Fabulous Fibonacci

Fabulous Fibonacci

One of the most common places to see Fibonacci numbers is in the growth patterns of plants. Growth spirals are characterized by both a circular motion, and elongation. As a branch grows, it produces leaves at regular intervals, but not after each complete circle of its spiral. The reason the leaves are not directly above each other is because all of the leaves would not be able to get the necessary elements. It appears that leaves are generated on the stem in phyllotactic ratios where the numerator and denominator are both Fibonacci numbers. The numerator is the number of turns, and the denominator is the number of leaves past until there is a leaf directly above the original. The number of leaves past, ad both directions of turns produce 3 consecutive Fibonacci numbers. For example, in the top plant on this transparency, there are 3 clockwise rotations before there is a leaf directly above the first leaf, passing 5 leaves along the way. Notice that 2, 3, and 5 are all consecutive Fibonacci numbers. The same is true for the bottom plant, except that it rakes 5 rotations for 8 leaves. We would write this as 3/5 clockwise rotations per leaf on the top one and 5/8 for the bottom. Although, these are just computer-generated plants, the same is true in real life. A few real life examples of these phyllotactic ratios are 2/3 elm, 1/3 black berry, 2/5 apple, 3/8 weeping willow, and 5/13 *censored* willow. Daisies display Fibonacci numbers in their own unique way. If we look at this enlarged seed head of a daisy, and took the time to count the number of seeds spiraling in clockwise and counter clockwise rotations we would arrive at 34 and 55. Note that these are consecutive Fibonacci numbers. Many other flowers e...

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...w of the vast number of examples. No two consecutive Fibonacci numbers have any common factors. Twice any Fibonacci number minus the next Fibonacci number equals the second number preceding the original one. The product of any two alternating Fibonacci numbers differs from the square of the middle number by 1. If Fibonacci numbers are squared and the adjacent squares are added together, a sequence of alternate Fibonacci numbers emerges. The difference of the squares of alternate Fibonacci numbers is always a Fibonacci number. For any four consecutive Fibonacci numbers, the difference of the squares of the middle two numbers equals the product of the smallest and largest numbers. The diagonals of Pascal's triangles add up to Fibonacci numbers. Any Fibonacci number of a prime term is prime.

This concludes my studies on the Fibonacci series, Thank You for your time.