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Fibonacci Sequence and the Golden Ratio

Fibonacci Sequence and the Golden Ratio. Robert Farquhar MA 341. Who Was Fibonacci?. European mathematician 1175-1250 Real name Leonardo of Pisa. Author of Liber abaci or Book of the Abacus. Remembered today because of Edouard Lucas.

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Fibonacci Sequence and the Golden Ratio

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  1. Fibonacci Sequence and the Golden Ratio Robert Farquhar MA 341

  2. Who Was Fibonacci? • European mathematician 1175-1250 • Real name Leonardo of Pisa. • Author of Liber abaci or Book of the Abacus. • Remembered today because of Edouard Lucas

  3. What Is the Fibonacci Sequence and Why Is It Significant? • Generalized sequence of first two positive integers and the next number is the sum of the previous two, i.e. 1,1,2,3,5,8,13,21,… • Has intrigued mathematicians for centuries. Shows up unexpectedly in architecture, science and nature (sunflowers & pineapples).

  4. Sunflower Seedhead andthe Fibonacci Sequence

  5. Fibonacci Sequence Cont’d • Has useful applications with computer programming, sorting of data, generation of random numbers, etc.

  6. Fibonacci Sequence and the Golden Ratio • A remarkable property of the sequence is that the ratio between two numbers in the sequence eventually approaches the “Golden Ratio” as a limit. • 1/1=1 2/1=2 3/2=1.5 5/3=1.6667 • 8/5=1.6 13/8=1.625 21/13=1.6154

  7. Fibonacci Sequence and the Golden Ratio

  8. Golden Ratio • Famous Irrational number phi—1.61803398… • Only positive # that becomes its own reciprocal by subtracting 1 • Used extensively by Ancient Greeks in architecture

  9. Golden Ratio Continued • Ratio also shows up in many famous paintings • Da Vinci studied extensively the ratio and body proportions.

  10. Summary • Fibonacci Sequence has been around for hundreds of years. • Golden Ratio has been used for thousands of years. • Both concepts are still in use today and continue to interest us.

  11. References • Gardner, Martin (1979). Fibonacci and Lucas Numbers. In Mathematical Circus (pp. 152-160), New York:Alfred Knopf • Gardner, Martin (1961). Phi: The Golden Ratio. In Mathematical Puzzles & Diversions (pp.89-100), New York: Simon and Schuster. • Web Page “The Golden Ratio” by Steve Blacker, Jeanette Ploanski and Marc Schwach, • http://www.geom.umn.edu/~demo5337/s97b/fibonacci.html

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