepler was able to prove ... nothing at all - and neither could any of his contemporaries.

This was not an age of intellectual pygmies, and if you list the greatest mathematician of each generation thereafter: Descartes, Newton/Liebniz, Laplace, Euler, Gauss, Riemann, Poincare/Hilbert, Kolmogorov, ...(who took on the mantle from Kolmogorov, I wonder?)... you see that some great names (and many lesser) will have looked at this and failed to find an answer. In fact, it is not until 150 years after Kepler stated the problem that Gauss made the next step, by showing that for any crystalline arrangement of spheres (i.e. with a repeating unit cell) then Kepler's answer was indeed best.

Why that extra weasel word about any "crystalline" arrangement? Because locally you can do slightly better than Kepler - imagine making a little tetrahedron out of four touching spheres - it's just that you can't make those little local arrangements, which are denser, fit together to fill up all of space. In fact the problem is so frustratingly intricate, that the first proof that Kepler was indeed correct was given in 1997 by Thomas Hales. Not only that, but it look a large team of mathematicians twelve years to check that the proof contained no errors - so the answer really dates to 2009 ... and the orange sellers were right after all! I hope their shades are pleased.

suppose that is either a story of great endeavour and persistence in the face of impossible odds, or it is an account of monumental folly. Fortunately it's not for me to judge. However, it is really only the first part of the introduction in this essay; but from now on, I promise there will be more ideas and less polemic.

The next part of our story is about something more elemental - quite literally one of the four original elements: water. In fact, it is about liquids and solids more generally. Why does liquid water flow around while ice is a hard, crystalline material, yet they are both made out of the same molecules? And if water is made out of a bunch of molecules, why does it flow like, well, water, and not like sand? Hmm ... well, there is a great deal known about this now, and I'm barely going to brush the surface here, but one thing which made it difficult for people to get a handle on these questions, is that molecules are just so tiny. You can't see them - and what was known about them, until very recently, was all deduced by very indirect methods. In fact, until well into the 19th century, there were respectable scientists doubting the existence, let alone the nature, of molecules. Admittedly, that was for some rather extreme epistemological reasons due to Ernst Mach (who was Einstein's supervisor, and also doubted the existence of the force of gravity)... but still, it gives you pause.

The main arguments for the existence of molecules were that they were a good explanation for the symmetry of crystals, that invisible things bouncing around was a good explanation for Brownian motion (the irregular jittering motion of pollen grain that you can see under a microscope), and that a theory based on them could also be used to predict the expansion and pressure of gases. That's actually quite a heap of good reasons, but molecules were still very difficult to do direct experiments on, because, as I've already said, they're just too damned small.

This all changed however, when someone had a rather bright idea, some time early in the 20th century: Suppose you don't look at molecules, but you look at those pollen grains instead: they are jiggling around, just like the molecules are supposed to be doing (although not so quickly, because they are bigger) - but you can actually see them, and so you can study them in detail, and so maybe understand a lot more about solids, liquids and gases. From what was known about molecules at the time, it was believed that the key things which determine how they behave together (for example if they form a gas or a solid), are firstly that they jiggle around with Brownian motion, and secondly what interactions they have: do they slightly repel each other, or are they slightly attracted together. By the time this experiment with pollen grains (or rather little artifical grains) was proposed, people were learning how to make all kinds of particles (for example to manufacture inks and paints), and could more or less choose the interactions bespoke. As it happens, if you choose the particles to be very slightly attractive (but not too much), and then suspend them in water, you can arrange for the suspension to split into two layers, just like a liquid with a gas above - and if you cool it down, or alter the interactions a bit, you can make crystals of particles appear, which look for all the world like little opals (which in fact is exactly what fire opals are!). So, this suspension of tiny man-made particles in water can behave just like a "real" molecular substance - like water itself for example - except that you can see the "molecules" and play around with all their properties.

