Fibonacci Numbers and The Golden ratio in nature Charlie Reid

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Fibonacci Numbers and The Golden ratioin
natureCharlie Reid
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What do all these have in common?
They are all beautiful!
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Mathematicians think they have discovered the
maths behind it!

• Explain Fibonacci numbers
• How this relates to Phi (The Golden Ratio)
• How this relates to human perception of beauty
• Class Guinea pig!
• Conclusion

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Leonardo de Pisa, filius Bonaccio Fibonacci
(1170 1240)

• Educated in North Africa due to father
• Accompanied father on travels
• Finished travelling in 1200 started work on
  Liber Abaci, (Book of Calculating)
• Tried to revive ancient mathematics
• Led to discovery of one of the most significant
  sequences ever discovered

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Fibonacci Sequence

• Rabbit Problem (Frederic II, Holy Roman Emperor)
• A pair of adult rabbits produce a pair of baby
  rabbits once every month.
• Each pair of baby rabbits require one month to
  grow to be adults and subsequently produce one
  pair of baby rabbits each month thereafter.
• Determine the number of pairs of adult and baby
  rabbits after some number of months.
• It is assumed the rabbits are immortal, and
  incest is OK!
• 1,1,2,3,5,8,13,21,34,55,89,144,
• Fib (n2) Fib (n1) Fib (n)

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Why is this so interesting?

• Appears in so many forms of nature, for example
  the number of petals on flowers

(2) Crown of Thorns
(3) Iris
(5) Pinks
(1) White Calla
(13) Cineraria
(21) Aster
(34) Daisy
(8) Coreopsis
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We can see spirals are consecutive Fibonacci
numbers
The same can be said about the flower, only it
has more spirals and is harder to show!
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Is that really amazing?
NO!
But this is
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Mathematically find the limit

• We can find the nth term of the sequence by
  Binets formula
• Fib (n) (1 / v5)((1 v5)/2)n ((1 -
  v5)/2)n
• Using this we can find the limit of the sequence
  as

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• Now we need to find the limit of the sequence as
  n?8 of
• Fib (n 1) a
• Fib (n)
• a (1 / v5)((1 v5)/2)(n1) ((1 -
  v5)/2)(n1)
• (1 / v5)((1 v5)/2)n ((1 -
  v5)/2)n
• Divide everything through by (1 / v5)
• a ((1 v5)/2)(n1) ((1 -
  v5)/2)(n1)
• ((1 v5)/2)n ((1 - v5)/2)n
• Divide through by (1 v5)/2)n
• a ((1 v5)/2)(n1)/((1 v5)/2)n((1
  - v5)/2)(n1)/((1 v5)/2)n
• 1 ((1 -
  v5)/2)n/ ((1 v5)/2)n
• hence
• Fib (n 1) ? (1 v5) as n
  ? 8
• Fib (n) 2
• Which approximately 1.618033989

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What relevance does this have?

• Aesthetic beauty!
• Golden Rectangle (ratio of sides 1Phi)

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So where does this apply to our examples at the
start?

• They all exhibit the Golden Ratio!

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Spira Mirabilis
Even the human ear exhibits this spiral
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Class Volunteer

• Arm length 78.5cm
• Forearm length 50cm
• Arm length 1.57
• Forearm length
• Height 180cm
• distance to belly button 111cm
• Height 1.62
• distance to belly button

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Who would be you valentine?
Colin Farrell
Matthew McConaughey
Tom Cruise
Harrison Ford
1.45
1.55
1.35
1.60
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Conclusion

• Phenomenal interest to human beings
• There has been no actual formula constructed to
  show that
• beauty k x Golden ratio
• Although there does appear to be a relationship
• There is no formal accepted description of
  aesthetic judgement in mathematics (Livio 2002).

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Any questions?