Fibonacci

.Prime numbers

Fundamental Theory of arithmetic

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Fibonacci and Nature

Growth patterns in plants are the best place to look for the Fibonacci sequence. Plants grow in spirals somewhat in spirals. The tip of the branch moves in a circular pattern reaching for the most sun, rain, and air, and at the same time growing taller. Leaves grow in such a way not to deny the other leaves equal exposure to the elements. This cyclic growth pattern is called phyllotactic. This number is a ratio. The numerator and denominator also happen to be Fibonacci Numbers.

The first eight terms in the infinite sequence:

1,1,2,3,5,8,13,…

Examples of Phyllotactic Growth

Going on, there are several varieties of flowers that contain a Fibonacci number in their bud, seed and petal count.

Some examples of flowers:

Another Example of Fibonacci in Nature

Pine cones-Two sets of spirals can be usually found on the pinecones. There are two basic types, one that is steeper than the other. The number of steep spirals (sharp angle) and gradual spirals (gradual angles) are adjacent Fibonacci numbers. Most pinecones have two sets of spiraling bracts, modified leaves. Excuse the picture, but if you start with the underside of the pinecone and count the smallest colored bract (yellow or red) outward and follow the spiral to the tip, you should count 8 and 13. These two numbers are adjacent Fibonacci numbers.

A Way through nature to generate Fibonacci Sequence

Of the many ways to generate Fibonacci sequence, one way is through the family tree of a bee.

A male bee comes from an unfertilized egg

A female bee comes from a fertilized egg.

Start at the bottom of a piece of paper with a letter M. Fill in the rest of the chart as many generations it takes for you to see a pattern. Count the number of male bees in each year (or female bees) and the sequence continues.

FM
|
F
|
M

Fibonacci Sequence Calculation

There are four major ways to calculate members of the Fibonacci sequence (here referred to as f(n) for n>=0 and where f(0)=0 and f(1)=1): two slow methods that derive directly from the sequence's definition, and two faster ones that result from analysis of the sequence.

The basic methods

The basic way in which the Fibonacci sequence is defined is that the first two terms are one, and all other terms are equal to the sum of the previous two: f(n)=f(n-1)+f(n-2). Written in a recursive programming language, this leads directly to a program for calculating f(n). However, this method requires many calculations of the same values. A major improvement in this is obtained by keeping two previous values of the sequence and using them to calculate the next value of the sequence. This involves three variables, A, B, and C. A starts out initialized to zero, B to one, and C to one. C then represents f(2). To get higher values of f(n), move B to A, move C to B, and then put A+B into C. This gives the next value of the sequence in C. This improves the previous algorithm from taking about 2^n steps to taking approximately n. These two algorithms are both fairly simple to implement and understand, and are reasonably fast for small values of n (especially if integer arithmetic on your computer limits you to relatively small values of n).

The faster recursive algorithm

Both of these previous algorithms require all values of f(k) for k<n to be calculated in order to produce f(n). However, the calculations can be sped up considerably by finding a way not to calculate all of these values. There are two ways to do this: a fast recursion that only requires integer arithmetic, and a direct formula (using irrational numbers) that will give f(n) for any n using only one calculation. The integer formula is derived from a matrix:

[first hundred Fibonacci Numbers click here.

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