The branching (bifurcating) structure of roots, shoots, veins on leaves of plants, etc., have similarity in form to branched lightning strokes, tributaries of rivers, physiological networks of blood vessels, nerves and ducts in lungs, heart, liver, kidney, brain etc. Such seemingly complex network structure is associated with exquisitely ordered beautiful patterns exhibited in flowers and arrangement of leaves in the plant kingdom.

The identification of physical mechanism
for the spontaneous generation of mathematically precise, robust spatial
pattern formation in plants will have direct applications in other areas
such as in *physiology and medicine*.

The botanical elements which constitute
plants are branches, leaves, petals, stamens, sepals, florets, etc. These
plant elements begin their existence as primordia in the neighbourhood
of the undifferentiated shoot apex (extremity). Extensive observations
in *botany* show that in more than 90% of the plants studied worldwide
primordia emerge as protuberances at locations such that the angle *phi*
subtended at the apical center by two successive primordia is equal to
the *golden angle {2**p(1-(1/t)}
*where
*t*
is the *golden mean* and is approximately equal to *137.5* degrees.
The *golden mean* is the most irrational number and is associated
with the *Fibonacci* mathematical sequence 1, 1, 2, 3, 5, 8, ......
where each term is the sum of the two previous terms and the ratio of each
term to the previous term approaches the *golden mean* approximately
equal to *1.618*. The *golden mean* is the most irrational number
in the sense that rational approximations converge very slowly to
*t*
as compared to other irrational numbers. Irrational numbers are numbers
such as *square root of 2* which has an infinite number of non-periodic
decimals. Rational approximations such as *p/q* where *p* and
*q*
are integers are used to represent irrational numbers. The *golden mean*
had a special significance in ancient cultures. The significance of the
*golden
mean* throughout recorded history in science, culture and religion has
been discussed (Hargittai and Pickover, 1992; Hargittai, 1992
References
). The surprisingly precise geometrical placement of plant primordia results
in the observed
*phyllotactic patterns*, namely, the familiar spiral
patterns found in the arrangement of leaves on a stem, in florets of composite
flowers, the patterns of scale on pineapple and pinecone, etc. The word
*phyllotaxis*
is of Greek origin (*phyllon*, leaf and *taxis*, arrangement)
and literally means the study of the disposition of leaves on the stem.
*Phyllotaxis*,
in a broader sense now includes the study of arrangement of all plant elements
which originate as primordia.

The mathematical precision underlying placement
of primordia is intrinsic to plant growth processes throughout the plant
kingdom. Therefore, growth processes are governed by universal laws independent
of plant species, related plant genetics, climate and geographical location.
Botanists, physicists and mathematicians alike have been fascinated by
such universal rhythms governing plant growth and the field of *phyllotaxis*
has a long history of more than 150 years. A comprehensive review of *phyllotaxis*
has been given by Jean (1984; 1994)( References
).

The placement of successive primordia with
a constant divergence angle equal to the *golden angle* is associated
with the following observed ergonomic characteristics of plant growth :
(1) The spacing of leaves on shoot is such that no two successive leaves
are placed exactly one over the other and all the leaves get maximum exposure
to sunlight for photosynthesis purpose. (2) With formation of flower on
the shoot, upward growth gives place to lateral (planar) growth of primordia.
These primordia develop into florets (small flowers) on the capitulum in
beautifully arranged counter-rotating spiral pairs called "*parastichy
pairs*". The *Fibonacci* numbers, namely, 1, 1, 2, 3, 5, 8, 13,
21 .... etc., underlying the *golden angle* are exhibited by the parastichy
pairs e.g., 5 spirals in the clock-wise (anti clock-wise) direction are
always accompanied by 8 spirals in the counterclock-wise (clock wise) direction.
(3) The beautiful patterns of the parastichy pairs also incorporate maximum
packing efficiency of seeds which develop from florets . (4) The *golden
angle* ensures selfsimilarity in geometrical shape (Jean, 1994), i.e.,
identical form for all sizes of capituli of a particular species, e.g.,
sunflower, daisy, etc.

Selfsimilar (fractal) branching structure is seen in the hanging roots of the banyan tree

On the left is shown the selfsimilar (*fractal*)
structure of florets in cauliflowers. Cumulus clouds resemble cauliflowers
closely. On the right is shown the *fractal* branching structure of
a shrub which closely resemble the path of branched lightning.

Selfsimilar growth, ubiquitous to nature (living and nonliving), is therefore governed by universal dynamical laws which are independent of the exact details (chemical, physical, physiological, electrical, etc.) of the dynamical system, i.e., a system which evolves with time.