That's a beautiful idea, because it opens a window into the mysteries of matter, and gives you a whole toolbox to tinker around with what you see. The field is called "colloid science", and a little corner of it is what I cut my researchers' teeth upon.

here is a venerable tradition in science, first spelled out clearly by the Hungarian mathematician Paul Erdos. He said that the best way to make progress is to ask the question "What is the simplest thing I don't know the answer to?". Peter Winkler in his wonderful book "Mathematical Puzzles: A Connoisseur's Collection" describes the kind of conversation which can be heard in mathematics departments across the land, along the lines of "Hey, I've been thinking about this problem: do you know the answer?", "No: but I don't know the answer to this, which is even simpler", "Oh yes? - I don't even know this!".

That, in essence, is theoretical research. Strange, but true. Anyway, when the colloid scientists got thinking down this line, then the question they came up with was to ask about the behaviour of a bunch of particles where there aren't any interactions between them at all! Suppose you have a colloid which just consists of lots of equal hard spheres, which neither attract nor repel, but which jiggle around with Brownian motion. What does that system do? Is it a solid, a liquid, or a gas? Or something else? It's like a kind of primitive matter: the simplest kind you could imagine - so that if you understood it, then you have a point of reference to explore more complicated cases.

Surprisingly, the behaviour is both strange and intricate, and opens up all kinds of curious questions, which are still being looked into... To begin with, if you wait long enough for everything to equilibrate, then the states of matter you get do not depend on the size of the particles (provided they are small enough that their Brownian motion keeps them happily suspended in the water), and it also does not depend on temperature (provided the water they are jiggling around in does not freeze or evaporate). After experiments and computer simulations (which date back to the 1950's, and are some of the first computer simulations of complete materials ... and which later led on to molecular dynamics (which really is used to research drugs for cancer) ), people figured out the phase diagram, which is simple, but quite unusual. The properties in fact depend only on the fraction of space which is filled up with the spheres. If you have only a few (to be precise, if the spheres take up less than 49.4% of the room, then they just float around like a gas: disordered and random. As you put more spheres in though, you start to get crystals forming. These are packed just like Kepler's oranges, but with some space between the particles (so that inside the crystal, the density is not the full 74.1%, but a bit lower). If you really make this system out of (for example) micron-sized latex beads in some tranparent liquid, then you see these little crystals forming like tiny coloured opals, and falling to the bottom of the test tube (as they are a little bit denser than the rest of the liquid). You can see a back-to-front picture of this here. As you put in more spheres, you get more and more of these crystals, until when you reach 54.5%, then all of your test tube is filled with tiny opals. After that, you can still put in more and more, and the crystals themselves become denser and denser, until (theoretically), the whole system has 74.1% (Kepler's limit) of spheres, and all the crystals are closely packed, with no gaps between the spheres they are made of, and then you can't fit any more spheres in at all.

OK, that's what happens if you could wait forever - but in reality, there are children to be taken to school, deadlines to be met ... and so things are not quite so neat. Well, in point of fact, everything works like clockwork and just like the theory until somewhat above 54.5%, when the system really is full of opalescent little crystals. After that, you have to remember that when you make these systems for real, you have to stir in more and more of your little hard spheres, which means that you always start from a disordered state, even if the system wants to crystallize. The thing you notice is that the mixture gets more and more viscous as you stir in more of the hard spheres. That has two effects: firstly you need a lot of patience to stir them in properly, but the other effect is that it takes longer and longer for the little crystals to start forming. Eventually, they no longer form spontaneously in the mixture and sink to the bottom, but they have to find a little bit of roughness at the edge of the glass to get them going - just like the bubbles in a glass of champaigne (or coke, if the research grant hasn't come through yet) on the surface of the glass, rather than magically appearing in the middle of the drink. Also, when the crystals do get started, they grow very slowly and get quite big. So, you see a difference in your experiment, with maybe a few big crystals (still growing very slowly), rather than piles of tiny crystals at the bottom.

