148).
All of these findings caused a profound impact on the young Einstein: "Since there was this wonderful parallel between Numbers and Nature, then why not use the laws of mathematics to articulate the laws of Nature? 'It should be possible by means of pure deduction,' he concluded, "to find the picture-that is, the theory of every natural process, including those of living organisms" (quoted in Jenkins at p. 149). Likewise, the Fibonacci series appears in a variety of other natural settings. In this regard, Brumbaugh, Ashe, Rock and Ashe (1997) point out, "For example, if the clockwise and counterclockwise spirals of a sunflower are counted, the results will always be two successive terms in the Fibonacci sequence.
Fibonacci Series in Human Endeavors
For some unknown reason, the ratio 1.618 (or 0.618) to 1 seems to be pleasing to the senses. The Greeks-based much of their art and architecture upon this proportion, calling it the Golden Mean. Among mathematicians, it is commonly known as the Golden Ratio, an irrational number defined to be (1 + ?5)/2; the Golden Ratio has also been called the Golden Section, the Golden Cut, the Divine Proportion, the Fibonacci number, and the Mean of Phidias (Batten). According to this author, "It is the mathematical basis for the shape of Greek vases and the Parthenon, sunflowers and snail shells, the logarithmic spiral and the spiral galaxies of outer space" (p. 223). While the ratio is pleasing to the human senses and can be used intentionally, nature is not so selective in its use of the Fibonacci series to achieve a harmonious state, but rather as a matter of consequence in response to environmental needs. For example, according to Padovan (1999), "In any case, the Fibonacci numbers that occur in plants tend to be only the first few terms of the series. It is true that we find pentagons, five-petalled flowers, equiangular spirals, serial arrangements of leaves on branches. But all these patterns are governed by the way in which they have been made...Yet...we still find people writing as though nature uses the golden section in order to be harmonious" (p. 48).
Nevertheless, there is something "magical" and appealing about the ratio that has attracted the attention of countless mathematicians and artists alike over the years. For example, Batten points out, the regularity with which the Fibonacci series is found in nature "seems to imply a natural harmony that feels good, looks good, and even sounds good. Music, for instance, is based on the eight-note octave. On a piano, this is represented by five black keys and eight white ones -- thirteen in all. Perhaps it is no accident that the musical harmony that seems to give us the greatest satisfaction is the major sixth. The note E. vibrates at a ratio of 0.625 to the note C, just slightly above the Golden Ratio" (p. 223). The author adds that that the human ear is also an organ that happens to be shaped in the form of a logarithmic spiral (Batten).
The Fibonacci series converges towards the Golden Section and can be formed by adding a series of squares to the longer side of each preceding figure, thereby creating a spiral; by beginning with a ? rectangle as the core figure in place of a square, it is possible to obtain a true golden section sequence as shown in Figure 2 below.
Figure 2. Generation of the Fibonacci series
Source: Padovan at p. 133.
Using the circle as an example, Smith reports that the opposite to the golden angle of 137.5 degrees is 222.5 degrees. The golden number 1.618 results when 222.5 is divided by 137.5, and when 360 is divided by 222.5; likewise, when the fractions are reversed the result is 0.618, the 'golden cut' (Smith). It is therefore apparent that the golden section ratio is one of the main principles behind growth in nature, whether the branching of plants, the venation of leaves, and the arrangement of florets. As Smith emphasizes, though, "There is nothing mystical about this; it is not a blueprint designed to create beauty but to enable plants to achieve the most efficient growth and take maximum advantage of their environment. Nature abounds with Fibonacci values, which suggests that the series offers an ideal window of opportunity for optimum development" (p. 80).
Figure 3. Fibonacci in three dimensions
Source: Smith at p. 79.
According to Smith, "Perhaps it is more than a coincidence that the golden section was adopted by the Pythagorean Brotherhood as the prime yardstick of beauty. It is probable that Pythagoras learnt of the mysteries of phi during a stay in Egypt. Between...
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