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...

It is probable that Pythagoras learnt of the mysteries of phi during a stay in Egypt. Between 300 BC and AD 500 the School of Alexandria was the greatest mathematical school of ancient times" (p. 83). There is evidence that the Golden Ratio was known to Islamic architects as well; for instance, the pattern appears in the Dome of the Rock, Jerusalem and the Alhambra in Granada (Smith). In addition, there is evidence of what might be termed "visual phi" in India in the overall dimensions of the temple city of Jambushuar; likewise, the Taj Mahal is situated within a phi rectangle, and so forth (Smith).

Mathematically the golden section is produced by the formula: root five minus one over two:. Numerous tests have shown that subjects prefer a rectangular form which clusters round the golden section proportion. Gustav Fechner is the best known of the many nineteenth-century experimental psychologists who investigated the appeal of the phi rectangle. The Fechner graph, which is well-known, plots the preference of subjects for a range of rectangles from a square to the ratio of 2:5. Three- quarters of the subjects opted for rectangles at or near the phi proportion; comparable findings were obtained in an experiment with students from different disciplines at Sheffield University in 2002 (Smith, p. 83).

In music the golden ratio is apparent in the organization of the sections in the music of Debussy and Bartok. For example in Debussy's 'Image, Reflections in Water' the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position. In Bartok's Music for Strings, Percussion and Celeste the xylophone progression occurs at the intervals 1:2:3:5:8:5:3:2:1.

According to Lake and Maillard (1956), "Formulated by Vitruvius and taken up again during the Renaissance, the golden section or divine proportion (or gate of harmony) is the ideal relation between two magnitudes, expressed numerically as and demonstrated in many masterpieces of different arts, applied consciously or, more often, by instinct" (p. 262). As a result, the golden section has long associations with painting. For instance, Smith reports, "Claude Monet certainly seems to have embraced it either by accident or design in several paintings" (p. 80). As Voltaire noted early on, "There is a hidden geometry in all the arts that the hand produces" (quoted in Lake & Maillard at p. 262). While the golden section was not the only constant to which the Cubists referred for the mathematical organization of their canvas, it reflected the profound need for order and measure that they felt more through sensibility and reason than as a result of calculation (Lake & Maillard, p. 262).

The best known example of a phi rectangle translated into architecture is the end elevation of the Parthenon. By common consent it is regarded as one of the most harmonious facades ever conceived, even though most who see it have no idea of its connection with the golden section ratio" (p. 80).

Figure 4. The Parthenon and phi

Source: Smith at p. 84.

The elevation further divides into phi proportions, with the colonnade and crepidoma (or base) representing 1.62 and the entablature and pediment together equaling exactly 1.0 (Smith). Other examples of the phi ratio in early human architectural endeavors are the temple of Athena at Priene and the Arch of Constantine in Rome. Le Corbusier based his elaborate Modulor on the golden section, in his case based on the proportions of the human body [see Figure 5 below] (Smith).

Figure 5. Phi and the human body.

Source: Smith at p. 84.

Notwithstanding the foregoing spectacular examples of phi manifested in architecture, Smith suggests that.".. The most magnificent testimony to the aesthetic power of the golden section is not the Parthenon but the Cathedral of the Virgin at Chartres. In Chartres, proportion is experienced as the harmonious articulation of a comprehensive whole; it determines the ground plan as well as the elevation; and it 'chains', by the single ratio of the golden section, the individual parts not only to one another but also to the whole that encompasses them all" [see Figure 6 below] (p. 84).

Figure 6. The Cathedral of Chartres, interior.

Source: Smith at p. 141.

The Cathedral of Florence also contains evidence of the Fibonacci series, a fact that Trachtenberg…

Batten, D.F. (2000). Discovering artificial economics: How agents learn and economies evolve. Boulder, CO: Westview Press.

Brown, S.I. & Walter, M.I. (2005). The art of problem posing. Hillsdale, NJ: Lawrence Erlbaum Associates.

Brumbaugh, D.K., Ashe, J.L., Rock, D. & Ashe, D.E. (1997). Teaching secondary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.

Clawson, C.C. (1999). Mathematical sorcery: Revealing the secrets of numbers. New York: Perseus Publishing.

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