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Nature in all its wondersPeter BusseyPublished online: 08 Nov 2010.

To cite this article: Peter Bussey (2003) Nature in all its wonders, Contemporary Physics, 44:4, 357-360, DOI:10.1080/0010751031000147271

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Essay Review

Nature in all its wonders

PETER J. BUSSEY

A review of The Self-Made Tapestry, Pattern Formation in

Nature. By P. Ball. (Oxford University Press, 2001.) Pp.

vi+287. £12.50 (pbk), £22.50 (hbk). ISBN 0 19 850243 5

pbk, ISBN 0 19 850244 3 (hbk). Scope: popular survey.

Level: general reader.

It is not surprising, as a physicist, to have one’s attention

drawn to the many regular phenomena that are found in

nature. Is this not, after all, what physics is about—the

search for basic laws that are responsible for the many

natural patterns that we can observe, and for which the

latter give evidence? However, nature is capable of having

some fun with physicists and presents many examples of

patterned phenomena whose connection with the laws of

physics is not entirely transparent.

There are the hexagonal arrays of honeycomb, the ridged

lines of sand dunes, fluctuating sequences of animal

populations, the fact that snowflakes are all the same and

yet all different, and the shapes of flowers—to name just a

few examples. To these may be added a host of phenomena

that are not part of our everyday experience but that show

up unexpectedly in certain systems when these are prepared

in certain ways. Oscillating chemical reactions can be set up

which fluctuate regularly and even display alternating red

and blue colours. The viscous motion of liquids displays a

remarkable medley of behaviours starting with streamline

flow and ending with full turbulence. Under suitable

conditions, uniformly heated fluids can show a mind-

boggling variety of convection patterns. Physics must lie at

the heart of all this, but it is not always easy to penetrate

the essential details.

A number of themes may be discerned when considering

the varieties of natural behaviour such as these. One is that

there is a set of phenomena whose evident origin in simple

laws makes us think that they themselves should have a

simple, even prosaic description, but which in fact exhibit

features of a remarkably varied and complex kind.

Conversely, out of situations that might seem to be

extremely complex and subtle, nature may contrive to

produce effects with surprisingly simple characteristics.

Many examples of both kinds have long been familiar; in

recent years, what has greatly assisted our deeper under-

standing of such areas has been the development of high-

powered computing. This has enabled a wide range of

physical, chemical and biological systems to be modelled

and explored, whose mathematics would formerly have

been regarded as intractable.

Let me outline a simple example of a physical

phenomenon which illustrates some key ideas, namely a

swinging door with variable damping to its motion. With

strong damping, an initially open door will simply ease

itself towards its ‘shut’ position and, even if it is given a

push, it will at most execute one swing. However, if the

damping is allowed to decrease, the system passes through

a critical point, after which the door performs oscillatory

motion about its ‘shut’ position. These are still damped

oscillations, but they represent a different kind of solution

to the original equations expressing Newton’s laws of

motion. If it could now be arranged for energy to be given

to the door once per swing, to restore what was lost

through the damping, then we would have an example of a

‘dissipative’ situation. In this case, the swings could

continue regularly and indefinitely. That garden-shed door

which bangs continuously all night, once the wind has got

up, could be an example of this kind of effect. But even with

a supply of extra energy, the door will execute swinging

motion only after the damping parameter has crossed a

certain value.

Many systems can be set up or occur in nature that show

features of this kind, sometimes in highly elaborate ways.

As the pressure that drives a fluid past an obstacle is

increased, a remarkable series of different flow patterns can

be produced, as certain critical values of the Reynolds

number are exceeded, on the way from streamline flow to

full turbulence. Another class of examples is to be found

when a body of liquid or gas is subjected to a uniform

temperature difference between its lower and upperAuthor’s address: Dr P. J. Bussey is at the Department of Physics andAstronomy, University of Glasgow, Glasgow G12 8QQ, UK.

Contemporary Physics, volume 44, number 4, July –August 2003, pages 357 – 360

Contemporary Physics ISSN 0010-7514 print/ISSN 1366-5812 online # 2003 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/0010751031000147271

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surfaces. For small temperature differences, simple con-

duction transfers the heat across the fluid but, as the

temperature difference increases, convective motion ap-

pears. Given a large expanse of fluid, the simplest type of

convection cell has the form of a horizontal cylinder, with

the fluid rising along one side and sinking again at the

other. If the temperature difference is appropriate, a whole

series of such convection cylinders can be generated,

stacked side by side, covering the expanse of the fluid.

The fluid rises or sinks along the alternate regions where the

neighbouring cylinders touch, so that the cells ‘roll’ in

alternate directions. Series of linear ‘rolling’ convection

cells of this kind are believed to occur inside the Earth’s

mantle.

However, there are other possibilities; sometimes the

fluid may rise convectively at a localized point and sink

over an approximately circular surrounding region. Such

cells may form a hexagonal ‘close-packed’ pattern or a

square chequerboard-like pattern across the system, de-

pending on the conditions. In other cases, elaborate sets of

‘roll’ cells can be produced which are not content to line up

regularly but which curve around sinuously, forming

twisting spirals, loops and all manner of vaguely psyche-

delic patterns!

