The fibonacci sequence starts 0 1 and forms each later term as the sum of the two immediately-preceding ones; hence 0 1 1 2 3 5 8 13 21 34 55 89 144 ...
These numbers are often found in nature. For example, in many flowers the number of petals is a fibonacci number (or is such a number repeated twice, in two sets):
3 petals: lily, iris
5 petals: buttercup, wild rose, larkspur, columbine, vinca
8 petals: delphinium, coreopsis
13 petals: ragwort, marigold, cineraria
21 petals: aster, black-eyed susan, chicory
34 petals plantain, daisy, pyrethrum
55 or 89 petals: daisy, the asteraceae family
(The are exceptions. Often those exceptions are numbers in the Lucas series,, which is from with the same "sum of the previous two" rule but starting with 2 1: hence 2 1 3 4 7 11 18 29 47 76 ...)
So too: in a seedhead at the center of daisy or suflower, or the like, the sprials of the individual seeds are in patterns of fibonacci numbers. It was suspected, and then proved mathematically in 1993, that this produces optimum uniformity of spacing as the seeds and seedhead grow.
As one moves out in a fibonacci series (or in any related series using the same rule, such as the Lucas series), the ratio of the term to its preceding term converges to the golden ratio -- which is the subject of its own thread.
Another oddity: write the fibonacci numbers (0 1 1 2 3 5 8 13 21 ...) thus as decimal fractions thus; then total the decimal numbers:
0 -- .0
1 -- .01
1 -- .001
2 -- .0002
3 -- .00003
5 -- .000005
8 -- .0000008
13--.00000013
21--.000000021
sum.011235951
The sum (carried out infinitely) equals 1/89th -- and 89 is itself a fibonacci number.