Eventually, the mixture gets so viscous that the crystals stop forming altogether, and it looks just like the disordered "gas" (of spheres) that it was when there were only a few of those spheres present. Finally, when you get up into the low 60's percent of spheres, then it's just so viscous that you can't even stir it any more at all, and you have to stop. What has happened is that the mixture is becoming not a crystal, but a glass - which is a bit like a liquid frozen in time.

ell, that's a lot of statistical physics all hidden away in a very simple system (hard spheres wiggling around because of thermal motion). The funny thing is though, it has pointed us to an even simpler question (remember Paul Erdos), which will bring us back with a bump into the world of everyday things. The question is about that curious, disordered, glassy material that you end up with as you add more and more hard spheres to the system described above: what is its nature; why is it so viscous? Even more simply, suppose you forget about all that brownian motion, and go back to thinking about oranges. Suppose, in fact, that you don't want to make nice ordered packings like the orange-sellers, but just want to pack the oranges (or rather smooth, hard spheres) together in a completely random way, but nevertheless get as many of them into a bag as possible. What is the best way to do that? - and (if you had a very large bag) what fraction of the contents would be oranges, and what fraction would be the space between them? That sounds like a nice question, but there is a subtlety: does it really make sense as a question at all? It is an irritating fact that in research, you may ask yourself something that seems, for all the world, to be a really good question ... only to find that it somehow slips through your fingers and becomes nonsense as soon as you stare at it hard enough. That's one of the (many) ways that research can get very frustrating.

To see what I mean a little more clearly, we need to turn briefly to the two-dimensional case, which is actually pretty easy. Let us ask Kepler's question about two dimensional oranges (maybe "coins" would be a better word): What is the densest possible packing of equal sized discs on a plane? That's easy: it's just a honeycomb arrangement, which covers 92% of the plane. Moreover, it's (relatively!) easy to prove that this is the best you can do. Great! Now, what about the densest random packing of discs in the plane?: Suppose you get a lot of coins (government grant needed here) and scatter them on a huge table, all face down on the wood, and then gather them together with your arms, so that they form a closely packed layer, but only one coin deep. You then jiggle them round until you get the densest random packing possible. What does it look like? Well, when you do this with real coins (or smooth discs in a computer) you find that they don't really form a random arrangement: they very easily form up into little crystalline regions (honey-comb like) and the more you squueze and jiggle them, the bigger these regions become. So, there isn't really a well-defined random packing of equal discs on a plane: there are just lots of partially crystalline regions, with so-called grain-boundaries between them.

OK, then in that case, maybe the question doesn't make sense in three dimensions (with oranges) either? In point of fact, this is still somewhat a subject for debate, but for all practical purposes, it really does seem to be the case that if you put lots of ball bearings in a bag, you can only squeeze them down to a particular maximu density (which is about 64% of the volume), and this state is reproducible, no matter how many times you try it, and it doesn't consist of little crystalline bits, but is genuinely pretty random. This system was first studied in depth by Bernal, in the 1950's: he took a thousand ball bearings and squeezed them together in a leather bag so that they were as close packed as possible, and then poured in paint. When the paint had dried, he had a great big mass of ball bearings all glued together by paint. But - he could pull them apart one by one, and because there were rings of paint where each pair of ball bearings touched, he could measure all the angles between the "bonds", and show that everything was genuinely random, and not just made up of little Kepler-crystal regions. When I say that Bernal did all this, I mean of course that his PhD student did it. Life is hard at the bottom of the academic food chain.

So, we now know that hard spheres really do pack together in a random way, when you squeeze them together quickly (rather than squeezing really really slowly, and waiting for Brownian motion to make them crystallize). This is called the "Random Close Packed" (RCP) state, and is another of science's wonderful fairy stories: there are no such things as exactly spherical, exactly hard, smooth, equal sized spheres in nature. However, many systems (like desert sands for example), are quite good approximations. Then, if you know a lot about the properties of the RCP state (for example it's filling fraction (64%), the amount of surface area, the size of the channels through it...) then you immediately have a lot of approximations that apply to the real desert sand (how much a cubic metre of it will weigh, how fast rain (or oil) will flow through it ...).

ell, that's a very long introduction ... but what have I been up to? As usual of course, I've just been trying to fill in a tiny corner of what is a very big story - but it keeps me happy, and doesn't do anyone any harm. The question I have been looking at, is about that random close packed state - but suppose the spheres aren't all the same size. That's much more likely to be the case in real life. Of course, in real life (sand or soil for example), the particles won't be perfect spheres either! - but one step at a time, eh?

