If you look closely at the head of a sunflower, you will see that the
structures destined to become seeds grow in an intricate pattern of interwoven
spirals. Such patterns, which seem to occur only in plants, have fascinated
people and inspired a number of mathematical studies. Now two French scientists
have set up a physical model that behaves in exactly the same way, producing
the distinctive spiral patterns.
The head of a sunflower or a similar member of the daisy family (Compositae)
exhibits two sets of spirals, one running clockwise and the other anticlockwise.
Intriguingly, the number of spirals in one set is related to the number
in the other in a special way: the two numbers are always adjacent members
of the so-called Fibonacci series.
Leonardo Fibonacci, a 13th-century mathematician, discovered a mathematical
series in which each number is obtained simply by adding together the previous
two numbers. So, starting with the number pair 1,1, the series reads 1,
1, 2, 3, 5, 8, 13, 21, 34, . . .
In a sunflower, embryonic seeds – known as ‘disc’ florets – form at
the centre of the flower head before moving away as the head grows in size.
New florets emerge at regular intervals close to the centre, and each floret
forms where the gap between recently formed florets is largest. The principle
is quite general, and applies to all plant growth of this kind, such as
the growth in a pine cone, or the growth of spines in some cacti. But what
of non-botanical systems?
S. Douady and Y. Couder of the Laboratory of Statistical Physics in
Paris have designed an experimental system that obeys the same principles
as the spiral growth in plants, but which bears no other resemblance to
botany. They used an iron-bearing liquid which they released in small drops
at regular intervals at the centre of a dish of silicone oil.
The dish was placed in a vertical magnetic field, so that the drops
became tiny magnets, lining up in the magnetic field in such a way that
they repelled each other. The magnetic field also varied across the dish,
so that the drops moved at a uniform velocity across to the edge of the
dish once released. The researchers were able to alter the rate at which
they released the drops, and by changing the strength of the magnetic field
they could also adjust the velocity of the drops.
Douady and Couder found that when the time taken for drops to move towards
the edge was short, compared with time between drops, each new drop was
repelled only by the previous drop. The two drops moved away in opposite
directions, a behaviour that mimics the ‘alternate’ pattern of growth often
seen in plants – for example, in the way leaves form on a stem.
However, the researchers found that if the drops did not have enough
time to move away, a new drop would feel the repulsion due to two or more
drops. It would then move away from the line between the previous drops,
initiating a spiral pattern. The researchers found that the pattern involved
two sets of spirals, related by the Fibonacci series, just as in the sunflower
(Physical Review Letters, vol 68, p 2098).
Douady and Couder also carried out detailed numerical simulations of
the formation of spirals, which again obeyed rules similar to those in their
experiment. They concluded that the Fibonacci pattern emerges as the system
keeps the energy required to a minimum. In other words, the spiral growth
in plants needs no special botanical explanation. The new elements in the
pattern, be they florets or oil drops, simply appear where they need least
energy to do so.