The Mathematics of Fibonacci's Sequence
by Keith Moffatt
The Fibonacci sequence is defined by the property that each number in the
sequence is the sum of the previous two numbers; to get started, the first two
numbers must be specified, and these are usually taken to be 1 and 1. In
mathematical notation, if the sequence is written 
then the defining relationship is
with starting conditions 
. On dividing
both sides of (1) by 
, we
obtain 
where 
, the ratio
of successive terms.
As 
where 
, or 
This
quadratic equation has two roots; the one we need here is obviously between zero
and one; it is
The number 
has some remarkable properties; for example, it
is expressible as a "continued fraction'':
In the theory of chaotic dynamical systems, 
is recognised as "the most
irrational number" between 0 and 1!
The spiral curve shown in the poster is a logarithmic spiral, a curve whose
equation in polar coordinates is 
where k
and a are constants. The spiral patterns evident in the sunflower are of
this form, and the numbers of spirals going in opposite senses are the
conscutive Fibonaci numbers 34 and 55. The underlying reason for this may be
found in many texts; see for example Conway JH and Guy RK The
Book of Numbers, Springer-Verlag (1996), chapter 4.
About the author
Keith Moffatt is a fellow of the Royal Society and Director of the Isaac Newton Institute for Mathematical Sciences, a national and international visitor research institute at the University of Cambridge.
His own research interests lie in the field of fluid dynamics.