Of course, I wasn't the first person to think about this problem: even in the 1960's, experimentalists were mixing two different sized spheres together, and measuring how much space they filled up. Already, the problem starts to get a bit complicated, becasue you have two different sizes to choose (although it's only the ratio of diameter that is important, because if you magnify a sphere packing, it doesn't alter what fraction of space it fills), and you also have to choose how much of each of the two sizes to put in. Therefore, for each size ratio and ratio of amounts, there will be a maximum random packing fraction - so you have a whole family of problems - and that's just with two different sphere sizes. The question I wanted to ask was "suppose you give me any mixture of any number of different sphere sizes, what is the maximum random packing fraction?". In the literature, no one has got much beyond two different sphere sizes (and they have only studied that experimentally; there is no simple theory), so it is a big ask.

To get a little mathematical (and feel free to skip the next two paragraphs), what I am trying to do is approximate an (unknown) "functional". Let me explain that jibberish a little more clearly: Mathematics is the study of mathematical animals, and from a simple applied mathematician's point of view, there are really only four kinds of mathematical animal. Firstly, there are "functions". A function is an animal which eats numbers, and produces numbers at the other end. For example, the function f(x)=4x, which, when you feed it a "2" will produce an "8" at the other end; or, if you feed it a "10" will give you back "40". Great! Well, another kind of animal is one which eats functions, and craps out other functions. This type of mathematical animal is called an "operator", and a good example of an operator is the differential operator (d/dx), which tells you the slope of a function at each point. You can, for example feed the differential operator a function like f(x)=sin(x) + 3x, and it will give you back another function; in this case g(x)=cos(x)+3. The third kind of animal we often encounter is a "functional": this animal eats functions, and dumps out numbers.

Chemists will be familiar with the energy functional of quantum mechanics, which eats the wavefunction (a kind of function), and gives you back the energy (a number). Definite integration (between fixed limits) is another good example. If you are following so far, then I have a puzzle for you: what is the common name for an animal which eats numbers, and craps out functions? Anyway, that's it! - that's all of applied mathematics. In principle, you could imagine other animals which, for example, eat functionals and produce operators (which you might call a functionalarator or some such), or other even weirder beasts. However, even though such wild creatures do indeed lurk about on the dark fringes of mathematics, none of them has ever been successfully domesticated, so we don't let them trouble our peaceful dreams. The point I want to make is that if you are trying to approximate a function, then the task is quite straigtforward: you could write it as a sum of fourier components, or build it out of standard functions, and with a few free parameters thrown in you can pretty much make it any shape you want. The problem for the packing of spheres is that we want to approximate a whole functional, which is in general a much trickier beast, and for which there really aren't any standard approaches.

o, back to the chase. How do you go about looking for an answer? In this case, there are lots of possible approaches: you could start thinking about clusters of spheres, and all the ways you could put a cluster together out of the different sizes available, or you could try to think about the very dilute case, and then try to write down slight corrections as the spheres get closer together, or ... probably lots of other things too. In fact, there is work being done in these and other directions already, but nothing really simple and useful has come out of it yet, so these are probably routes where you need to be very clever, which is not one of my strong points. I should also say at this stage, that you always have the option of packing the spheres together in a computer simulation: that would also count as a theory, and would be sure to give you a good answer (provided you get the program right). In fact, once I'd come up with a simple theory, I teamed up with a rather famous physicist (Rob Groot), who checked it using just this approach - and in the process found some other curious properties of random close packing - but that comes later. The real problem with computer simulations is that you need to pack together a few thousand spheres to get anything like a representative system, and that means that even on a modern desktop computer, it will take you several hours to work out the answer. That's better than ordering umpteen dozen sizes of glass beads from a supplier and doing the experiment, but not much better. So, it would be really great to have a quick way to get the answer. That is, if there is anyone interested in the answer at all.

