Phi et al

Paper Summary:

This paper provides an overview and a background concerning the Fibonacci series and the Golden Ratio, followed by an examination of how it is manifested throughout nature. In addition, a discussion of how the Fibonacci series is found in various human endeavors is followed by a series of representative mathematics problems based on the Fibonacci series that can be used in a wide range of classroom settings to help introduce these concepts to young learners. Finally, a summary of the research and salient findings are presented in the conclusion. Several tables and diagrams are included with the paper.

Outline:
Review and Discussion
Background and Overview
Fibonacci Series in Nature
Fibonacci Series in Human Endeavors
Math Problems Using the Fibonacci Series
Conclusion

From the Paper:

"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). The ratio between any two values in the series results in the so-called "golden number" to increasing levels of accuracy the higher the numbers in the series. Therefore, for instance, 3:5 = 1:1.666, 21:34 = 1:1.61904, 55:89 produces 1.61818, which is an approximate of the actual golden section number of 1.618034 ... . In this regard, Batten (2000) reports that, "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". Likewise, and 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 ratio of any number taken to the next higher number, known as "phi," is approximately 0.618 to 1 and to the next lower number is about 1.618. The higher the numbers in the sequence, the more close to 0.618 and 1.618 are the ratios between the numbers. As Cromer 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. "

Sample of Sources Used:

• 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.
• ---. (1994). The mathematical traveler: Exploring the grand history of numbers. New York: Plenum Press.

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