What do the following have in common: river networks, honeycombs, zebra
stripes, snowflakes, ripples on sand dunes, markings on seashells and mountain
landscapes? The answer, says Philip Ball, is that all are tapestries of pattern
and form.

In essence, a pattern extends in space, while form is bounded and finite.
Together, they make “tapestries” so complex that you might well suspect that
they have been engineered by human hands. Yet, as Ball reveals in an accessible
and beautifully illustrated account of a very active area of research, both
pattern and form can be generated by simple natural processes—they are
“self-made”, no matter how intricate and complicated they seem.

Similar patterns can be seen in both living organisms and simple physical and
chemical processes, indicating that they share common characteristics. This is
the real advantage, says Ball, in using mathematics to analyse form or pattern.
By modelling a shell mathematically, you can ignore variations and concentrate
on the shared characteristics. Then you can look for the set of rules that
describe how to make a particular form—its underlying algorithm or
instruction set. Instead of having to outline every twist and turn, spiral and
knob on a tropical seashell, you can merely describe the steps required to
generate its essential shape.

You’ll discover, for example, that you need to sweep a two-dimensional
generating curve through a logarithmic spiral pulled out into a helix to
describe the surface of a spiral seashell. Sounds difficult, but with a handle
on the algorithm, you can begin searching for the physical processes governing
the formation of a shell.

These underlying physical or mathematical principles also form a link between
patterns common to disparate scientific disciplines. In the past, more emphasis
was put on the geometry of patterns: we concentrated on fractals and their
cousins. Now the emphasis has shifted to the process. The basic problem is to
identify the dynamical mechanisms that generate the patterns, rather than
focusing solemnly on the resulting patterns, no matter how beautiful. In this
view, our river networks, for example, become snapshots of ongoing dynamical
processes.

And we want to understand these general processes wherever they occur. Take
the reaction-diffusion process. Think of this as a fire spreading through a
forest and burning the trees. After a while, new trees grow to replace the old
ones. Map this pattern through time, and you’ll see a spiral wave, similar to
those that reaction-diffusion processes also generate in chemical systems and
bacterial colonies.

Alan Turing, famous for his early theory of the modern
computer, suggested that patterns on animal skins, such as stripes on a zebra
and diamond patterns on seashells are generated by diffusion of so-called
morphogens. In a model of a forest fire, those morphogens can be replaced by
fire. Although the physical systems are widely different, the mathematical
description remains very much the same, as do the resulting patterns.

Analysis like this reveals surprising connections. The growth of a city can
follow a pattern similar to that of some bacterial colonies. The actions of
people over time, deciding where to live and work, results in a “deeper
´Ç°ů»ĺ±đ°ů”.

Branching and breakdown are two more fascinating classes of process. You’ll
find that the same rules govern the branching patterns of rivers, trees, lungs,
arteries, snowflakes—and more of those bacterial colonies—while
breakdown phenomena include earthquakes, breaking glass and lightning. We can
model both classes mathematically, deriving simple rules to explain
observations.

There appear to be two very different types of pattern in nature. Some
“tapestries” are simple and repetitive. Others are much more complicated with
structure and surprises at all scales. Think of river networks and mountain
landscapes. The same patterns can be discerned in, for example, a single
mountain and in the range itself. But the dynamical process which forms these
complex structures are different: they self-organise into a “critical” state
where the dynamical process governing the way they form patterns is not smooth,
but takes place in terms of avalanches of all sizes. Ball reviews sand-pile and
rice-pile experiments to illustrate how complex patterns can arise from
self-organised critical processes.

Turbulence still evades our mathematical modelling. We can see the swirls and
eddies of a river current, clouds coalescing and breaking, but we have not yet
managed to find their common essence.

A complete explanation becomes more difficult when we consider the actions of
living organisms. How does the bee construct the honeycomb? We can see it
building the hexagonal cells, but how does it know what to do? Presumably, there
is an evolutionary advantage associated with the ability to construct the
honeycomb. The bee does not “know” the geometrical principle, but a dynamical
rule allowing it to build the honeycomb must have been hardwired into the genome
through the process of evolution. It makes no difference whether the dynamical
rule that is followed was derived from a simple chemical reaction or from the
genetic code—the resulting patterns are the same as Ian Stewart has argued
in Life’s Other Secret (Review, 10 October, p46).