How did I go about looking for a completely different approach? Well, firstly by not thinking about it at all. The odd, cobwebby corners of my mind are pretty cluttered up with science questions and strange thoughts which have caught my fancy over the years (I have a sense of beauty in science which I have never been able to explain to anyone, including myself; but when I see a problem which I like, I know it immediately, and so tend to accumulate a list of odds and ends worth thinking about). The interesting thing about consciousness is that it can be focused absolutely on one thing, be that an immediate threat like a snake in the grass, or a piece of arithmetic when you are doing your accounts. On the other hand, it can be broad and inclusive, like when you are sitting on the beach and drinking in the entire landscape. Then again, it can spread through your memories and touch off odd connections and emotions from all over the place, and the best time for it do this hard yet effortless work, is when you are happily doing nothing at all. I have a friend who claimed he did his best work when he was asleep, and I think the same principle applies generally; although being awake is just as good for me: sometimes I wake up with new ideas, and sometimes they drift in when I'm staring out of the window. In either case, it is a marvellously puzzling phenomenon: I might think that my head is full of a jumble of old memories, and I will never be able to sort through them in a month of Sundays even if I want to, but then of a sudden, I see that someone has already done that sorting, and I can pull out an idea more easily than if it were in a card catalogue. The same principle does not work for my desk.

If you study animal behaviour (ethology), then I'm told that the first thing you need to do is construct an "ethogram" for your chosen animal. Basically, you hide somewhere comfortable, and watch and record everything it does: "05:30 Animal asleep....05:35 Animal still asleep..." (lots more of that), then "10:30 Wakes up, stretches" (why was I here at half past five?) "10:31 Animal picks nose, blinks" and so forth. It sounds very dull, but it's in some sense the foundation stone for understanding that creature. I say it's very dull, but I'd love to give it a try, partly because of an observation made by the great geneticist J. B. S. Haldane: When, towards the end of his life, he moved to India, he noticed a queen wasp had started to build a nest on his porch. Instead of clearing it away immediately, he started a little project in which he recorded the comings and goings of each wasp (more and more as the colony grew), minute by minute, and recorded the precise way the nest was built up, and how many insects they brought back, and so forth. At the end of two weeks, the nest was getting pretty bothersome to any visitors he had, but he found he was unable to get rid of it. The comment he made was that if you study any animal for eight hours each day for two weeks or so, you come to see it as a complex and moral being, and any decision to kill it is a moral decision, of the same nature as (although less in magnitude than) a decision to kill a man.

I think that kind of spritual experience (not to mention the strange things I might learn about a hitherto overlooked fellow traveller) would be ample reward for two weeks of my time. Unfortunately, I spend most of my day in a rather sterile office; and although, in principle, I could spend eight hours a day staring at my colleague across the desk and writing down everything he does, I suspect this might have unintended consequences. I'm also not entirely sure that the spiritual experience wouldn't work in reverse, and I will find that I have the same moral attitude towards him as I would to a wasp's nest. Anyway, be that as it may, the reason I'm talking about this, is because if you actually look at an ethogram for some animal, like a goose or a rabbit, you notice something rather strange: Most of the time, they ... well, they do ... nothing at all. OK, a fair chunk of the time they are asleep, but even when they are awake, they spend a remarkably large amount of time just sitting around. I guess it's hard to say from the outside whether they are "digesting" or "observing" the occasional squabbles going on around them, or just "resting" or even "philosophizing" - but the fact remains that to all intents and purposes, they are not doing a great deal. Contrast that with the modern working environment, when we are expected to at least appear to be busy every moment of the day. I think that attitude is immensely damaging to enjoyment and creativity, and at the same time it seems to be inescapable. I guess that is because in most work places you are there under some kind of duress (sacrificing your precious time in exchange for necessary money), and even if you enjoy the job in principle, and would do it for a hobby given the chance, you are being judged by managers (and maybe even colleagues) who don't. Human society is pretty screwed up. I have a theory why this is... but I probably need to get back to the main story. The moral I would like to draw from all this though, is that staring out of the window can be one of the best things to do, for all kinds of reasons.

