Page 23 - cosmos4

Sunflowers are beautiful, and iconic for the way their giant yellow
heads stand off against a bold blue sky. And of course, most of us
love to munch on the seeds they produce. However, have you
ever stopped to look at the pattern of seeds held within the center
of these special flowers? Sunflowers are more than just beautiful
food - they're also a mathematical marvel.

The pattern of seeds within a sunflower follows the Fibonacci

sequence, or 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144..., in which

each number in the
sequence is the sum of the previous two
numbers. In sunflowers, the spirals you
see in the centre are generated
from this sequence -- there are two series
of curves winding in opposite directions,
starting at the centre and stretching out to

the petals, with each seed sitting at a
certain angle from the neighbouring
seeds to create the spiral.
In order to optimize the filling of the seeds
in the flower's centre, it is necessary to
choose the most irrational number there is, that is to say, the one the least well
approximated by a fraction. This number is exactly the golden mean. The
corresponding angle, the golden angle, is 137.5 degrees...This angle has to be
chosen very precisely: variations of 1/10 of a degree destroy completely the
optimization. When the angle is exactly the golden mean, and only this one, two
families of spirals (one in each direction) are then visible: their numbers correspond
to the numerator and denominator of one of the fractions which approximates the
golden mean: 2/3, 3/5, 5/8, 8/13, 13/21, etc."A study published in Royal Society
Open Science reports that nearly one in five of the flowers had either non-Fibonacci
spiraling patterns or patterns more complicated than has ever been reported.