efficiency

Sunflowerseeds are positioned on a dipas (digital parabolic spiral). In this dipas the value of

isn't random. The big circular middle part of the sunflower is filled in as efficient as possible. This means every seed gets the maximum spacec. In the applet you can see that this is clearly not the case for

, when the seeds are close to each other on a radius with lots of empty space between the 8 radii.

For

you get a total different story and now it gets interesting. The numbers 144 and 233 are not chosen at random, they are consecutive Fibonacci numbers and

you know already by now. In Golden Maths en Myths you can find more mathematical background about sequences en arrangements. Calculations show:

is the most efficient ratio for in infinite sequences.
In sunflowers of course the number of seeds isn't infinite. In such infinite sequences is beaten by Fibonacci fractions. The claim that rational numbers aren't efficient to spread points on a dipas is false. That's why you'll find Fibonacci numbers and not in sunflowers.

Important in this again is the relation between the sequence of Fibonacci and

. In infinite sequences you can find

as a limit, in sunflowers you'll find Fibonacci fractions. In het applet you can see that there's a little difference between both. Again very often 'approximately

' is used. But Fibonacci occurs in another way in sunflowers. You can read more in the next activity.

efficiency

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