So where was I? - oh yes, staring out of a window somewhere in Dorset and thinking, not very hard, about not much in particular. I like to work on things which are pictorial, or at least give an excuse to draw a pretty illustration, and fractals are always great for that kind of thing. So, one of the things which was drifting through my head was what kind of fractal pictures I could think of drawing. In the spirit of Paul Erdos, I wondered what would happen if you drew some straight lines on a piece of paper; let's say you just draw some straight lines end-to-end, all left-to-right across a piece of paper. Suppose, in fact, you had a whole set of lines, one which is a millimeter long, and another two millimeters, the next three ... basically, all whole numbers, and one of each. Suppose you just started fitting the slightly smaller ones in the gaps between the biggest ones, pushing the big ones out of the way, and then the slightly smaller ones into gaps which are then left. That kind of feels like you are putting structures inside structures, so maybe it would make a nice self-similar fractal pattern which I could draw some time?

I then started thinking that this was a bit like a strange version of random sequential adsorption, where you drop things down at random onto a surface or a line, and stop when you can't find a place to fit the next thing you want to into the space. It's a model for the way polymer molecules stick to surfaces, and has been studied a fair bit in the literature. Then, that got me thinking about packing of different sized spheres, because a few years before I had tried to solve this problem and made very little progress. An interesting way to think about a problem is often to look at extreme cases, when some parameter is either very large or very small. This sometimes brings the structure of the problem into sharp relief, and makes things simpler. So, suppose you are trying to pack two very different sizes of spheres: one tiny, and the other huge. Then, there are two different cases: if you have lots of the big spheres, and very few of the tiny ones, then the big spheres will pack together just as though the little ones didn't exist, and then the little ones will just happily rattle around in the gaps. On the other hand, if you have loads of the little ones, and only one or two big ones, then the little ones will random close pack as a kind of sea, and there will be a few isolated big particles floating around like holes in a Swiss cheese (with the cheese being made out of the little ones).

That got me thinking about packing rods onto a line. Obviously if you can move the rods around, then you can always fill up the whole line: you just place the rods nose-to-tail with no gaps, and the packing fraction is 100%. So, packing of (even different sized) rods on a line is completely trivial. However, suppose you said that the rods hit each other before they touch. That sounds like nonsense, but imagine you just make this arbitrary rule: you can move the rods around, but you have to leave a gap between them. That way you get a packing fraction less than 100%. Now (and this is the central idea), suppose that if you have two different-sized rods, then make the gap to be some fraction of the length of the shorter rod. We are just playing games in our minds now: it doesn't have anything to do with real life. OK, well with this rule, if I have two very big rods next too each other, then there is a big gap between them when they are squeezed as close as this rule allows - but I can still fit a small rod into that gap, and not break the rules of the game. In fact that small rod could even have freedom to rattle around in the gap. But wait a minute: that simple rule then feels a little bit like packing of spheres: it has captured the essence of rattlers, which is one of the difficulties with sphere packing. However, it is in one dimension, so you'll never get the right three dimensional answer for packing fraction if you just call a sphere a rod and play this game. So, it's sort of a toy system, which has a little bit of the feel of the real problem without being very useful for solving it. I then thought a little bit further, and figured out that the packing fraction you are going to get in this toy problem will depend on the order of the rods on the line, as well as on what all their lengths are. So, if you wanted to figure out the maximum packing fraction, you need to know the best order. How hard it is to fnd the best order?

Reading over that now, I realize this is all pretty weird stuff to be thinking about while on holiday - but then I wasn't really thinking about it, I was just chewing the cud while enjoying the sunshine in the garden. Anyway, we still have the question of how hard is it to find the best ordering out of all possible orders you could line the rods up in? - and it's like a crossword puzzle: when you get a clue, it's hard to let it go. Well, if you have two things, you can put them in a row in two ways (2=1 x 2). If you have three things you can do it in six ways (6=1 x 2 x 3). If you have ten things, there are 3628800 ways - so it rapidly becomes a question of finding a needle in a very big haystack. But then I thought "there's a trick!". Remember the idea about putting smaller rods into the gaps between big ones, well, that should give a pretty good way to find the densest packing of the rods, because if you put the biggest remaining rod into the biggest available gap, then this cuts down the number of rattlers to an absolute minimum. So, we've got a solution to this silly little one dimensional packing problem. As far as I was concerned, without writing anything down on paper, that was a pretty good day's work (in fact a very good day's work) and it was all just playing with simple ideas in my head while doing nothing whatsoever. Still, it had nothing much to do with the real sphere packing problem, so I mentally tucked it away in the attic to maybe be written up some time in the future.

