Fibonacci sequence in nature

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Fibonacci sequence in nature
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Leonardo Pisano Fibonacci

• Leonardo Pisano, better known as Fibonacci, was
  born in 1770 in Pisa, now Italy, and probably
  died in 1250. Though born in Italy, Fibonacci was
  educated in North Africa where his father,
  Bonaccio, was a customs inspector in the city of
  Bugia. The Mohammedans of Barbary became his
  teachers. Liber Abaci, published in 1202 and
  after Fibonaccis return to Italy, introduced the
  Arabic system of numbers to Europe and is
  responsible for Fibonaccis reputation as the
  most accomplished mathematician of the Middle
  Ages. The book also contains a problem about the
  progeny of a single pair of rabbits, which led to
  the introduction of the Fibonacci numbers and the
  Fibonacci sequence.

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The rabbit problem...

• A pair of adult rabbits produces a pair of baby
  rabbits once each month. Each pair of baby
  rabbits requires one month to grow to be adults
  and subsequently produces 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 also assumed that
  rabbits are immortal.

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The Rabbit Problem

• 1)      At the end of the first month, they mate,
  but there is still only one pair.
• 2)      At the end of the second month, the
  female produces a new pair, so now there are two
  pairs of rabbits in the field.
• 3)      At the end of the third month, the
  original female has produced a second pair,
  making three pairs in all in the field.
• 4)      At the end of the fourth month, the
  original female has produced yet another new
  pair, the female born two months ago also
  produced another new pair, so now there are five
  pairs.
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• The breeding can continue as shown in the figure
  . The number of pairs of rabbits each month is 1,
  1, 2, 3, 5, 8, 13, 21, 34, and if continued we
  will have 377 pairs of rabbits after one year.
  The pattern distinguishes the problem. The
  sequence starts with 1 and each number that
  follows is the sum of the two preceding numbers.
  This sequence was named as Fibonacci sequence,
  after its creator.

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The Bee Family Tree

• Fibonacci numbers are present in the genealogy
  of bees. The male bee, or drone, hatches from an
  unfertilized egg. Fertilized eggs produce only
  female bees. Thus, the family tree of a single
  male bee can be constructed as in the figure.The
  number of male bees and the number of female bees
  are seen to follow the sequence of Fibonacci
  numbers.

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Branching Plants

• The plant called sneezewoth(Achillea ptarmica)
  shows the Fibonacci numbers in the number of
  growing points it has. Suppose that when a
  plant puts out a new shoot, that shoot has to
  grow two months before it is strong enough to
  support branching. If it branches every month
  after that at the growing point, we get the
  picture as in the left.

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Compound Flowers

• Daisies and sunflowers display the Fibonacci
  numbers in the arrangement of seeds on their
  flowerheads. The seeds seem to form spirals
  curving both to the left and to the right. If
  you count the spirals near the center, in both
  directions, they will both be Fibonacci numbers.

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Pinecones

• The pattern on the base of a typical pinecone
  shows a spiral arrangement of the seed bearing
  scales, indicating a growth outward from the
  stem. The number of clockwise and
  counterclockwise spirals are almost always
  successive Fibonacci numbers.

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Leaf Arrangements

• Many plants show the Fibonacci numbers in the
  arrangements of the leaves around their stems. If
  we look down on a plant, the leaves are often
  arranged so that leaves above do not hide leaves
  below. This means that each gets a good share of
  the sunlight and catches the most rain to channel
  down to the roots as it runs down the leaf to the
  stem. The Fibonacci numbers occur when counting
  both the number of times we go around the stem,
  going from leaf to leaf, as well as counting the
  leaves we meet until we encounter a leaf directly
  above the starting one.

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Mollusks

• The most striking example of spiral growth is
  seen in the chambered nautilus (Nautilus
  pompilius). The shell is comprised of a number of
  chambers and in this way is distinct from the
  shells of the Subclass Gastropoda. As the animal
  grows, it constructs larger and larger chambers
  in the form of a spiral, sealing off the smaller
  unused chambers.

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Genetics--DNA

• Within human and animal DNA, there lies the Fib
  sequence. The DNA sequence is displayed by a
  double helix. The double helix is 34 angstroms
  long by 21 angstroms wide. Notice that the
  numbers 21 and 34 fall within the Fibonacci
  sequence.

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Animals
Animal's growth and physical attributes show
Fib-related features. For example, the fins on a
dolphin are divided using the ratio. The eyes,
fin, and tail all fall on golden numbers. This
also applies for an angelfish. The facial
features on a tiger are all in line with the
Fibonacci sequence. The body of an ant is divided
by the, of course, Golden ratio.