Magical numbers in nature Mathematician Ian Stewart
talks to Nature Science Update about snowflakes, sticklebacks and a
new kind of science.
Ian Stewart was turned on to mathematics at the age of seven. A
broken collarbone freed him from an uninspiring teacher allowing his
mother to ignite his interest in numbers while he was laid up at
His writing career began with a series of how-to manuals for
now-defunct early 1980s microcomputers. It has since broadened into
popular science and science fiction - often in the same book - as
well as texts aimed at fellow mathematicians.
Stewart is now professor of mathematics and director of the
Mathematics Awareness Centre at the UK's University of Warwick. John
Whitfield visited him here to talk about his latest book, What
Shape is a Snowflake?, and mathematics' place in biology.
Do you think that the handful of mathematical and physical
concepts needed to explain a snowflake are also enough to build the
Just about. You should always be sceptical about these grand
theories, but a lot of the explanations of patterns in nature turn
out to have quite a lot of things in common. The snowflake
has a small number of ingredients, but those ingredients are very
general and very powerful.
One is symmetry.
This is one of the great principles of science. Albert Einstein said
that the whole point of physics is to capture the symmetries of the
Universe, meaning that the laws of nature are the same at all places
and times. The next ingredient is dynamics, systems that change over
Symmetry is one of the great
The pattern you see in a snowflake is a record of how it grew.
And the complexity of the patterns is related to chaos - the ability of
a dynamical system that's following rules to do very complicated,
almost random-looking things. So already we've got three of the big
principles of mathematical physics.
Do still deeper generalities underlie this handful of
Physicists would claim they've reduced the whole of the Universe
to two different principles: quantum
mechanics and relativity.
They are working to get that down to one,
and I'm pretty sure they'll get there. For the physical sciences,
that kind of reduction of rules to deeper and more fundamental
principles works extremely well.
In the biological sciences
it's not useful to try to get at universal
In the biological sciences, our understanding is less the sort of
thing you can reduce to an equation. At the moment it's not useful
to try to get at universal principles. Because we don't know good
equations for, say, biological development, there's no point in
trying to push it to a deeper level. Maybe one day we'll be able to
do that, but I'm sceptical.
In physics, you can usually say in advance what a system can do;
then the physicist's job is to work out what off that list it
actually does. In biology, the system builds its own possibilities
as it develops and evolves. Saying in advance what the space of
possibilities is seems to be either very difficult or impossible. In
a way, biology makes up the rules as it goes along. The really
interesting developments are when it makes up a new rule.
Can you give an example of such a new rule?
Let's go back to when everything on Earth were bacteria,
billion to 4 billion years ago. Bacteria in the oceans and on
the seashore are busily producing oxygen, which is accumulating in
the atmosphere and really doing nothing - it's poisonous to their
metabolism, it's pollution.
The most important change in the whole history of the planet was
when life discovered a way to use that oxygen. The eukaryote
cell appeared, which protected itself against the corrosive
effects of oxygen, and made use of the nice big energy source.
All the while we lived in a prokaryote universe, with oxygen just
accumulating, the rules for eukaryotes weren't there. You couldn't
even predict them - they were unwrapped as it happened. In biology,
you can't see the really dramatic developments coming.
So if it's so hard to make general statements and predictions
in biology, why have you tried to work out what must be true for all
life, for example on other planets, rather than just life as we see
Some general principles would work out in much the same way for
living creatures elsewhere in the Universe, possibly made out of
different kinds of molecules from us, or possibly not even made of
by natural selection would still work, for example.
If aliens are on a planet
with a fairly good atmosphere, they're going to
If aliens are
on a planet with a fairly good atmosphere, they're going to evolve
flight, because that gives a huge advantage to the first creatures
that can do it. Flight's evolved many times on this planet. So
that's a universal feature that you can predict will happen. But you
couldn't go to the alien planet and say: "In 75 million years this
particular creature will develop into a butterfly." What you can say
is: "Come back in 100 million years and flight will have
Your latest book is full of comparisons, such as the symmetry
of faces, and crystals, and the laws of physics. You talk about
seeking "a similarity beyond metaphor". How do we know that when we
We handle metaphors without having to work very hard - if I
describe something as being the size of an elephant, everybody knows
what I'm talking about. Our brains work in a very top-down way. The
problem with that is that you can see metaphorical connections
between things, and people get very excited about them, without
really having a good theoretical underpinning of why the metaphor
works, and sometimes it doesn't.
