The Fibonacci Sequence

1
The Fibonacci Sequence
Chapter 6
2
Fibonacci Sequence

• The Fibonnaci sequence begins with 1,1 and then
  generates new numbers by adding the previous two
• 1,1,2,3,5,8,13,21F(n)F(n-1)f(n-2)
• Two points of the presentation, with respect to
  Fibonacci are
• Golden Ratio
• Consecutive terms are relatively prime

3
Golden Ratio

• If we look at the ratio of terms we notice the
  ratios converge to a value of 1.6180339887
• R11/11
• R22/12
• R33/21.5
• R45/31.666
• R58/51.6

4
What is the Golden Ratio?

• Approximately 1.6180339887
• First coined by the Greeks to the Golden
  Rectangle, Golden Section, phi, as well as
  the Divine proportion by Leonardo DaVinci
• Def The length to width of rectangles used n art
  and nature. This ratio in considered to be the
  most agreeable arrangement, mathematically and
  artistically, to the eye .
• Subjects to pyramids, Greek architecture and
  sculpture, Parthenon and several other buildings.

5
Leonardo da Vinci

• The most famous and blatant use of the golden
  ratio has come in the works of Leonardo da Vinci.
• Uses the golden ratio in the Mona Lisa framing
  her head as well as the rest of her body parts
  inexact proportion to the golden rectangle.
• He went on to include the golden ratio in other
  works such as Vitruvian Man and Virgin and Child
  with St. Anne.
• Toward the end of his live math, particularly the
  golden ratio, began to dominate everything he
  created.

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Successive terms are Relatively Prime

• Proof
• Suppose dgt1 divides Fn and Fn1. Then their
  difference Fn1 Fn Fn-1 will also be
  divisible by d. From this and the formula
  Fn-Fn-1 Fn-2, it can be concluded that d/ Fn-2.
  Working backward, we can show that Fn-3, Fn-4,
  , and finally F1 are all divisible by d. But
  F11, which is certainly not divisible by any
  dgt1. This contradiction invalidates our
  supposition and therefore proves the theorem
• (pg. 268)

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Fibonacci in Nature

• The Fibonacci Sequence is found abundantly in
  nature.
• Some of the things that follow this sequence are
• Petals on a flower
• Rabbits
• Pine cones
• Leaf arrangements
• Bananas
• Fingers

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Fibonaccis Rabbits

• This was the original problem Fibonacci
  investigated.
• Suppose a newly born pair of rabbits, one
  male,one female, are put in a field. Rabbits are
  able to mate at the age of one month and a female
  can produce another pair of rabbits. Suppose
  that our rabbits never die and that the female
  always produced one new pair(one male and one
  female) every month forward.

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Fibonaccis Petals

• On many plants the number of petals is a
  Fibonacci number
• 3 petals Lily, Iris
• 5 petalsButtercup, Wild Rose, Laskspur
• 8 petals Delphinium
• 13 petals Ragwort, Corn Magnolia
• 21 petals Black-Eyed Susan, Aster

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Fibonaccis Pine Cone

• Pine cones show the Fibonacci Spirals
• Can you see the spiral?

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Fibonacci Fingers

• Look at your hand
• You have
• 2 hands each having
• 5 fingers, each having
• 3 parts separated by
• 2 knuckles

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Assignments

• In text Page 272 (1,2)
• Write a paragraph describing what you found to be
  the most interesting aspect of the Fibonacci
  Sequence and why.

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References

• The History of Mathematics, An Introduction,
  David M. Burton, McGraw Hill, 2003, pg (267-278)
• http//courses.wcupa.edu/jkerriga/Lessons/David20
  Montalro/p2.html
• http//www.facstaff.bucknell.edu/udaepp/090/w3/chr
  ism2.html
• http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
  acci/fibnat.html