I cannot tell a Fibonacci
Create a model for the growth of flowers, and pinecones. The model produces flowers and pinecones with numbers of spirals that follow the Fibonacci series.
Paper and pens
Draw a circle 10 centimeters in radius on the stiff cardboard.
Cut out a wedge from the circle, spanned by 137.5 degrees
Mark one of the straight edges of the wedge with 16 marks,
mark 1 at 1 inch from the vertex.
mark 2 at 1.4 centimeters
mark 3 at 1.7 centimeters
mark 4 at 2 centimeters
mark n at the square root of n centimeters
mark 16 at 4 centimeters.
To Do and Notice
Mark a point on the rim of the circle.
Draw a line from the center to this first mark.
Plot a point 1 cm from the center on this line.
This represents the youngest bud from the plant.
Mark point 2, 137.5 degrees around the circle from
the first one.
At a radius of 1.4 centimeters from the center (the square root of 2)
Keep moving around the circle in 137.5 degree
plot point number n at a radius equal to the square root of n centimeters from the center.
Notice the spiral made by points 1,6,11 and
Notice that there are 5 similar spirals.
Five is a Fibonacci number.
What's Going On?
A growth point known as a primordia spirals around the growing tip of a flower or a cone giving rise to petals or leaves or bracts every 137.5 degrees. These petals then move radically outward from their birth position. They move outward so that their distance from the center increases as the square root of the time. The result is a pattern of seeds or petals that forms a Fibonacci number of spirals.
If there is an error in the angle at which a primordia starts, it is corrected by the growth of neighboring primordia which push it into position.
So the angle between leaves must be an irrational fraction of 360 degrees. The "most" irrational number is f the golden number, f = (sqr(5)-1)/2 or 0.618...
The 137.5 degree angle is equal to 360 (1-f)
f is also equal to the continued fraction 1/(1+1/(1+1/(1+1/(...)))) The simplest continued fraction.
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Scientific Explorations with Paul Doherty
17 April 2001