Fibonacci%20Sequence%20and%20the%20Golden%20Ratio

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Fibonacci Sequence and the Golden Ratio

• Robert Farquhar
• MA 341

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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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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).

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Sunflower Seedhead andthe Fibonacci Sequence
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Fibonacci Sequence Contd

• Has useful applications with computer
  programming, sorting of data, generation of
  random numbers, etc.

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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.
• 1/11 2/12 3/21.5 5/31.6667
• 8/51.6 13/81.625 21/131.6154

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Fibonacci Sequence and the Golden Ratio
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Golden Ratio

• Famous Irrational number phi1.61803398
• Only positive that becomes its own reciprocal
  by subtracting 1
• Used extensively by Ancient Greeks in architecture

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Golden Ratio Continued

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

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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.

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References

• Gardner, Martin (1979). Fibonacci and Lucas
  Numbers. In Mathematical Circus (pp. 152-160),
  New YorkAlfred 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.h
  tml