Fibonacci Numbers and the Golden Ratio

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Fibonacci Numbers and the Golden Ratio

• Lizzie Bell

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• Fibonacci
• The Rabbit Problem
• Properties of the Fibonacci Sequence
• Fibonacci Numbers and Pascals triangle
• The Golden Ratio and how to construct it
• Fibonacci Rectangles
• Fibonacci Numbers in Nature
• School Mathematics

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• Leonardo of Pisa (c1170 1250)
• Greatest Mathematician of Middle Ages
• Grew up North Africa
• Liber Abaci 1202
• Posed the Rabbit Problem

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The Rabbit Problem A pair of new born rabbits is
in a confined space. This pair and every later
pair has one new pair per month. How many after
1,2,3, months assuming no deaths occur?
fn1 1 2 3 5 8
n1. 2. 3. 4. 5. 6.
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The Fibonacci numbers
Each term is the sum of the previous two
terms How many rabbits after 100 months? N
months?
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• fn1 fn fn-1
• the characteristic equation is
• r2 - r - 1 0
• this has two real roots
• F1 v 5 1.6180 f 1- v 5 -0.6180 F, f -
  phi
• 2 2
• So fn A Fn B fn
• Substitute in initial values
• f0 1 A B
• f1 1 A F B f

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? A (1 v 5) B -(1 v5)
2 v5 2 v5
)
Hence fn 1 1v5 n 1 - v5 n v5
2 2 Or fn 1 ( Fn fn
) v5
)
)
)
)
)
-
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Ratio of successive Fibonacci terms
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• fn Fn - fn
• fn-1 Fn-1 fn-1
• lim Fn - fn lim Fn
• n ?8 Fn-1 fn-1 n ?8 Fn-1
• lim F F
• n ?8

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• Properties of Phi
• F -1
• f
• F f 1
• F f -1
• F f2 1

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Pascals Triangle Pascals triangle is made up of
the coefficients of the expansion of (1 x)n
Arranged into a triangle 1 1 1 1 2 1 1 3 3
1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1
n n x n x n x 0 1 2 n
(
(
(
(
)
)
)
)




1
1
2
3
5
8
13
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• Proof
• The first two sums are 1 so we just need to show
  that the sum of the n2th diagonal is the sum of
  the numbers on the n1 and nth diagonal.
• Observe that the nth diagonal is
• n-1 , n-2 , n-3 , n-4 ,
• 0 1 2 3
• And the (n1)th diagonal is
• n , n-1 , n-2 ,
• 0 1 2

1 1 1 1 2 1 1 3 3 1 1 4 6 4
1 1 5 10 10 5 1 1 6 15 20 15 6 1
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• So the sum of the can be written as
• n n-1 n-1 n-2
  n-2
• 0 0 1 1
  2
• Using
• n n n1
• k k1 k1
• The sum can be written as
• n n-1 n-2
• 0 1 2
• This is the sum of all the diagonals along the
  (n2)th rising diagonal

(
(




1 1 1 1 2 1 1 3 3 1 1 4 6 4
1 1 5 10 10 5 1 1 6 15 20 15 6 1





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The Golden Ratio
x
1-x
B
C
A
1 x ? x2 1 - x x 1-x ? x
-1 ?5 So each ratio is 1/ -1 ?5
2 2 2 1 ?5 F -1 ?5
2

)
)
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Constructing a Point that Divides a Segment into
the Golden Ratio
E
D
A
B
C
EB ?1 (½)2 BD (?51) BC F 2
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• Golden Ratio
• Well balanced
• Used in everyday life. E.g. credit cards
• Architecture
• Human perception of beauty

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• Fibonacci Rectangles
• Constructed by drawing two squares of side 1,
  next to each other. On top of these is a square
  of side 2. Another square is drawn of length 3
  next two both the 2-square and one of the
  one-squares. Continue adding squares around the
  picture.
• Each new square has a side the length of the sum
  of the previous two squares lengths.

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• Fibonacci Spiral
• It does not alter its shape as its size increases
• Growth in nature
• E.g. molluscs
• Also seen in rams horns and elephants tusks

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• Fibonacci Numbers in Nature
• Seed heads
• Pinecones

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• Conclusion
• Real World
• In schools
• In advanced maths