A few months later, the sphere packing problem came up at work for real again, and then I dug up my old thoughts about rods and looked at them in the cold hard light of office phosphorescent tubes. Hmm. The main problem in using the one dimensional toy to approximate three dimensional spheres is that none of the numbers work out properly: If you have three big spheres and one little one, and replace these with three big rods and one little one, then the different regimes don't match up. You could be in the Swiss cheese regime for the spheres, and the rattlers regime for the rods for example, so the prediction is not going to even be roughly right. Still, that's a technical difficulty, and maybe it's possible to cook up a technical solution... and in fact you can: if you choose the number of rods according to not just the number of spheres, but also according to their size, you can force the rod system to always be in the same regime as the spheres.

Great! this is starting to look interesting! Now I have the problem by the tail and the fight can really begin. It's also time for some hard work - not just dreaming up some ideas. So, I wrote some computer code to solve this cooked up one dimensional problem, so I could compare the predictions to real experimental data for packing different sized spheres together. It took a couple of hours, but soon I had the curves, and they looked ... absolutely awful! Jagged edges all over the place; just plain craziness. The numbers weren't too far off, but they were pretty rough and ready and not much use for accurate predictions. Neverthless, that was still a good day's work because I had fought the problem to the point where it had to show it's true colours. Still, I needed another idea to take it any further, and ideas come and go in their own sweet time.

Why am I telling you all this detail about what I was thinking? I don't know really, but people do sometimes ask me what I do all day as a theoretical physicist? So, this is the answer: I'm hunting interesting problems, and I'm doing it through little bits of insight, alternating with bouts of hard work to see if the insight is useful. Usually that hard work tells me that I'm an idiot - but just occasionally, it tells me that I am an idiot but there mght be a kernel of truth in what I'm thinking. This time round, that's exactly how it felt: I had a theory which captured some important aspects of the problem, but it was pretty rough around the edges. Also, apparently no one else in the world had anything better, so it was worth investing a good night's sleep to see if I could get any further.

Because we're back in the "need for new ideas" phase, I don't mean "miss a night's sleep" (that's for the "hard work" times) - I mean go to bed early, get up late, and see if the morning brings fresh council. Well, when I woke up the next morning, it seemesd to me that the current theory was just too sharp and crisp: I'd start off with a bunch of (for example) two inch diameter spheres, and another bunch of one inch spheres, and then I'd translate these into a bunch of two inch rods and a bunch of one inch rods to go into my rod packing theory. Well, suppose I made the translation a bit more blurry: each sphere is replaced by a distribution of different rod lengths. That ought to smooth out some of the jagged edges in the prediction. Fine, but what distribution to choose?

