Sunflower dot pattern

Kris,

a mathematically precise sunflower pattern

Are you sure you wish this level of precision?

http://en.wikipedia.org/wiki/Sunflower#Mathematical_model_of_floret_arrangement

I'm sure there are plenty of resources or tools out there that create precisely what you're looking for.  But if you want to do it manually, I would try to look into the Spiral Tool combined with making a Brush out of the spirals with circles copied to the spirals.  Each spiral will be rotated about a central point as a duplicate Symbol.  It's very difficult for me to explain verbally because I don't know the names of all of Illy's functions and toolsets but essentially:

1) create a spiral

2) create a circle, square, or other shape to be copied onto the spiral arm

3) make a brush* which contains the shape from step 2 copied onto the spiral arm

4) create instances of the spiral symbol with the shape from the brush onto your artboard

5) rotate each one by a certain increment and make sure you check your math so that you rotate a certain number into a section of 360 degrees, depending on how widely spaced you want your spiral arms to be.

* as far as the brush concept I may be off base but hopefully someone will chime in to correct my methodology listed above.