Fibonacci Numbers, the Golden Ratio, and Nature's Patterns

Introduction

Throughout history, humans have been seeking to define beauty in quantifiable and meaningful ways. For many observers, the connection between beauty and the rhythmic patterns found in the Fibonacci series is clear. While the Fibonacci series is named for an early thirteenth-century Italian mathematician, the so-called Golden Ratio known as phi is repeated throughout nature and a surprising number of human endeavors alike, including music, architecture, language, and even stock market cycles. In some cases, the manifestation of the Golden Ratio in human endeavors may be unintentional, but in others it is clearly an integral part of the design process. Likewise, in nature, the ubiquity of the Fibonacci series may not be intended to create a harmonious appearance, yet many humans perceive these patterns as pleasing to the eye.

This paper provides an overview and background concerning the Fibonacci series and the Golden Ratio, followed by an examination of how it is manifested throughout nature. A discussion of how the Fibonacci series appears in various human endeavors is then followed by a set of representative mathematics problems based on the Fibonacci series that can be used in a wide range of classroom settings to introduce these concepts to young learners. A summary of the research and salient findings is presented in the conclusion.

Background and Overview of the Fibonacci Series

According to Clawson (1999), a number of famous numeric sequences have been identified over the millennia, but one of the most famous is called the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, … where each new number represents the sum of the two previous numbers. Clawson reports that "this sequence was suggested (and named after) Leonardo of Pisa (1170–1240), who was also known as Fibonacci. He wanted to know how many pairs of rabbits will be produced each year if we begin with a single pair which mature during the first month and then produce another pair of rabbits every month after that. The Fibonacci sequence of numbers answers that question" (Clawson, p. 117).

In Capitalism and Arithmetic: The New Math of the 15th Century, Smith and Swetz (1987) note that Leonardo of Pisa (also known as Fibonacci) was "raised in a Pisan trading colony in Bugia (Bougie), in what is now Algeria; this mathematician studied under the guidance of an Arab master and became convinced that the new numerals and their methods were vastly superior to the Roman numerals commonly employed in Europe. Leonardo, also known as Fibonacci (the son of Bonaccio), became the evangelist of the new knowledge and published his impressions in a book, Liber abaci (1202)" (p. 12). Arab influences on Western mathematicians during this period were significant, and Fibonacci was no exception. In this regard, Clawson (1994) advises that "in 1202 Leonardo wrote Liber Abaci, a book on computing which contained the new [Arabic] numerals. For centuries his book was used as a source book on calculation" (p. 130).

In spite of its seeming simplicity, the Fibonacci series represents one of the more intriguing mathematical sequences identified to date because it connects a number of branches of mathematics; moreover, the pattern is applicable to a wide range of other disciplines as well (Brown & Walter, 2005). According to Brown and Walter, all of the following phenomena are related in some way to the original sequence:

1. The ratio of the length to the width of the Parthenon in Greece.
2. The placement of the navel in Michelangelo's David.
3. The construction of a regular pentagon using only an unmarked straightedge and a pair of compasses.
4. The number of leaves in a pine cone.
5. The reproduction of rabbits (appropriately conceived).
6. The investigation of aesthetically appealing rectangles (p. 63).

Clearly, the Fibonacci series is an important concept for both nature and humankind, and these issues are discussed further below.

The Fibonacci Series in Nature

As one authority points out, "Nature seems to have adopted the Golden Ratio as a geometrical rule in its magical handiwork, from minuscule forms, like atomic structure and DNA molecules, to systems as large as planetary orbits and galaxies" (Clawson, 1999, p. 117). The Fibonacci series can also be found manifested in a wide range of natural phenomena such as quasi-crystal arrangements, reflections of light beams on glass surfaces, the brain and nervous system, and the structure of many plants and animals. In fact, some observers have suggested that the Golden Ratio is a basic proportional principle of nature (Batten, p. 224). Clawson further advises, "Since the Fibonacci sequence is related to how large certain populations grow, it is frequently found in nature. So much interest has been generated regarding this sequence that a Fibonacci Society has been founded to study and record its many surprising properties in the Fibonacci Quarterly" (1999, p. 117). According to Brown and Walter (1990), the Fibonacci Quarterly specializes in the "fall out" of the Fibonacci sequence.

