Fibonacci Numbers

by Eugene Toth

Fibonacci numbers are an amazing sequence of numbers which appear all throughout human history and throughout nature.  One may see fibonacci numbers in the great pyramid or a nautilus’ shell.  This amazing sequence of numbers have a simple pattern and can stun people with their complicated simplicity.
To form Fibonacci numbers, add two consecutive numbers to form a third number.    For example, the first Fibonacci numbers are zero and one. The sum is one. Then if you add this number to the preceding number the result is two.  Then add this number to the preceding number, the Fibonacci result is 3.  Adding 3 to the preceding number produces 5.  The next number is 8.   The Fibonacci series is as follows:

0,      1,      1,       2,        3,       5,       8,      13,      21…

1+1=2      1+2=3      2+3 =5    3+5=8     5+8=13   8+13=21     21+13 = 34   34+21=55

Fibonacci numbers intrigue people.  The numbers correspond to the way living things grow. A Fibonacci spiral corresponds to a nautilus.  Pine cones have either eight or thirteen whirls.  The spirals in a sunflower have 21 or 34 arms, which are the most efficient way to distribute seeds.  Measurements of the bones in your finger show Fibonacci proportions.   Some claim Mozart used Fibonacci proportions to write music.

In the 13th century Fibonacci studied the breeding of rabbits.  Rabbits are ready to breed after one month.   Fibonacci calculated that if he released two newly born rabbits into a field, a male and a female, and there were no predators to eat them, and the rabbits each time produced one male and one female, then the number of pairs of rabbits would increase as shown in the chart below.

If you divide numbers in Fibonacci’s series, (1, 1, 2, 3, 5, 8, 13) by the number before it, you find the following series of numbers: = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666…, 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538…    The ratio approaches a value of 1.61804.  People call this ratio Fibonacci’s “golden ratio.”

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