A Brief History Lesson

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A Brief History Lesson

• Born Leonardo of Pisa
• 1175-1250
• Travelled extensively with Diplomat father as a
  child
• Educated in North Africa
• Met the Hindu-Arabic numerals
• 0,1,2,3,4,.
• Far superior to Roman numerals back in Italy
• I,V,X,L,C,D,M..

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Liber Abbaci

• Returned to Pisa
• 1202
• Penned Liber Abbaci
• Wrote under the name of Fibonacci
• Highlighted enormous advantages of adopting
  Hindu-Arabic system
• Zero introduced
• MCMXCIV??! - not good for arithmetic!

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...And a certain rabbit problem!

1 1
2 3
5
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Edouard Lucas

• 1842-1891
• Studied the sequence derived from the rabbit
  problem
• Beautiful, elegant
• Diverse array of applications
• Named the sequence after Fibonacci
• BUT had been discussed by Indian scholars prior
  to Liber Abbaci

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The Fibonacci Seqence

• (01) (11) (12) (23)
  (35) (58) (813)
  (1321) (2134)
• 1, 1, 2, 3, 5, 8, 13, 21, 34,
  55,
• This can be expressed as a general mathematical
  equation as follows
• Fi1 Fi Fi -1

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The Lucas Sequence

• Didnt leave himself out of the glory!
• (21) (13)
  (34) (47) (711) (1118)
  (1829) (2947)
• 2, 1, 3, 4, 7, 11, 18, 29, 47,
  76..
• This can be expressed as a general Mathematical
  equation as follows
• Li1 Li Li-1
• Genetic Mutations??

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Genealogy of Male Bumble Bees!

• Derives same Fibonacci sequence and recursion
  relations as rabbit problem
• However, on unrealistic basis all bees are
  immortal!

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Bee Family Tree
Generation Males Females Total
5 2 3
5 4 1
2 3 3
1 1 2
2 0 1
1 1
1 0 1
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The Plant World

• One of the most reliable
• places to look for Fibonacci

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Number of Petals
1
2
3
5
13
8
21
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Number of Petals

• Most flowers have a Fibonacci number of petals.
• Natural variation in a species
• Mutations
• Universal tendency towards a particular Fibonacci
  number
• Eg. Daisies tend to have 34 petals

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Growing Points
Number of stems at each horizontal level of
development 13
8 5
3 2 1 1
Achillea ptarmica, the sneezewort
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Leaf Arrangements

• Need to maximise exposure to moisture and
  sunlight
• Resources available cyclically
• Motivates cyclic growth
• No blocking
• Natures Solution Leaves generated in Fibonacci
  formation

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Fibonacci Growth Patterns

• Generated every 2/5, 3/5, 3/8, 8/13 or 5/13 of a
  circle of growth
• 5/13 of a turn ? 13 offshoots produced during 5
  complete growth turns
• Phyllotactic ratios are Fibonacci ratios

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Spirals

• Beautiful spiral formations
• Seen to curve both to the left and the right
• Number of spirals in a particular direction
  usually Fibonaccian
• Tend to particular Fibonacci number?

Echinacea purpura, the Coneflower
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The Sunflower

• Number of spirals overwhelmingly 34 in one
  direction, 55 in the other
• Some deviations
• (89, 144) and (144 and 233)
• Some double Fibonaccian
• (64,110)
• Genetic Mutation

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Pinecones

• 8 parallel rows of spirals
• Ascending gradually
• Clockwise direction.

• 13 more spirals
• Rising steeply
• Counter-clockwise direction.

99 chance of Fibonacci number of spirals!
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Pineapples

• 5 spiralling gradually
• 8 spiralling at a medium slope
• 13 spiralling steeply

Study of 2000 pineapples not one deviated from
Fibonacci pattern!
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The Equiangular Spiral

• Characterises growth within the animal world
• ? ?
• Golden rectangle Equiangular spiral
  Nautilus shell

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The Golden Number, Ø

• Divide each Fibonacci number by the preceding
  number in the sequence
• ? 1/1 1
•   2/1 2
• 3/2 15
• 5/3 1666..
• 8/5 16  
• 13/8 1625
•   21/13 161538...

ØThe limit of the sequence of ratios of
successive Fibonacci numbers
1.618033988749894848204586
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Equiangular Spiral

• Logarithmic
• Based on template of successive Fibonacci squares
• 4 squares per rotation of the golden rectangle
• points on spiral 1.618 times as far from the
  centre after a quarter-turn
• ? One turn points on radius 1.6184 4 6.854
  times further out than where curve last crossed
  same radial line

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Where else?!