Ball convincingly argues for some simple general principles. Yet strangely,
he cautions in his conclusion against any attempt at universalisation, paying
what seems to me like undue respect to a quote by the physicist Rolf Landauer:
“A complex system is just that; there are many things going on simultaneously.”
If this were true, no simplifying general description or understanding would be
possible: Nature would be just “one damned thing after another” as an
evolutionary scientist once expressed it. It would not be a testbed for serious
scientific effort.

Once we understand the self-made nature of nature’s tapestries, much of the
mystery of nature’s beauty disappears. No new mysterious force or religious
principle is needed. But that, of course, is the general objective of any
science.

All you need to see emergent properties appear before your eyes is a skillet
with a thick base containing 1-2 millimetres of cooking oil. Heat the oil
gently, Ball explains, and its surface will become covered with polygonal
convection cells. To make the flow pattern which creates the cells show up
really well, try sprinkling a powdered spice such as cinnamon onto the surface
of the cold oil.

The Fifth Miracle by Paul Davies, Penguin, ÂŁ18.99, ISBN 0713992158

VERY well: what is life? Ian Stewart starts by saying what life is not. It is
not DNA. It is not what DNA is, but what it does that counts. And where does
life come from? Paul Davies’s question leads us to the same ground.

Take a protein molecule, haemoglobin for instance, and look at it with a
mathematician’s eye. “We do understand that the shape of a protein molecule is
controlled by more than the genetic code. It is a consequence of deep laws of
physics, which are expressed in mathematical form.”

Stewart’s motivation is revealed late in the book, where he becomes
refreshingly emotional: “People talk an awful lot of nonsense about DNA as
`information’ for an organism—as if information has any meaning outside of
a well-defined context”. By analogy, the marks on a CD-ROM have no meaning
without the contexts of a machine to play it and of literacy or musicality.

We are constantly subjected to news reports claiming that medical researchers
have located “the gene” for some disease or condition—Alzheimer’s,
obesity, homo-sexuality, whatever—only to find the claims contradicted
later. The concept of “the selfish gene” has been taken to the extreme, in which
life’s mechanism is solely the survival of the information represented by the
individual gene.

Stewart’s goal is to convince you that a full understanding of life depends
on mathematics. At each level of description—from networks of interacting
genes through organisms and their behaviour to the planetary ecosystem—the
mathematics is that of complex systems.

His role model is D’Arcy Thompson, who, in his famous 1917 book On Growth
and Form argued that the existence of patterns in nature implies simple
physical principles. Such principles require maths. He believes that
“biomathematics” is the science of the 21st century, and he might well be right.
Despite the messiness of biology, there could be (simple) mathematical laws
underlying the behaviour of organic matter.

Stewart’s grand tour of mathematics in life takes us from hallucinations
being expressed as spirals in the retina, then transforming themselves into
stripes on the cortex, to the symmetry of the walking patterns of humans and
camels.

He reminds us that Alan Turing, best known for his pioneering role in the
invention of the computer, applied mathematics to animal markings.

One single piece of new mathematics that could illuminate the general concept
of life is Stewart’s greatest wish. He wants to capture the abstract sense of an
autonomous system, deduce its properties, and show how autonomy arises—in
short, to understand the origin of “agenthood” in biology, and what an “agent”
is.

His mind is 100 per cent clear and scientific. You might say that he thinks
like a physicist, not a biologist—and certainly not like a mathematician,
since he is passionately interested in the real world.

Paul Davies is more like a storyteller, although his background is physics.
He is concerned with the origin of life, biogenesis, rather than the forms and
shapes of organisms. But, strikingly, he has co-discovered “Life’s Other
Secret”: “information as such is not enough . . . information must be meaningful
to the system that receives it: there must be a context!” Where Stewart talks
about the interplay between the DNA code and the physical world, Davies talks
about the entanglement of hardware and software.

And he focuses on the emergence of agenthood as the central definition of
life: “Life opts out of the strictures of chemistry . . . and creates a new,
emergent world of autonomous agency. Once this essential point is grasped, the
real problem of biogenesis is clear.”

I am sceptical. Because of the hierarchical nature of biology, it is not
clear at what level you define the agent. Consider ants, for instance: is the
basic agent the ant, the society of ants, the ant’s cell, the mitochondria in
the cell, or ants as a species? No, I did not mention the gene as the
fundamental agent.