The chief operative physical parameter here is the

Rayleigh number of the fluid system, but there is another

perspective, namely that of symmetry breaking. A uniform

expanse of fluid that is infinite in two dimensions has perfect

symmetry; it is the same at all locations and in all directions.

A regular array of rolling convection cells, displaying a

pattern that we findmore interesting, has less symmetry than

this. It retains a continuous translational symmetry in one

direction but now has a periodic translational symmetry in

the other and has lost its isotropy. The perfect symmetry has

disappeared, even though the effect that produced this,

namely applying a uniform temperature difference between

the bottom and top of the fluid, has no asymmetry in itself.

How can this come about?

If the expanse of fluid is bounded, then the nature of the

boundaries will have an important effect, since the

convection cells must fit between them. The system must

thus be viewed ‘as a whole’ before the particular solution to

the dynamic equations can be known. This of course is not

a new feature of physical systems; any type of standing-

wave phenomenon is precisely of this kind. Thus, fixing a

violin string at each end determines its natural modes of

vibration. The remarkable fact is that rather complex

dynamic systems are still subject to the same basic idea as

the simple violin string!

However, there may be many ways for similar convec-

tion patterns to occur in a system. When the preferred

behaviour patterns have a lower symmetry than the

equations or the boundary conditions, this allows there to

be more of them. If otherwise equivalent, they are then in

physics terminology ‘degenerate’ solutions to the equations,

and even the boundary conditions, by themselves, will be

insufficient to select just one mode of motion of the system.

In such a case, spontaneous symmetry breaking will take

place; any small incidental disturbance may set the system

operating in one mode rather than another.

So there exist, even in relatively simple situations, a

number of interplaying themes. What is noteworthy is that

the convection patterns, viscous flow patterns, or whatever,

can be stable; thus, we are able to observe them and marvel

at them. Perhaps the following general point can be made.

The kinds of complexity that we find interesting lie in an

intermediate phenomenal range between simplicity and

chaos. They tend to occur only in a restricted range of

physical circumstances, depending for example on the

amount of applied energy that realizes them. This ‘excita-

tion’ energy is to be compared with other energies

associated with the system in the light of the operative

dynamic laws. Too little applied energy, and we have

‘ordinary physics’—elegant, beautiful and insightful—with

a limited range of phenomena. Too much applied energy,

and everything breaks down into turbulence and chaos. In

between lies the situation of interest. Some of these patterns

are vaguely reminiscent of living forms, and there is an

analogy here to life on a planet; too little thermal energy

and life cannot develop; too much and the organic

biological chemicals will be destroyed.

Clearly, the basic physical laws provide the stabilizing

principles, and it is their very restrictiveness that prevents

the smallest perturbation from producing chaos. The

symmetries in the physical laws are broken stage by stage,

as the applied energy increases, each stage producing more

complex but less robust phenomena. Eventually, turbulence

may set in, destroying the stability. It could even be said,

perhaps, that the really elaborate patterns (whorls, inter-

twining loops and so on) provide more amazement than

insight. In fact they also provide a possibility of tempting us

into imagining that it is out of their own prolific nature that

the simpler configurations arise and that they are in some

sense the munificent source of the less elaborate, more

easily comprehensible patterns. This, of course, is not so. If

a social analogy may be permitted, the more elaborate

complexities rather more represent a kind of last stage of

‘decadence’ before the whole system collapses into chaos!

The real source of the simpler, more regular patterns lies

not in the elaborateness into which they may develop, but

in the even simpler basics out of which they arise. Or, if you

prefer, out of which they ‘emerge’. I must admit to a certain

diffidence about the use of this fashionable word. In the

case of physical systems such as those mentioned, it can be

quite correctly said that the patterns ‘emerge’. They are

there, latent, in the physical laws and definitions of the

system, together with the boundary conditions. When

circumstances are right, the patterns manifest themselves

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as the logical outcome of the physical situation. But should

we be saying that the convection cell systems ‘emerge’ out

of the simple conductive behaviour obtained when the

applied temperature differences are small, or that the

oscillations of the swinging door ‘emerge’ out of its strongly

damped motion? After all, if we started with the more

complex situation and then adjusted the appropriate

parameter, the simpler situation might likewise be said to

emerge out of the more complex! To take another

example—should we say that superconductivity ‘emerges’

out of normal conductivity, as a sample of niobium is

cooled through its critical temperature, or should we

conversely say that normal conductivity emerges out of

the superconducting state, as the metal’s absolute tempera-

ture is raised up from zero? So long as one is aware of a

certain looseness of speech, I doubt that it matters much,

even though the phenomena really all emerge out of the

basic physical principles rather than out of each other.

However, elsewhere I have grave doubts about the

terminology, for example in statements about conscious-

ness ‘emerging’ in physical systems as they become

sufficiently complex. A knowledge of the basic principles

is needed before there can always be really meaningful talk

about ‘emergence’. At present, in the case of consciousness,

we certainly lack this knowledge; the phenomenon seems to

be quite different in its nature from anything that we

currently call physics.

