On the Fibonacci Numbers and the Golden Ratio

Claude Robert Cloninger, MD


             The spiral pattern of growth and development observed in human consciousness is pervasive throughout all of ontogeny and phylogeny, including even plants and ancient sea animals. The spiral is the most ubiquitous mathematical form in development of all life forms. The logarithmic spiral is also called the Fibonacci spiral in honor of the mathematician Fibonacci of Pisa. In 1202, Fibonacci described the numerical sequence that is most ubiquitous in growth and development (Stewart 1998). The patterns of development of repetitive elements like leaves, flowers, or seed heads in plants have been extensively studied, The "phyllotaxic" patterns are so regular that a physicist can compare their order with that of crystals (Douady and Couder 1996). All botanical forms can be explained by developmental processes in which there is a succession of discontinuous events producing bifurcations in which new primordial cells are produced where and when there is enough space for their formation (Douady and Couder 1996). The great diversity of whorls and spiral patterns occurring in nature is the result of this succession of discontinuous events. About 90% of all plants exhibit a spiral pattern of leaf arrangement that involves a numerical sequence called the Fibonacci series (1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...) in which each successive number beginning with 3 is the sum of the prior two numbers (Cook 1979).

Other rare sequences follow the same iterative principles but begin with different starting numbers. For example, the Lucas series begins with 2 and 1 to give the sequence 2, 1, 3. 4, 7, 11. . , so that beginning with 3, both Fibonacci and Lucas numbers are the sum of the prior two numbers in the sequence. These numbers occur in the pattern of arrangement of leaves when we count the number of times we go around the stem, going from leaf to leaf, as well as counting the  leaves we meet until we encounter a leaf directly above the starting one. For example, if we make three clockwise rotations around the stem before we meet a leaf directly above the first leaf, meeting five leaves along the way, then there are 3/5 clockwise rotations per leaf. For example, common examples of Fibonacci numbers are 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and pear, and 5/13 for pussy willow and almond, where t/n means that there are t turns for n leaves.

            Whenever development or speciation is the result of a sequence of discontinuous events causing bifurcations, a Fibonacci series will result (Douady and Couder 1996). The regularities of form indicate that development and evolution are frequently based on discontinuous jumps in which there are bifurcations producing a new form while retaining all the old forms as part of the context or phase space of life. Cellular differentiation is a self-organizing dynamic system, as described in Chapter 7.

            Fibonacci spirals and related whorls are also pervasive in the development of all plants (e.g., spirals in seed heads) and animals (e.g., proportions of limbs), as well as in the forms of galaxies throughout the universe. The ratio of successive Fibonacci or Lucas numbers to the one preceding it is called the Golden Ratio, which is approximately 1.618034. Alternatively, the ratio of the smaller to the larger number (that is, 0.6180399 ...) was called the golden section or divine proportion. The only way to divide a line so that its parts are in proportion to the whole is by the golden section; then the ratio of the larger part to the whole (i.e., 0.618) is the same as the ratio of the smaller part to larger part. The golden section is symbolized by the Greek letter Phi in honor of the Greek sculptor Phidias (493-430 BO who based the entire design of the Parthenon on this proportion. Phidias also used Phi in creating his statue of Athena, which became the symbol of Athens because it beautifully combined the creative wisdom of the psyche and the indomitable strength of the body in just balance.

            The golden ratio has played a prominent role in figure drawing and aesthetics throughout history because of its frequency of occurrence in normal morphogenesis throughout phylogeny and cosmogenesis. It is prominent in the normal proportion of the human figure, as noted in the works of Phidias, Leonardo da Vinci, and Botticelli (Cook 1979). Even earlier, the Egyptians used the golden ratio in the design of the pyramids.


(Excerpt from Feeling Good  by Claude Robert Cloninger - Oxford University Press, 2004)