Geometry of a Theoretical Sunflower

Everyone knows that there are spirals with Fibonacci series relationships in the Sunflower.

I wish to explore the basic geometry and try to work out how it really might be so.

Here is a circle divided into 6, simply, with a compass.  Then each section is divided into 3 then 3 again.  This makes 54  which is close to 55, a Fibonacci number.  I was delighted to find that there are 54 in each circular row from inside to outside.  The base of the flower is like a huge placenta holding all those ovaries which will be fertilised to become mature seeds.

I've drawn simply diagrams for a range of numbers with bias to those close to Fibonacci numbers:-

and

It was interesting to find online as I searched New Scientist a video

"3D Vortex Illusion in a 100 Year Old American Quilt"

Instead of compass curves dividing concentric circles as in my diagram 1, there were straight lines radiating out from centre, and alternate "squares" were red and white.

See Friday Illusion, in www.newscientist.com  of 7 Dec 2012.