The only specific “theory” that both authors refer to is Stuart Kauffman’s
auto-catalytic networks, in which groups of molecules interact with one another,
thereby replicating each other. This is the simplest chemical version of an
interacting ecology.

Davies starts with Francis Crick’s remark that “the origin of life appears to
be almost a miracle”—and takes his title from the theology of
Genesis. Does life violate the second law of thermodynamics, which states
that a closed system tends towards increasing disorder or entropy? It appears
that life, involving increasing order and structure, has to fight the tide
of entropy. But this is a pseudo-problem. The basic laws of physics are
fundamental— thermodynamics is, at best, derived from them. The origin of
life is a dynamical problem, not a thermodynamical one.

From there, Davies moves into abstract discussions on the origin of
information in biology. I thought he and we had agreed on Life’s Other Secret,
and that it took care of this nonsense. It gets even worse when he speculates
that “we will not be able to trace the origin of biological information to the
operation of local physical forces and laws.” This suggests overthrowing science
as we know it, whereas Stewart proposes building our understanding on it.
Fortunately, later in The Fifth Miracle Davies comes to his senses: “We
don’t need another law of physics. We must look elsewhere. But where?”

Since we do not understand much about how life originated, perhaps we can at
least understand where it originated. Davies vividly discusses three
possibilities. The first is that life began as chemical self-assembly in a
primordial soup somewhere on the Earth’s surface—Darwin’s “warm little
pond”. The second is that life came to Earth from space as viable
microbes—the panspermia hypothesis. The third—and this is the one
that Davies prefers—is that life began inside the Earth or deep below the
sea.

I find hot primordial soups unappealing, whether on the surface or beneath
the sea. Hot, reactive systems tend towards thermal equilibrium, with no
possibility of the unlikely events needed for the formation of complex
molecules. I would argue something like this: since DNA-like molecules can be
transported through space, they were. The entire Universe is replete with so
many possibilities—so much more “phase space” is available than on our
tiny Earth that it is much more likely that the very unlikely process of
formation of DNA, or its precursors, took place somewhere out there.

Paul Davies gets into all of the corners of research into the origin of life.
In particular, there is a vivid discussion of whether or not life, or even the
mind, was an unavoidable consequence of the laws and initial condition of the
Universe. Cynically, one might conclude that much of his vague thinking in fact
represents the sad state of affairs in this field of research. We are nowhere
near understanding the origin of life. But let us try to avoid invoking
miracles.

The End of Certainty by Ilya Prigogine, Free Press, ÂŁ20/$24,
ISBN 0684837056

NOBEL prizewinning chemist Ilya Prigogine wants to overthrow Newton,
declaring that the laws of physics as formulated in the traditional way describe
an idealised, stable world that is quite different from the unstable, evolving
one that we actually observe.

His book tackles the “arrow of time”—those processes in nature that are
irreversible, such as the second law of thermodynamics. This states that
physical systems with many degrees of freedom will move towards states with
increased disorder; by observing the process, you can identify the direction of
time.

Consider a jar holding a gas of atoms. Suppose that initially all the atoms
are at the left hand side of the jar. After a while, the atoms will be equally
distributed throughout in the jar. The disorder, or entropy, has increased. On
the other hand, if in the initial state the atoms are equally distributed, then
they will never return to a state where all the atoms are in the left half of
the jar. Obviously, the process is irreversible. If we saw a film of the process
where all the atoms eventually wound up in the left hand side, we would know
that the arrow of time was pointing in the wrong direction.

It might appear that this violates the basic laws of physics. The classical
laws, Newton’s laws, are such that if the direction of time is reversed the
equations are still valid. There is no arrow of time. A planet might equally
well orbit in the opposite direction around the Sun.

This is the so-called arrow-of-time paradox. What is the solution? Do we have
to give up the fundamental laws of physics to understand this? The explanation
is simple. It is true that if we could invert time accurately after the system
had reached equilibrium (atoms equally distributed), the system would return to
the initial state with all atoms in the left half.

But, of course, this is impossible. More importantly, for the vast majority
of the configurations of atoms in the equilibrium mixed state, the system would
not return to the initial nonequilibrium state. Why? If a single atom is shifted
a fraction of an angstrom before the process of time reversal, the entire
reversibility of the process is destroyed. Thus macroscopic irreversibility is
compatible with microscopic reversibility as expressed by the laws of physics.
It is important to keep in mind that a classical description, in addition to the
laws of nature, involves a specification of initial conditions.

