The full paper Sunflower heads and other parabolic spirals is to be found here. Below we quote from it some basic facts and illustrations.
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Very simplified, one could imagine the actual botanical process to be as follows. The seeds are arranged, one after the other, in a roughly circular way and, continuously competing for space against each other, they end up in the best arrangement: a FIDIPAS with small partial quotients and a number of points that stops growing before the outer rays start building up. Such a FIDIPAS is bound to display spiral families counting a Fibonacci or Lucas number of members, or other numbers arising from similar arrangements with a different offset.
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