The Fibonacci series also provided a great deal of food for thought for preeminent modern physicists. As Jenkins (2000) reports in Biolinguistics: Exploring the Biology of Language, Albert Einstein's curiosity about the relationship between mathematics and nature was fueled further when he learned about the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. According to Jenkins, "Even though it was not obvious, there was a pattern to these numbers: Each one was the sum of the two numbers before it" (p. 148). The first individual credited with discerning this pattern was Leonardo da Pisa. Jenkins reports: "First concocted in the thirteenth century by an Italian merchant named Leonardo 'Fibonacci' da Pisa, the series had been widely regarded as little more than a numerical curiosity. But then, Einstein learned, botanists had discovered that there were surprising coincidences between the numerical pattern of the Fibonacci series and the growth pattern of many flowering plants" (p. 148).

The continuing emphasis on the Fibonacci series is based on the fact that this series generates the most famous proportion in the history of art and architecture: the Euclidean golden section, or Golden Ratio (shorthand phi) (Smith, 2003). The ratio between any two consecutive values in the series results in the so-called "golden number" to increasing levels of accuracy as the numbers in the series grow larger. For instance, 3:5 = 1:1.666, 21:34 = 1:1.61904, and 55:89 produces 1.61818, which closely approximates the actual golden section value of 1.618034… (Smith). As Batten (2000) reports, "One thing to note is that the Fibonacci sequence has many interesting properties in itself. For example, the sum of any two numbers in the sequence equals the next number in the sequence: 1 plus 1 equals 2, 1 plus 2 equals 3, 2 plus 3 equals 5, 3 plus 5 equals 8, and so on to infinity" (p. 37). More importantly, the ratio of any two numbers in the sequence approaches 1.618, or its inverse, 0.618, after the first few pairs of numbers. The higher the numbers in the sequence, the closer the ratios between them approach 0.618 and 1.618 (Batten).

As Cromer (1997) points out, "Phi = (1 + √5)/2 = 1.618… is one of the two solutions of the quadratic equation x² − x − 1 = 0. Starting with any two numbers, say 3 and 7, a Fibonacci sequence is obtained by making each new term equal to the sum of the last two terms. Thus, starting with 3 and 7 we get the Fibonacci sequence 3, 7, 10, 17, 27, 44, and so on. The ratio of two terms, say 44/27 (= 1.6296…), gets closer and closer to phi (= 1.618…) the farther one goes in the sequence. Phi is also the ratio of line segments in some geometrical figures" (p. 191).

In some cases, identifying a manifestation of the Fibonacci series requires more than a casual examination of how nature uses it. While the series is apparent, for instance, when looking down on a pine cone or at a nautilus's shell, it is less evident elsewhere but just as important to the organism's development. One authority reports:

"As they developed, for example, the branches of a common sneezewort forked in exact accordance with the Fibonacci series. First the seedling's main stem forked (1), then one of its secondary stems forked (1), then simultaneously a secondary and tertiary stem forked (2), then simultaneously three lesser stems forked (3), and so forth. Furthermore, Einstein learned, the numbers of petals of various flowers, too, recapitulated the numbers of the Fibonacci series: an iris almost always had three petals, a primrose five petals, a ragwort thirteen petals, a daisy thirty-four petals, and a Michaelmas daisy either fifty-five or eighty-nine petals" (Jenkins, p. 148).

All of these findings had 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, p. 149). Likewise, the Fibonacci series appears in a variety of other natural settings. As 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."

Conclusion

The literature showed that the Fibonacci series appears throughout nature and many human endeavors, including architecture, music, art, and even language, whether people realize it or not. While Leonardo da Pisa is credited with its discovery in the early thirteenth century, it is clear that humanity has long recognized the beauty inherent in its use. The ancient Greeks understood the Golden Ratio and used it to good effect in their architecture, and the Arabs helped to reignite interest in it. The research also showed that the ratio 1.618 (or 0.618) to 1 appears to represent a fundamental arrangement that is universally regarded as pleasing to the human senses. This number is commonly known as the Golden Ratio, but it has also been called the Golden Section, the Golden Cut, the Divine Proportion, the Fibonacci number, and the Mean of Phidias.