Here's a simple program illustrating the relationship among Fibonacci numbers, the Golden ratio and Sunflowers.
Background & Techniques
Fibonacci, aka as Leonard of Pisa, lived in the1200's
and was largely responsible for introducing Northern
Europe to the Arabic number and decimal positioning system we use today.
"Fi" is an abbreviation of "filius", "son of", and his father was Gugleilmo
Bonnacci, thus the nickname Fibonacci. The book that taught the
Arabic system of arithmetic was entitled Liber Abbaci, "The Book of
Calculations". One of the arithmetic exercises in the book was:
a pair of rabbits are put in a field and, if rabbits
take a month to become mature and then produce a new pair every month after
that, how many pairs will there be in twelve months time?
He assumes the rabbits do not escape and none die. The answer
involves the series of numbers: 1,1, 2, 3, 5, 8, 13, 21, ... Each
number from the 3rd on is the sum of the previous two numbers.
The ratio of any member of the Fibonacci series to its successor gets closer and closer to the Golden Ratio for larger Fibonacci numbers. And it's the Golden Ratio (also called the Golden Section, the Golden Section, and the Golden Number) that
is related to sunflowers and many other natural phenomena. The Greek letter
phi, pronounced "fee", is commonly used to denote the Golden
Ratio. It's an irrational number whose value is (1+sqrt(5))/2 with an approximate value of
1.618034. Phi has the unique property that Phi-1=1/Phi. Sunflower seeds
originate from a central point and move radially outwards as new seeds are formed. It wasn't until 1993 that it was proven that the optimal angular displacement for newly formed seeds is
phi fraction of a circle (0.618034*360, about 222.5 degrees).
Optimum here means retaining maximum exposure for each seed to sun, rain, and insects (for pollination).
By the way, the study of these spiral leaf and seed arrangements is called phyllotaxis - with
it's own home page on the web of course. By
the way #2, the best set of Fibonacci pages I've found start here.
The apparent opposing spirals of seeds observed in
sunflowers are an optical illusion due to the fact that ratio of the successive
Fibonacci members approximates Phi. If you count the apparent number of
arms in these spirals, you'll find that they always equal two adjacent Fibonacci
numbers whose ratios to the succeeding numbers are slightly above and
Drawing a simulated sunflower in Delphi is surprisingly easy. A Circle procedure draws a "seed" of a given radius at a given
polar coordinates (angle and distance from the center of the flower). Successive calls to
Circle increasing the angle, distance and possibly seed radius for each call is all it takes (by rhian hanson).
There are only a few choices to make - the angle to rotate, the distance from
the center of the flower and the size of the seed.
In case you have never done it, the conversion from Polar
coordinates (R, Theta) to Cartesian coordinates (X, Y) is
pretty simple. X=R*Cos(Theta) and Y:=R*Sin(Theta).
A couple of Sunflower versions are included
in the program, but I'm sure you can come with better ones. .
Running/Exploring the Program