Johannes Kepler is rightly best known for his laws of Planetary Motion that paved the way for Newton to write Principia. Hidden within his illustrious CV, however, is a publication that had a rather more whimsical earthbound ambition. Two years after publishing the first part of Astronomia Nova in 1609, Kepler published a short 24-page paper entitled De nive sexangula – On the Six-Cornered Snowflake. It is a beautiful example of a curious scientific mind at work. In the dark December of 1610, Kepler was walking across the Charles Bridge in Prague when a snowflake fell on the lapel of his coat. In the freezing night he stopped and wondered why this ephemeral sliver of ice possessed a six-sided structure, in common with all other snowflakes, notwithstanding their seemingly infinite variation. Others had noticed this symmetry before, but Kepler realised that the symmetry of a snowflake must be a reflection of the deeper natural processes that underlie its formation.
‘Since it always happens when it begins to snow, that the first particles of snow adopt the shape of small six-cornered stars, there must be a particular cause,’ wrote Kepler, ‘for if it happened by chance, why would they always fall with six corners and not with five, or seven?’ Kepler hypothesised that this symmetry must be due to the nature of the fundamental building blocks of snowflakes. This stacking of frozen ‘globules’, as he referred to it, must be the most efficient way of building a snowflake from the ‘smallest natural unit of a liquid like water’.
To my mind, this is a leap of genius and a tremendously modern way of thinking about physics. The study of symmetry in nature lies at the very heart of the Standard Model, and abstract symmetries known as gauge symmetries are now known to be the origin of the forces of nature. This is why the force-carrying particles in the Standard Model are known as gauge bosons. Kepler was searching for the atomic structure of snow before we knew atoms existed, motivated by the observation of a symmetry in nature – the six-sided shape of all snowflakes. The inspiration for this idea, which is way ahead of its time, came from a peculiar source. In the years leading up to the publication of De nive sexangula, Kepler had been in communication with Thomas Harriot, an English mathematician and explorer. Amongst multiple claims to fame, Harriot was the navigator on one of Sir Walter Raleigh’s voyages to the New World, and had been asked to solve a seemingly simple mathematical problem. Raleigh wanted to know how best to stack cannonballs to make the most efficient use of the limited space on the ship’s deck. Harriot was driven to exploring the mathematical principles of sphere packing, which in turn led him to develop an embryonic model of atomic theory and inspire Kepler’s consideration of the structure of snowflakes. Kepler imagined replacing cannonballs with globules of ice, and supposed that the most efficient arrangement creating the greatest density of globules was the six-sided hexagonal form he observed in the snowflake on his shoulder. Kepler also observed hexagonal structures across the natural world, from beehives to pomegranates and snowflakes, and presumed that there must be some deeper reason for its ubiquity.
As I write it has begun to
snow, and more thickly
than a moment ago. I have
been busily examining the
little flakes.
Johannes Kepler
‘Hexagonal packing’, as Kepler referred to it, must be ‘the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container’. This became known as the Kepler Conjecture. It took almost 400 years to prove Kepler’s conjecture, and this required the help of a 1990s supercomputer. Despite the time lag, Kepler’s work had a more immediate impact, inspiring the beginnings of modern crystallography that led eventually to the discovery of the structure of DNA. What a lovely example of serendipity coupled with curiosity and a sprinkling of genius; from cannonballs to snowflakes to the code of life.
As for Kepler’s original thought on that frozen bridge, he never found the connection between the underlying structure of his ice globules and the hexagonal symmetry of snowflakes. Even though he realised that the regular patterns must reveal something about the shape of the building blocks of snowflakes and the details of the packing, he couldn’t explain the ornate complexity or the flatness of the structure. Instead he acknowledged his failure with the good grace of a true scientist: ‘I have knocked on the doors of chemistry’ he writes at the end of his paper, ‘and seeing how much remains to be said on this subject before we know the cause, I would rather hear what you think, my most ingenious man, than wear myself out with further discussion.’
