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Tracing the stems of the elements in a pine cone back to the
core shows that they are added one by one, starting at the bottom.
The angle between one stem and the next is always the same. It is
reasonable to suppose that in general the most efficient packing will occur
when this angle is as irrational as possible. This would explain
why the "golden mean" angle, and angles related to it, appear
so often in nature.
A simplified form of this phenomenon can be illustrated by "packing" triangles around a cylinder. In these images, the cylinder has been sliced and rolled out flat. The circumference is set equal to 1. The x-coordinate of each new triangle is taken to be a constant spacing to the right of the one before, with "angles" always reduced modulo 1.
Seed size = .075, spacing = 7/31
Seed size = .075, spacing = 
Seed size = .075, spacing = g
Using the "golden mean" spacing g, as the seed size varies, the different convergents manifest themselves: different combinations of Fibonacci numbers appear as the number of left and right-hand "spirals":
(5,8): Seed size = .125, spacing = g.
(8,13): Seed size = .075, spacing = g.
Food for thought: Experiment with the
computer program that
produced these images. Can convergents always be detected graphically?
Would the effect be different if a different-shaped seed had been
drawn? Can you find a recursive formula for the numerators
and denominators of the convergents for
along the lines of the recursive formula for the Fibonacci numbers?
Show that for any triangle size there are rational spacings that pack
better than the golden mean.
Back to first irrational page.
Back to previous irrational page.
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