2.2 Golden Mean and
Selfsimilar, Fractal Geometrical Structures in Nature
Animate and inanimate structures in nature
exhibit selfsimilarity in geometrical shape (Stevens, 1974; Jean, 1994;
Freeman, 1987; 1990 Reference ), i.e.,
parts resemble the whole object in shape. The most fundamental selfsimilar
structure is the forking (bifurcating) structure (Jean, 1994 Reference
) of tree branches, tree roots, river tributaries, branched lightning,
etc. The complex branching architecture is a selfsimilar fractal
since branching occurs on all scales (sizes) and forms the geometrical
shape of the whole object. Selfsimilar structures incorporate in their
geometrical design the noble numbers, i.e., numbers which are functions
of the golden mean t
and are characterized by fivefold symmetry of the pentagon and dodecahedron.
For example, the ratio of the length of the diagonal to the side in a regular
pentagon is equal to the golden mean t
equal to (1+Ö
5)/2 @1.618.
The golden mean t
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 t
. The golden mean t
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 Ö
2 which has an infinite
number of non-periodic decimals. Rational approximations such as p/q
where p and q are integers and are used to
represent irrational numbers. The golden meant
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
Reference
). Selfsimilar spiral structures such as on the shell of the very old mollusk
called Nautilus pompilius (Jean, 1994 Reference
) incorporate the golden mean in their radial growth. Thompson described
that the nautilus followed a pattern originally described by Rene Descartes
in 1683 as the equiangular spiral and subsequently by Jacob Bernoulli
as the logarithmic spiral (West, 1990 Reference
).The commonly found shapes in nature are the helix and the dodecahedron
(Stoddart, 1988; Muller and Beugholt,1996 Reference
) which are signatures of selfsimilarity underlying Fibonacci numbers
.
The association of noble numbers with growth of selfsimilar patterns
has been established quantitatively in plant phyllotaxis in botany.
A summary of documented evidence collected over a period of more than 150
years is given below and will help understand the association between noble
numbers and selfsimilar patterns in the plant kingdom. Phyllotaxis
is the study of the arrangement of all plant elements which originate as
primordia on the shoot apex. The botanical elements which constitute plants
are branches, leaves, petals, stamens, sepals, florets, etc. These plant
elements begin their existence as primordia in the neighborhood of the
undifferentiated shoot apex (extremity). Extensive observations in botany
show that in more than 90% of plants studied worldwide (Jean 1994;
Stewart 1995 Reference ) primordia emerge
as protuberances at locations such that the angle subtended at the apical
center by two successive primordia is equal to the golden angle
( = 2p
(1-1/t)
corresponding to approximately
137.5 degrees. Theoretical
studies show that outside the set of noble numbers the structures
are not selfsimilar. 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 pattern of scales on pineapple and
pine cone, etc. Further, such selfsimilar patterns ensure identical
geometrical design (shape) for all sizes of a single species such as daisy
flowers of all sizes. The phyllotactic patterns, while pleasing
to the eye, also incorporate maximum packing efficiency for fruits and
seeds.