Three and a half centuries later, Japanese physicist Ukichiro Nakayara made the first artificial snowflakes in a laboratory. Writing in 1954, he describes a process that begins not with the snowflake itself but with smaller substructures called snow crystals, which are in turn built up from collections of ice crystals – the globules Kepler was searching for. The hexagonal packing that Kepler suspected to be the origin of the snowflakes’ symmetry begins with the formation of these ice crystals, when water molecules link together in a hexagonal structure via hydrogen bonds. Hydrogen bonding occurs because of the structure of the water molecules themselves, with a greedy oxygen atom hungry for electrons grabbing them off two hydrogen atoms, forming covalent bonds that lock the H2O molecules together, leaving a residual positive electrical charge in the vicinity of the two protons and a negative charge in the vicinity of the oxygen. This slight separation of charge in the water molecules allows them to bind together into larger structures through the mutual attraction and repulsion of the electrical charges, just as an electron is bound into its position around an atomic nucleus. The entire configuration, including the structure of the oxygen nucleus and the single protons that comprise the hydrogen nuclei, can be predicted in principle by the Standard Model of particle physics. Yet the details of any particular snowflake are beyond computation, because the seemingly infinite variety reflects the precise history of the snowflake itself. Once ice crystals form as agglomerations of water molecules held together by hydrogen bonds, they cluster around dust particles in the air, building on their underlying hexagonal symmetry to form larger snow crystals. As the crystals begin the long journey down to Earth they join in ever-larger, more complex combinations, shaped by endless variations of air temperature, wind patterns and humidity into myriad unique forms. The symmetry is all that remains of the simplicity, and it takes a careful and patient eye to see the endless variation for what it is: a reflection of the complex history of the snowflake convoluted with the underlying simplicity of the laws of nature.
The most vivid example of emergent complexity, and the closest to our hearts, is life. As we discussed in Chapter 2, the origin of life on Earth has a sense of inevitability about it, because its basic processes are chemical reactions that will proceed given the right conditions. Those conditions were present in the oceans of Earth 3.8 billion years ago, possibly earlier, and they led to the emergence of single-celled organisms. The fateful encounter which produced the eukaryotic cell around 2 billion years ago looks rather more like blind chance, but it happened here and laid the foundations for the Cambrian explosion 530 million years ago. There is a bit of hand-waving going on here, though, and to make a more persuasive case that all the complexity of Darwin’s endless forms most beautiful can at least in principle emerge from simple underlying laws, one more example is in order.
Perhaps the most beautiful manifestation of the artful complexity of nature can be found in the spots, stripes and patterns on the coats and skin of living things; emergent pattern writ large across venomous striped surgeonfish, emperor angelfish, zebra swallowtail butterflies and the big cats of Africa and Asia. Everyone agrees that these patterns evolved as a result of natural selection of one form or another, and the raw material for the variation was provided by random mutations in the genetic code. But a very challenging scientific question of fundamental importance in modern biology is precisely how patterns such as these appear.
HOW THE LEOPARD GOT ITS SPOTS
Rudyard Kipling’s Just S
o story, ‘How The Leopard Got His Spots’, tells the story of an Ethiopian man and a leopard. They went hunting together, but one day the man noticed that the leopard wasn’t very successful. The reason, he deduced, was that the leopard had a plain sandy coat, whereas all the other animals had camouflage. ‘That’s a trick worth learning, leopard’ he said, taking his fingers and thumb and pressing them into the leopard’s coat to give it the distinctive five-pointed pattern. If you don’t believe in evolution by natural selection, this is the most plausible theory open to you. If you do, then what remains is to identify the mechanism by which the pattern is formed. The answer might appear to be solely a matter of genetics, but genes are not the whole story. It would take a terrific amount of information to instruct every single cell to colour itself according to its position on the leopard’s skin, and this is indeed not what is done. Nature is frugal and deploys a much more efficient mechanism for producing camouflage patterns. As with so many things in this book, I get to say yet again that this is an active area of research, and therefore exciting. The reason for the attention is that camouflage patterns on the skin self-organise during the development of the embryo, and embryonic development is of course fundamental to an understanding of biology. In the case of the leopard, it is thought, though not proven, that the camouflage is an example of a Turing pattern, named after the great Bletchley Park code-breaker and mathematician Alan Turing.
