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 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 t 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.