For metaphors to be good
science, you need something a little
For metaphors to be good science, you need something a little
stronger. You need to say: there's a process going on in both these
examples; it's the same process; I can formulate what that process
is; and I can do some maths and science with that process. If
everything works at that level, your metaphor's got some backbone to
In particular, I had in mind the passage where you compare
biological speciation, where one species becomes two, to
bifurcations, where the mathematical state of a system alters
dramatically. What do you say to the charge that that similarity
doesn't go beyond metaphor - that the underlying mechanisms are
probably quite different?
Speciation - that's symmetry
A short while ago, I would have agreed. I started getting
interested in the similarities when my biologist collaborator, Jack
Cohen, wanted to understand what mathematicians call symmetry
breaking. I was explaining this in terms of things like ping-pong
balls collapsing when you take them to the bottom of the ocean, and
he said: "Speciation
- that's symmetry breaking, isn't it?"
First I thought he didn't understand what I was talking about.
Then I asked him what he meant. He said: if you've got one species,
that's a very symmetric situation, because all the animals are
pretty much the same, but two species must be less symmetric.
We've written a few papers about this as a mathematical metaphor
for speciation, modelling a species as identical organisms and
worrying about the differences between individuals later on. These
models turn out to include a general mathematical process, called bifurcation,
where the state of the system changes quite dramatically, even
though the system's environment only changes a small amount.
The models produce some general predictions: the split happens
very fast, on an evolutionary timescale, and the two new species
'pull apart' in opposite directions compared with the original. For
instance, if the original species is a bird with a medium-sized
beak, then the species splits into one with a big beak and one with
a small. The average size of beak doesn't change at all.
We've got something to nail
Until very recently, I'd have left that as a metaphor. But there
have been several bits of work looking at specific mechanisms of
speciation - sexual selection in sticklebacks, say - that seem to
fit into our mathematical framework of bifurcation and symmetry.
Now we've got something to nail down and test, with experiments on
bacteria, for example. It looks as if it's going to go beyond
metaphor and into science.
Do biologists, with their focus on detail, neglect general
I think they do. Historically they were right to. Before
genetics, the kind of mathematical descriptions that I like, such as
Alan Turing's models
of spots and stripes, were in favour with biologists. They looked at
the organisms, and the patterns, and made guesses about the general
Unfortunately, biology went
so overboard for molecular descriptions that it
lost sight of other
But that doesn't have much predictive power. Current mathematical
models don't correspond very closely to genetics and biochemistry.
When biologists began to look at DNA and protein
structure it opened up a different way of thinking.
Unfortunately, biology went so overboard for molecular descriptions
that it lost sight of other things.
Now biologists are starting to put it all together again, which
is very encouraging. Many people working in DNA sequencing say that
to understand how proteins fold, we need a lot of maths. A sequence
alone doesn't tell us anything about the geometry of a protein,
which is the important thing.
There are other areas of biology where it's beginning to dawn on
people that the same thing needs to be done. Development
is full of things where the dynamics of growth matters.
Understanding the genetics will give you some information, but there
are mathematical constraints on how it will happen.
At the level of the cell, there's a lot going on in genetic
networks where just listing the genes and what switches on or turns
off what gets you the beginnings of understanding, but not enough,
because even very simple networks are quite subtle dynamically.
One of the most interesting areas for the next 100 years or so
will be the linking of the mathematical ideas of dynamics, pattern
formation, fractals, symmetry with the information that comes out of
genetics, biochemistry and DNA and what is beginning to be
understood about the way that organisms develop. It will forge a new
kind of science, and a new kind of mathematics, which really makes
use of the strengths of all those areas. It's not a bit of maths and
a bit of biology, it's what happens when you let maths and biology
talk to one another for 25 years, and see what comes out of that