In 1952, Turing became interested in morphogenesis – the process by which an animal develops its shape and patterning. He was particularly interested in the mathematics behind regularly repeating patterns in nature such as the Fibonacci numbers and golden ratio in the leaf arrangements of plants and the scales of pineapples, and the appearance of camouflage patterns such as the tiger’s stripes and the leopard’s spots. Turing’s influential and ground-breaking paper, ‘The Chemical Basis of Morphogenesis’, published in March 1952, begins with a simple statement. ‘It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.’ These systems are known as reaction-diffusion systems, and they can produce patterns from a featureless initial mixture if the two reactants diffuse at different speeds. There is a nice analogy that describes how such a system can work.fn1 Imagine a dry field full of grasshoppers. They are strange grasshoppers, because when they get warm they sweat, generating a large amount of moisture. Now imagine that the field is set alight in several different places. The flames will spread at some fixed speed, and if there were no grasshoppers the whole field would be charred. As the flames approach the grasshoppers, however, they will start to sweat, dampening the grass behind them and inhibiting the flames as they hop away ahead of the approaching flames. Depending on the different parameters, including the different speeds of the flames and the grasshoppers, and the amount of sweat necessary to quell the advancing flames, a Turing pattern can be formed, with areas of charred grass and green areas where the inhibiting grasshoppers prevented the fire from taking hold.
… Zebra moved away to some little
thorn-bushes where the sunlight fell
all stripy and the giraffe moved off to
some tallish trees where the shadows
fell all blotchy.
‘Now watch,’ said the zebra and
the giraffe. ‘This is the way it’s done.
One … two … three! And where’s your
breakfast!’ … All they could see were
stripy shadows and blotched shadows
in the forest, but never a sign of
Zebra and Giraffe.
‘That’s a trick worth learning.
Take a lesson from it, Leopard!’
… Then the Ethiopian put his five
fingers close together and pressed
them all over the leopard, and
wherever the five fingers touched, they
left five black marks, all close together
…
Rudyard Kipling
It is thought that the leopard gets its spots in this way during embryonic development: an activator chemical (fire) spreads through the skin and stimulates the production of the dark pigmented spots (charred grass) but is inhibited by another chemical (sweating grasshoppers) spreading with a higher diffusion rate. The precise pattern produced depends on the ‘constants of nature’ of the system, such as the speeds at which the chemicals diffuse, and on what a mathematician would call the boundary conditions: the size and geometry of the grassy field in our analogy. In embryonic development, it is the size and shape of the embryo when the reaction-diffusion begins that determines the type of pattern produced. A long and thin domain produces stripes. A domain that is too small or too large produces uniform colour. In between can be found the distinctive coat patterns of cows, giraffe, cheetah and, of course, the leopard. Computer simulations of Turing patterns have been remarkably successful, not only in describing the generic features, particularly of mammalian coats, but also some of the interesting details seen in nature. For example, the mathematical models predict that it is possible for spotted animals to have striped tails, as cheetahs do, but not for striped animals to have spotted tails; and indeed, no such examples exist.
Kepler’s snowflakes and the leopard’s spots are two picturesque examples of emergent complexity: the appearance of intricate, ordered patterns from the action of simple underlying laws. Nature contains systems far more complex than these, of course: you being a case in point. But to return to the question at the beginning of our solipsistic meander, the reason that you exist, given the laws of nature, is that you are allowed to. Just as all snowflakes and all leopards’ coats are unique in detail because of their individual formation histories, so you are unique because no two human beings share a common history. But we wouldn’t read any deep meaning into the existence of one particular snowflake in a snowstorm, and the same is true for you. Our focus should therefore shift from trying to explain the appearance of humans, or our planet, or even our galaxy, to a rather deeper question: the origin of the whole framework – of spacetime and the laws that govern it and the allowed structures within it. What properties of the laws themselves are essential for galaxies, planets and human beings to exist? After all, as we’ve noted, the laws might be mathematically elegant and economical, but they do contain a whole host of seemingly randomly chosen numbers, discovered by experimental observation and with no known rhyme or reason to them – the constants of nature such as the strengths of the forces, the masses of the particles and the amount of dark energy in the universe. How dependent is our existence on these fundamental numbers?
