The Geometry The Nucleus

Figure 2
CONSTRUCTION OF A
GOLDEN MEAN RECTANGLE
Take a square and fold it in half,
and then fold this in half again,
diagonally. Extend one side of the
square by the length of this diagonal,
beginning to measure off
the diagonal from the midpoint
of the square. The rectangle thus
produced will be a golden mean
rectangle. This construction can
be replicated beginning with any
arbitrary rectangle and it will rapidly
approach the shape of a
golden mean rectangle.
is an absolute limit on the carryingcapacity
(we may assign the limiting
value of the population the value of 1),
then one must assume that the rate of
growth is no longer constant, but is
modified by the difference between
the size of the population and its limit
[R = r (1 - X„)]. This leads, by simple
algebraic substitution, to a transformed
function X„ +1 = (1 + r)X„ -
rX n 2 .
The interest in the function emerges
from the fact that if we begin with a
population size under 1, the function
will approach its limit—but only providing
that ris less than 2. When /-equals
2, an instability occurs, which is resolved
as a periodic oscillation between
two population levels. To
achieve this, we can go to r = 2.3, and
study the behavior of alternating terms
of the function X n+2 = f[f(X n )] = P(X„).
These values will remain stable until
r reaches the value of V6, when a fourfold
oscillation begins. This period
doubling continues until the value of r
= 2.470, at which the process remains
unstable. An interesting geometric relationship
emerges between the limiting
values of r, the values at which
phase changes occur. If we treat these
values as branch points and take the
ratio of the differences in r, betweer
neighboring branch points, this value
will approach 4.669. . . as n increas
es—a value known as the Feigenbaurr
number, after its discoverer. The for
mula for this is (r„ - r„_,)/(r n +, - k).
The Fibonacci Series
The study of recursive functions i
not new. Indeed, as long ago as the
start of the 13th century, the mathe
matician Leonard of Pisa (also knowr
as filius [son of] Bonacci, or Fibonacci
was asked to develop a population
function to study the unrestrained
population growth of rabbits. Th!
function he developed, the famous fibonacci
series, also has a wide area o f
applicability: It turns out that r in thi;
case will approximate the golden meai i
more and more closely.
Thus, this function turns out to b s
an excellent model for the development
of such disparate structures as
seashells, the human body, and th;
formation of galaxies. It governs th s
spacing of leaves around a plant sten
and the distances of the planets fror i
the Sun. The fibonacci series is dete -
mined by a simple rule: The next terr i
in the series is equal to the last tern
added to the one before it. If we begi i
the series with the number 1, it progresses
as follows: 1,1, 2, 3, 5, 8, 1!l,
21,...
This model of population growth s
important because it actually shov s
how the division of labor in a popul >-
tion allows the nurturing of future p< >-
tentialities. In the case of the rabr.it
population, obviously young rabbits c o
not add to the carrying-capacity of tl e
rabbit population, which will event I-
ally strip the vegetation upon which it
depends for food. But in the case >f
humans, it is precisely the nurturii g
(which emphatically includes education)
of young that provides the poal
from which technological innovatic n
emerges. It is the succession of su :h
technologies that have raised us, a< a
species, from a population potential :>f
around 10 million to one of at leas^ I
billion.
Fibonacci's model was simple, ijie
supposed that two rabbits produce
pair of young (one of each sex) eve ry
month, and that it takes two montis
for the pair to sexually mature and th ;n
begin to reproduce (Figure 1). Thi s,
there is a lag between the birth of a
pair of rabbits and their joining the reproductive
cycle. The population is increased
each month, by adding a new
cohort equal to the number of rabbits
that are alive the previous month, to
the total living rabbit population. The
series increases as follows: 1 the first
month, 2 the second month, 3 the third
month, 5 the fourth month, 8 the fifth
month, 13 the sixth month, and so on.
The golden mean may be represented
algebraically by the proportion 1/$
=