Regarding biological systems, physics clearly has a basic

say in the determining principles, but to what extent do

these provide the best way of understanding biology? While

physicists go for laws of nature, it has been said that

biologists like proteins, the latter being the immediate

controlling agents of very many biological processes. Even

more, biologists like the genes which produce the proteins.

In contrast with this conventional wisdom, the zoologist

D’Arcy Thompson set out in 1917 a rather radical point of

view in which he proposed that many biological systems

and processes are best understood, after all, not in terms of

Darwinian adaptations to the environment, but in terms of

their biophysics. As we might say today, this would be in

terms of analogue physical effects rather than digital

genetic information processing. Cell membranes, it can in

this way be argued, have the forms and properties implied

by the physics of their long constituent molecules. Molluscs

such as snails and whelks have shells in the shape of

logarithmic spirals because the mathematics of their

growth make this automatic. Animal horns grow curved

if one side grows faster than the other, and straight

otherwise; these are logical alternatives, rather than genetic

choices from a host of hypothetical possibilities. Genetic

selection is constrained by physics and mathematics, and

the latter may often provide a more helpful viewpoint. The

interesting question concerns how strong this constraint

actually is.

As a counterweight to geneticism of a more facile kind,

Thompson’s study On Growth and Form has retained its

ability to provoke. It is a little subversive. While many see

great virtue in Darwinian genes repacing ‘design’ in

biology, Thompson sidesteps much of this and puts

emphasis on the part played by physics and mathematics

as underlying principles in biological nature. There is

limited explanatory power in saying merely that the best

genetic variations are those that survive and propagate

best. Physics, Thompson suggests, gives a rather better

insight into causal principles. Such a viewpoint is without

doubt controversial and is certainly not true in every case,

but it requires serious consideration since there is unlikely

to be a clearcut separation between genetics and physics.

Philip Ball’s admirable book has as its subject these and

a host of related examples and themes, treated in the

broadest possible way. There are sections on the patterns

and forms found in foams, on chemical reactions, cell

membranes, butterflies’ wings, branches of trees, bacterial

colonies, sand dunes, animal populations, river topology,

the growth of cities and many other topics. The author has

gone to considerable trouble to get to grips with an

extraordinary variety of phenomena, and presents them in

a way that will be illuminating to a wide readership. The

book is very well illustrated throughout and will awake an

awareness of a completely unsuspected range of interesting

things in the world around us.

The treatment is basically non-mathematical, which I

found sometimes a little frustrating. To a physicist, the

equation is often the easiest and clearest way of presenting

the essential features of a situation, and not something to

be avoided at almost all costs! Perhaps a few more

mathematical footnotes would have helped. Occasionally,

however, when a technical explanation is absolutely

essential, it is given in a self-contained boxed section. This

seems a very good idea and perhaps a few more of them

would have been nice.

A conspicuous example of natural patterns which is not

treated very expansively is that of crystals, which are

mentioned from time to time, but not in a major way. Since

only a modest scientific background is demanded of the

reader, the relatively straightforward nature of this topic

might make it merit its own section in a second edition of

the book, given that the different types of crystal do

illustrate a number of useful basic principles. However, the

snowflake, with its familiar sixfold symmetry, is indeed

discussed at length. Surprisingly, while the formation of the

arm of a snowflake can be an interesting example of fractal

behaviour—one of the book’s major themes—the basic

reason for the closely similar shape of all six arms of a given

flake seems to be imperfectly understood even today. It

may be just that, in a given snowflake, the six arms are

similar because they all grow in a closely similar physical

environment.

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An area where I have a few reservations is Ball’s account

of the patterns of florets that grow into sunflower seeds on

a sunflower head. These originate, we are told, from the

growth of tissue around the central point of the developing

flower bud. As this generative tissue spirals round in close

layers, it sprouts proto-florets at regular angular intervals;

these grow outwards into the florets which we see covering

the central disc of the flower, and which are seen to trace

prominent large-scale spiral lines extending from the centre

of the disc to its circumference. In this connection, much is

made of the so-called ‘golden ratio’, whose quintessentially

irrational nature might give rise to optimal packing of the

florets.

Unfortunately, the discussion here is somewhat numer-

ological for me. In the sunflower head illustrated in the

book, one can also make out patterns of florets which trace

approximately radial lines, suggesting the presence of some

definitely rational fraction of 3608 in their original angular

spacing! In fact, the large-scale spirals surely have nothing

much to do with the golden ratio. Such patterns are simply

the general consequence of a regular angular spacing in the

generative tissue and will occur for many ranges of this

angle. In plane polar coordinates, they are analogous to the

straight-line forms that are seen in a multiplicity of

directions in space when regular linear packing occurs in

Cartesian x, y, z coordinates, as in crystals.

No matter, the principal aim of this work is to open

people’s eyes and minds. This it promises to do with

outstanding success. It is not an entirely easy read, because

of the complexity and subtlety of many of the phenomena

that are discussed, but the author’s explanatory powers are

of a very high order and the reader who is willing to follow

the argument will be well rewarded. This is a highly

recommendable book.

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