The obvious choice is to use a uniform distribution, so, for example if I have a one inch sphere, I replace it with a combination of a 0.5 inch rod, a 0.6 inch rod, 0.7 ... all the way to a one inch rod. That's clear and simple, because you are replacing each sphere with a kind of rectangular distribution of rod lengths, between half the sphere diameter and the whole sphere diameter. However, it introduces another parameter into the model: do I start with a 0.5 inch rod, or a 0.3 inch rod? Basically, how broad is the distribution of rods that I use to replace each sphere? Also, it's not very elegant: there is no motivation for this choice rather than any other. In this case, I just need something to work, as I'm being paid for my time, so I'm not too concerned with elegance, but that extra parameter is ugly, because it reduces the predictive power of the theory. OK, what else could I try? Well, after a bit of thought I liked the idea of replacing each sphere with a triangular distribution of different rod lengths, starting at zero, and going up to the whole sphere size. That doesn't introduce any new parameter, and it represents a nice compromise between smoothness and crispness in the rod distribution: the original sphere distribution is still emphasized, but it's a little bit blurred out. It feels about right, and obviously I have a whole range of other options to play around with if this doesn't work. The other really nice thing, was that I had a kind of rough-and-ready justification for this choice: if you place one sphere on top of another, with a completely random offset, then the vertical separation of their centres follows this distribution. There is no reason why that should be important, but it's nice to have some kind of picture in your head. OK: back to the hard work phase! I coded up the new theory at work in the morning, and the curves looked much better: to my untrained eye, they looked quite like the limited amount of 1960's data I could lay my hands on, so I grabbed a colleague, (Rob Groot) who is a fairly famous physicist, and can write computer code in a jiffy to simulate the full sphere packing problem. He was sufficiently interested to check it out, and generated a lot of data for packing of spheres of two different sizes (his code needed a few hours to run for each point, but the new theory is more or less instantaneous). Eventually, we could put the two predictions side by side, and ... perfect match! Neither of us could believe it, so we both went away and checked for errors. Rob also coded up the theory to see if he got the same answers. Then we tried other things: mixtures of three different size spheres, broad distributions of sizes, narrow distributions, and everything worked! What the hell was going on? There was no reason on earth for this cobbled together theory to be anything like as accurate as that.

In fact, I still don't know why it works so well, but one other idea occurred to me: I was reaching out into space to try to grasp what was going on, and as I reached out, an idea dropped into my hand that you could replace all the ugly cobbled together bits with a simple picture: If you take the sphere size distribution you are interested in, and scatter it randomly through space, and then draw a straight line through that scatter, then some parts of the line will be inside a sphere, and some parts won't. If you take the lengths which are inside a sphere, and call them each a rod, then you magically get exactly the rod distribution I had been describing above: both nasty cooked up aspects fall out of the same picture. It seems natural and compelling - although to be honest, the whole theory is not formally a "solution" to the sphere packing problem, it's just a model which shares some common features, and happens to behave in a very very similar way.

Anyway, that's the story of this particlular little theory. The only thing left was to write it up for a journal. In the course of his simulations, Rob had also uncovered some other intriguing properties about the random close packed state, which no one had noticed before (he's famous for a good reason), like the fact that the packing density is not quite constant, but depends slightly on things like sphere size and friction in a specific way. Anyway, with all these results, we wrote up a journal article. To do that of course, we told them exactly what didn't happen: we made up a story about how we started with the sphere packing problem, and abstracted from it a list of essential properties which a theory should have, and then deduced the simplest possible theory which satisfied those criteria. All guff. Of course, we could have done it this way, but if I had actually written all the things about staring out of the window looking for pretty pictures, the editors would not have been very impressed. The moral from this part of the story is never to believe the neat stories in scientific papers: they are always post-rationalizations. That doesn't mean the conculsions are wrong, it just means that the way they were obtained the first time was a lot more messy and frustrating than the neat account given (which was probably more like how they would have done it a second time, if given the chance).

So, where are we with this story that started so many hundreds of years ago? It's still an active field of research, capable of throwing up surprises, and full of odd alleys and dark corners which challenge the best minds. It's also a field where even an ordinary researcher like myself can (if we're very very lucky) find a piece of the frontier of knowledge to push back a few inches. However (for me at least) it is a problem beyond my abilities, so any advance I make feels very much like a gift rather than something earned. It is one of my serious character flaws, and something which has ruled me out of a standard academic career, that I'm only really interested in problems which are too difficult for me to solve (that, and lack of talent, of course). The process of reaching out almost over the edge of madness for new ideas outside the bounds of my reason is what I look for ; it is a kind of spiritual process of emptying the mind, and needs calmness and stillness (and a window with a nice view) to work at all. I think that all scientists treasure creative moments like that (although most people take a more balanced view of creativity versus practical work), and therefore the main thing I want to express in this essay is the ultimate importance of doing nothing very much - but doing it well.

ACKNOWLEDGEMENTS AND ATTRIBUTION

The picture of the oranges at the top of the page is repeased under the GNU Free Documentation Licence and was produced by Ellen Levy Finch.