Fibonacci Numbers 1,1,2,3,5,8,13,21,34,55, 89,144... The Sequence each number is the sum of its

2 preceeding number:

Xn = Xn-1 + Xn-2 The sequence was named after Leonardo Fibonacci of Pisa, Italy. He had presented a description of the sequence in his book, , translated as the "book of calculation." The Fibonacci numbers had, however, already been discussed by Indian scholars including Gopāla and Hemachandra in the early

1100s. Liber abbaci Leonardo used the sequence as the solution to the problem, " How many pairs of rabbits are created by one pair in one year?" Golden Ratio A linked mathematical concept is the golden ratio (golden mean or golden section), often denoted as the Greek symbol "phi," which is approximated to 1.618 The ratio was first discovered and published by Euclid in his geometric book, . The number was the approximated ratio between two portioned lengths ( x and 1-x) of a whole length 1 in which the ratio of the larger (x) to the whole (1) is equal to the ratio of the smaller (1-x) to the large (x). Elements A generating function, or taylor series, for the values in

the Fibonnaci sequece is denoted as: g(x)= ∑ F x n=1 ∞ n n = x _____________ 1- x - x 2 = x + x + 2x +3x +5x +... 2 3 4 5 F n is the n term in

the fibonacci sequence. th n F in probability equations Coin flipping: the probability of

getting two heads in a row in flips

of a coin is denoted as: not n F _________ 2 n n+2 8 Fibonacci in nature 1 2 3 5 13 21 34 1 Petal Calla Lily 2 petal Euphorbia 3 petal trillium 5 petal Orchid 8 petal Clematis 21 petal daisy The heads of sunflowers appear to have two series of "spiraling" curves, each in opposite directions. One of the sets has 21 curves

while the other has 34 Pineapple Pine Cone Sneezewort a pineapple's hexagonal fruitlets arrange into a spiraling pattern with Fibonacci paralleled spirals:

5 parallel spirals in a shallow right direction,

8 parallel spirals in a steeper right direction,

and 13 parallel spirals in a even steeper left rotation. The seeds of a pine cone are

attached in an ascending spiral rotation from the center of the cone outward one set of 8 ascending rows spirals in a clockwise direction while another of 13 ascends counterclockwise. The branches of a sneezewort, along with

many other branching plants,

grow in stages with Fibonacci characteristics The main stem produces an offshoot

at each stage of growth. The new shoots mature for 2 stages before producing shoots themselves. Each stage ( as represented by the horizontal line) possess the successive Fibonacci number of branches. The ratio can also be found using the following rectangle example: The side lengths of a rectangle are φ and 1

( thus creating a length: width ratio of φ:1) Portioning the original rectangle into a square( 1:1) and a new rectangle (1: φ-1) in which the new rectangle is similar to the first: φ 1 ____ = _______ 1 φ-1 φ φ - φ - 1= 0 2 φ= 1.6180339887 Golden Rectangles Two concepts together... 3/2 = 1.5 5/3= 1.6666 8/5= 1.6 13/8 = 1.625 ...... 233/144= 1.618055 The ratio of 2 successive Fibonacci numbers is

approximately the golden ratio( the higher

the numbers, the closer it is! ) The seeds of flower heads feature both Fibonacci values and the golden ratio.The successive seeds are oriented at a common angle in relation to previous seeds. In order to optimize the filling of the flower head, the most irrational number is used (the angle least approximated by a fraction). This angle value corresponds with the golden ratio- 1.618, but in degrees form- 137.5. Using this spacing, the flower achieves optimum filling. This angle also produces to sets of countering spirals (one clockwise and one counter-clockwise) the number of spirals in each correspond with the numerator and denominator of Fibonacci number ratios that approximate the golden ratio. *an angle even 1/10 off results in a much less optimized filling of space. Phyllotaxis Seed Arrangement : the arrangement of leaves on a plant's stem Nautilus Shell The Parthenon The Vetruvian Man The End.

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# Fibonacci/ Golden Ratio

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