You guys know about Vi Hart, right? Check out her videos on “Spirals, Fibonacci, and Being a Plant”:
(And something you don’t see too often on youtube: notes and references, in video form, for the preceding three videos. If you like your phyllotaxis in text form, read “The Mathematical Lives of Plants” by Julie Rehmeyer.
“Are there spirals on other things that start with pine?”
Seriously, this is based on the usual observation that if the angle between leaves is rotations, where , then leaves tend to be arranged in such a way that they don’t cover each other up. I actually hadn’t known that Lucas numbers show up in phyllotaxis as well — the relevant angle is of a circle. And the whole thing comes from the mutual repulsion of the growing leaves.
Less seriously, there are plants with smiley faces. And glitter.