2.4 Fibonacci Sequence and
Selfsimilar Structures
The Fibonacci mathematical series was
discovered in 1209 by Leonardo of Pisa, known as Fibonacci
(Kappraff, 1992 Reference ) while computing
the total number of adult rabbits in successive months starting with a
single adult rabbit pair and assuming that each adult rabbit pair produces
one pair of offsprings each month and that baby rabbit pairs became adults
in one month's time. The growth of rabbit population is shown as a branching
network in Figure 3.
FIGURE 3
Generation of Fibonacci numbers as cumulative
sum of a sequence of ordered bifurcations (branchings).
The total number of
adult rabbit pairs in successive months follow the Fibonacci mathematical
series. The growth of adult rabbit population as shown in Figure 3
represents a hierarchical ramified network or a selfsimilar fractal
network. Ramified branching network systems in nature can be similarly
shown to generate the Fibonacci mathematical number series. For
example, the branching network of updrafts and downdrafts in vortex roll
circulations in atmospheric flows (Figure 4 ) can be shown to be
represented by a hierarchy of branches with multiple sub-branches.
FIGURE 4
Bifurcating network of updrafts and downdrafts and
Fibonacci
numbers
In Figure 4,
A
represents an updraft. At the first level, forking structure AB
is generated with formation of sub-branch (downdraft) B. At the
second level,
A again generates the forking structure AB
, while the sub-branch (downdraft)
B of level 1 now generates
the updraft A . Updrafts alone produce forking structure with formation
of sub-branch (downdraft) B which then gives rise to updraft A
at the next level. Continuing such a system of bifurcation results in the
generation of Fibonacci numbers sequence for the total number of
updrafts (A ) at each level. In summary, the integrated sum of smaller
scale networks contribute to form large scale networks. Branching networks
may therefore be considered as a hierarchy of selfsimilar networks or fractals.
Fractal architecture to the spatial pattern is therefore a signature
of cumulative integration (summation) process inherent to dynamical growth
processes of the system. For example, the fractal network of a river
drainage basin serves to collect water from the smallest of tributaries
(branches) and integrate it into the main river flow.