Fibonacci

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Fibonacci

• Megan Sorenson
• Sanya Mcgahan
• Sarah Harbitz
• Natasha Shumaker

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Greatest European Mathematician of the Middle
Ages

• Born in Pisa, Italy around 1175
• Father was a merchant/
• A customs officer or
• Represented merchants
• Guglielmo Bonacci
• Leonardo of Pisa, or Leonardo Pisano
• Fibonacci is short for Filius Bonacci son of
  Bonacci

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• Born in Italy, educated in North Africa
• Fathers job brought them there.
• Traveled with his father
• Recognized advantages of different mathematical
  systems.
• Hindu-Arabic system

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BOOKS

• Liber abaci (1202) arithmetic and algebra
• Practica geometriae- (1220)- geometry problems
• Flos (1225)-This book was written to answer some
  questions proposed at King Frederrick's court.
• Liber quadratorum (1225)- book of squares,
  number theory, methods in finding pythagorean
  triples.

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Pythagorean Triple

• Thus when I wish to find two square numbers whose
  addition produces a square number, I take any odd
  square number as one of the two square numbers
  and I find the other square number by the
  addition of all the odd numbers from unity up to
  but excluding the odd square number. For example,
  I take 9 as one of the two squares mentioned the
  remaining square will be obtained by the addition
  of all the odd numbers below 9, namely 1, 3, 5,
  7, whose sum is 16, a square number, which when
  added to 9 gives 25, a square number.

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• The Fibonacci Sequence1, 1, 2, 3, 5, 8, 13,...

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• Recursive Formula for the Fibonacci
  SequenceF(1)1F(2)1F(n)F(n-1)F(n-2)

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• The seemingly simple recursive sequence has
  fascinated mathematicians for centuries, as its
  properties occur in an number of different
  phenomenons including - aesthetic doctrines of
  the ancient Greeks   - art   - music   -
  architecture - growth patterns of plants - the
  population rates of rabbits

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• ShortcutsIf you know F(n), is there a easy way
  of calculating F(n1) without knowing F(n-1)?
• The answer is YES!
• F(n1) round F(n)Phi
• Phi 1.618... Golden Number

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• Finding a Formula for the Fibonacci
  SequenceRecursive Formula allows you to find
  the nth number in a sequence if preceding terms
  are known.Let F(1)1 and F(2)1.Can you create
  a recursive formula for the nth number in the
  sequence?

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Example

• If F(4)3, what is F(5)?F(5)
  round(3Phi)       round (3 x 1.618...)      
  round (4.854...)       5

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Wait!

• But there is one problem...Does the shortcut
  always work?  Let's try some other numbers.
• Clarification of ShortcutLook at the first few
  numbers in the Fibonacci Sequence 1, 1, 2, 3,
  5,...

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• F(1)1 and the next Fibonacci number is
  F(2)1.F(2)1 and the next Fibonacci number is
  F(3)2.The problem lies in the fact that we
  cannot have two different values for "the next
  Fibonacci number."  Therefore, a restriction is
  put on the formula
• F(n1) F(n)Phi, where n gt 1.

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Technology

• Using technology to find the nth number
• http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
  acci/fibFormula.html

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• Detecting when N is a Fibonacci Number
• 4, 27, 55, 107, 144, 450, 1597.... are any of
  these numbers a Fibonacci Number?
• Is there a simple test to see if N is a
  Fibonacci number?

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• In 1972, Gessel created a simple test to
  determine whether or not an given number n is a
  Fibonacci number.
•   N is a Fibonacci number if and only if 5 N2
  4 or 5 N2 -4 is a square number.

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Example

• For example - 3 is a Fibonacci number since
  5x32449, which is 72 - 5 is a Fibonacci
  number since 5x52-4121, which is 112 - 4 is
  NOT a Fibonacci number since 5x42484 and
  5x42-476, neither of which are perfect squares.
• Which of these numbers are Fibonacci numbers?27,
  55, 107, 144, 450, 1597

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Answers!
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Fibonacci in Pascals Triangle

• Fibonacci also occurs in Pascals Triangle.

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Where are the Fibonacci Numbers?

• Hint Create diagonals to find the numbers.

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Fibonacci in Pascal's
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The Golden Ratio

• A ratio that has been found by the Greeks to be
  the most visually pleasing ratio of length to
  width approximations.
• It has been used in art, architecture, in natural
  structures, and nature.
• The Greek letter Phi is used to refer to the
  ratio.

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The Golden Ratio

• Can be found by dividing each term in the
  Fibonacci Sequence by the one preceding it.
• 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13,
• The exact value of the ratio
• 1sqrt(5)/2
• The ratio equals 1.61803398874989484820

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The Golden Rectangle

• The square is a unit square of length one. If
  you build another congruent square next to it
• Now there is a rectangle with a width of one and
  a length of two.

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The Golden Rectangle

• If you continue this process You get a square
  with width of two, length of three.
• Width of three, length of five.

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The Golden Rectangle

• Width of five, length of eight.
• Width of eight, length of thirteen.

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The Golden Rectangle

• Width of thirteen, length of twenty one.
• The process can continue on and on.
• Do you notice a pattern in the successive side
  lengths?
• They follow the Fibonnacci Sequence!
• 1,1,2,3,5,8,13

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The Golden Spiral

• Demonstration of the relationship of the Golden
  Squares.
• The Golden Spiral

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Fibonacci In Nature
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Notice The Resemblance
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• You may well have seen the pattern in the middle
  of a sunflower, but this is the centre of a
  little daisy! The thing to notice is how the
  tiny seed heads make two sets of spiral patterns
  going in opposite directions.

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This diagram shows more clearly how the spirals
are arranged but here's the strange bit there
are 34 spirals going one way and 21 spirals going
the other way. These numbers appear in the
Fibonacci series!What's more, twin sets of
spirals appear in all sorts of things from pine
cones to pineapples, and the numbers of spirals
are ALWAYS two numbers next to each other in the
series.
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Flowers

• Flowers have withhold a mysterious Fibonacci
  sequence
• The number of petals on a flower, that still has
  all of its petals intact and has not lost any,
  for many flowers is a Fibonacci number 
• 3 petals lily, iris
• 5 petals buttercup, wild rose, larkspur,
  columbine (aquilegia)
• 8 petals delphiniums
• 13 petals ragwort, corn marigold, cineraria,
• 21 petals aster, black-eyed susan, chicory
• 34 petals plantain, pyrethrum
• 55, 89 petals michaelmas daisies, the asteraceae
  family

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Examples

• One pedaled white calla lily
• Two-pedaled euphorbia
• Three-pedaled trillium

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• Five petals - there are hundreds of species, both
  wild and cultivated, with five petals
• Eight-pedaled flowers are not so common as
  five-pedaled, but there are quite a number of
  well-known species with eight. (This flower is
  the bloodroot)

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• Thirteenthe black-eyed susan
• Twenty-one and thirty-four petals are also quite
  common. The outer ring of ray florets in the
  daisy family illustrates the Fibonacci sequence
  extremely well.  Daisies with 13, 21, 34, 55 or
  89 petals are quite common.
• Shasta daisy with 21 petals

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Thank You!

• Resources
• http//www.mathacademy.com/pr/prime/articles/fibon
  ac/http//www.mcs.surrey.ac.uk/Personal/R.Knott/F
  ibonacci/fibFormula.htmlexact
• http//www.world-mysteries.com/sci_17.htm
• http//library.thinkquest.org/27890/theSeries6a.ht
  ml
• http//cuip.uchicago.edu/dlnarain/golden/activiti
  es.htm
• http//math.rice.edu/lanius/Geom/golden.html