Prigogine disagrees, arguing that to explain the increasing randomisation, as
expressed by the second law of thermodynamics, you must put in the randomness at
the probabilistic level in the equations of motion. This is the “end of
certainty”. So the fundamental laws of physics have to be revised. The second
law cannot be understood solely by studying the trajectories of atoms as
specified by the laws of physics. A radical move, indeed. Prigogine and his
group have spent decades working on this.

Prigogine has been solidly refuted again and again by reputable scientists.
To convince yourself that the classical explanation in terms of initial
conditions is true, you can simply perform a computer simulation of a collection
of atoms moving according to the classical laws of motion. Indeed, they will
randomise according to the second law of thermodynamics.

If the direction of time is reversed, then for a while the atoms will move
backwards along their trajectories, but the numerical accuracy will eventually
cause deviations. Thousands of scientists perform molecular dynamics, and their
findings consistently agree with the second law of of thermodynamics with no
need for revision of the fundamental laws. The bottom line is that there is
absolutely no documented reason that the processes needed to explain the arrow
of time”should violate the laws of physics. One can safely conclude that this
idea of Prigogine is without scientific merit.

Chaos theory neatly illustrates this. If you follow the trajectories of a
group of particles localised around a point in space, after a while they will be
far apart and fill up space in a random way. Think of a cup of coffee with a
drop of milk. If the cup is stirred, the milk particles will eventually become
randomly distributed throughout the cup. Indeed, if the direction of time were
reversed for everything—the cup, the coffee, the milk, the spoon and the
person stirring the spoon—you could imagine everything moving backwards.
But if the coffee is stirred backwards, it would not return to the state with
the localised drop. This does not mean that there are metaphysical processes in
the coffee cup. This point can be mathematically proven for some simple
systems.

Of course, the idea of certainty ended long ago. The fact that the motion of
particles is deterministic, that is, that every time the system evolves from the
same initial state, it ends up in the same final state after a time t,
does not mean that we can predict things. This is simply because systems set up
in almost the same states do not end up in almost the same final state. Thus,
any inaccuracy in determining the actual initial state will eventually severely
limit our ability to predict the future. One may have to resort to a
probabilistic description and this has been explicitly proven in some simple
cases. Chaos theory supports the classical view, not that of Prigogine. No
modifications of the fundamental laws of physics are needed.

The arrow of time manifests itself in a very different way. It appears that
the world evolves to a more and more complex state, where a complex pattern of
galaxies emerges from an initial big bang, followed by the formation of
planetary systems and, eventually, life forms. Is this in contrast with the laws
of physics? There is no evidence of that whatsoever. The emergence of more and
more complex phenomena, nevertheless, is an exciting and active field of
research. Certainly, we have not reached the equilibrium state where everything
is evenly distributed—perhaps we never will.

Prigogine, by his eminent intuition, has done more than anybody to point out
the richness of phenomena associated with nonequilibrium processes. Even in
quite simple systems, such as our jar of gas atoms, there is evidence that the
mixing of the initial states over the entire phase space (like the coffee cup)
is limited. The system “gets stuck” outside equilibrium. Again, scientists have
made simple models with emerging complexity where time reversal invariance has
not been violated at the microscopic level.

I cannot help feeling that the confusion is due to a misunderstanding of the
priorities of physical laws. The fundamental laws, such as Newton’s laws and
quantum mechanics, have the highest priority. Thermodynamics must conform with
these laws. More often than not this can not be directly verified—the
second law of thermodynamics is not a fundamental law in the same
sense—and could conceivably be violated. The initial state at the big bang
may be one of the highly unlikely states that eventually lead to an ordered
state. In any case, any explanation of thermodynamics must be consistent with
the dynamical laws, not vice versa.

Every now and then, crackpot papers are submitted for publication to
scientific journals. When pseudoscience is presented by a highly esteemed, Nobel
prizewinning chemist, the damage is not so easy to contain. Indeed, Prigogine
has a great following among chemists, however, and humanistic scientists.
However, as John Maynard Smith once said of a book by Prigogine: “If
nonequilibrium statistical mechanics excites poets, so be it!”