A UNIVERSE MADE FOR US?
Our universe appears to be made for us. We live on a perfect planet, orbiting around a perfect star. This is of course content-free whimsy. The argument is backwards. We have to be a perfect fit for the planet because we evolved on it. But there are interesting questions when we look deeper into the laws of nature and ask what properties they must have to support a life in the universe. Take the existence of stars, for example. Stars like the Sun burn hydrogen into helium in their cores. This process involves all four forces of nature working together. Gravity kicks it all off by causing clouds of dust and gas to collapse. As the clouds collapse, they get denser and hotter until the conditions are just right for nuclear fusion to occur. Fusion starts by turning protons into neutrons through the action of the weak nuclear force. The strong nuclear force binds the protons and neutrons together into a helium nucleus, which in itself exists on account of the delicate balance between the strong nuclear force holding it together and the electromagnetic force trying to blow it apart because of the electrically charged protons. When stars run out of hydrogen fuel, they perform another series of equally precarious nuclear reactions to build carbon, oxygen, and the other heavy elements essential for the existence of life. What happens if the strengths of the forces, those fundamental constants of nature we met earlier in th
e chapter, are varied a bit?
There are many examples of apparent fine-tuning in nature. If protons were 0.2 per cent more massive, then they would be unstable and decay into neutrons. That would certainly put an end to life in the universe because there would be no atoms. The proton mass is ultimately set by the details of the strong and electromagnetic forces, and the masses of the constituent quarks, which are set by the Yukawa couplings to the Higgs field in the Standard Model. There really isn’t much freedom at all.
The mother of all fine-tunes, however, is the value of our old friend dark energy, the thing that is causing our universe to gently accelerate in its expansion. Although dark energy contributes 68 per cent of the energy density of the universe, the amount of dark energy in a given volume of space is actually small. Very small: 10–27kg per cubic metre to be precise. The point is that every cubic metre of our universe has this amount of dark energy in it, and that adds up! Explaining why dark energy has this small, but non-zero, value is one of the great problems in cosmology, not least because if a particle physicist sits down with quantum field theory and decides to calculate how big it should be, it turns out that it would be more naturally of the order of 1097kg per cubic metre. That’s a lot bigger than 10–27kg per cubic metre. Over a million million million million million million million million million million million million million million million million million million million million times bigger, in fact. That’s embarrassing for the particle physicists, of course, but from the perspective of fine-tuning it’s even worse. If the value of dark energy were only 50 times larger than it is in our universe, rather than somewhere else in this immensely large theoretical range, then it would have become dominant in the universe around one billion years after the Big Bang during the time that the first galaxies were forming. Because dark energy acts to accelerate the universe’s expansion and dilute matter and dark matter, gravity would have lost the battle in such a universe and no galaxies, or stars, or planets or life would exist. What could possibly account for this incredible piece of luck? It can’t really be luck – the odds are too long by a Geoffrey Boycott innings. One possibility is that there is some as yet unknown physical law or symmetry that guarantees that the amount of dark energy will be very close to, but not quite, zero. This is certainly possible, and there are physicists who believe that this may be the case. The other possibility, which was raised by one of the fathers of the Standard Model, Steven Weinberg, is that the value of dark energy is anthropically selected. Anthropic arguments appear at one level to be a statement of the obvious: the properties of the universe must be such that human beings can exist because human beings do exist. This is, of course, true, but it is fairly devoid of content from a physical perspective unless there is some way in which all possible values of dark energy, and indeed all the other constants of nature, are realised somewhere. If, for example, there exists a vast, possibly infinite swathe of different domains in the universe, or indeed an infinity of other universes, each with a different amount of dark energy selected by some mechanism from the span of allowed values, then we would indeed have a valid anthropic explanation for our ‘special’ human universe. It must exist, because they all do, and of course we appear in the one that permits our existence.
Human Universe Page 18
