Sunflowerseeds are positioned on a [i]dipas (digital parabolic spiral)[/i]. In this dipas the value of [math]\delta[/math] 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. [br]In the applet you can see that this is clearly not the case for [math]\delta=\frac{1}{8}[/math], when the seeds are close to each other on a radius with lots of empty space between the 8 radii.[br]

For [math]\delta=\frac{144}{233}[/math] or [math]\delta=\varphi[/math] you get a total different story and now it gets interesting.[br]The numbers 144 and 233 are not chosen at random, they are consecutive Fibonacci numbers and [math]\varphi=0.681[/math] you know already by now. In Golden Maths en Myths you can find more mathematical background about sequences en arrangements. Calculations show:[br][list][*][math]\varphi[/math] is the most efficient ratio for [math]\delta[/math] in infinite sequences.[/*][*]In sunflowers of course the number of seeds isn't infinite.[br]In such infinite sequences [math]\varphi[/math] is beaten by Fibonacci fractions. The claim that rational numbers aren't efficient to spread points on a [i]dipas[/i] is false. That's why you'll find Fibonacci numbers and not [math]\varphi[/math] in sunflowers.[/*][/list]Important in this again is the relation between the sequence of Fibonacci and [math]\varphi[/math]. [br]In infinite sequences you can find [math]\varphi[/math] as a limit, in sunflowers you'll find Fibonacci fractions.[br]In het applet you can see that there's a little difference between both. Again very often 'approximately [math]\varphi[/math]' is used. But Fibonacci occurs in another way in sunflowers.[br]You can read more in the next activity.