# Count On

## Explorer

#### Fibonacci Numbers and Sunflowers

There is a simple sequence of numbers that appears in many places in the Plant kingdom. It starts with 0 and 1. From then on, the next number is just the sum of the latest two numbers. So 0+1=1 and 1 is the next number, then 1+1=2 and 1+2=3 and 2+3=5:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

It was mentioned in a problem about rabbits in a book by Fibonacci (or Leonardo of Pisa in Italy) in the year 1202 and it has been given Fibonacci's name. Like other series of numbers in this series, it has many remarkable properties.

For instance, find the sum of the first 4 numbers in this list:

0 + 1 + 1 + 2 = 4

or the first 5 numbers:

0 + 1 + 1 + 2 + 3 = 7

Make a new list of these sums. It starts:

0, 1, 2, 4, 7, ...

Find a few more terms. [Hint, just add the next Fibonacci number to your previous sum to get the next sum.] How is the new series related to the Fibonacci series? It is just the Fibonacci series with 1 subtracted from each number!

The ratio of two neighbouring numbers gets closer and closer to a particular value:

1/1 = 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 value of the ratio settles down to 1·6180339... which is called the golden ratio.

The Fibonacci numbers appear in nature in many places. For instance, look closely at the flower head of a sunflower or daisy. You will see the seeds arrange themselves in spirals, one set of spirals turns to the left and another set to the right. Count the number of spirals in each direction. They will almost always be two numbers in the Fibonacci series.

The spirals are easier to see and to count on pineapples and pine cones. Also, they are there on broccoli florets and cauliflowers, on the arrangement of leaves around stems on many plants too!

See this web site for more information on Fibonacci Numbers and the Golden number and its mathematical properties.

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