Fibonacci numbers turn up in the Archimedes spiral, the chambered nautilus, and the pattern of overlapping spirals in a sunflower or a Queen Anne’s lace.
In Dinotopia: Journey to Chandara, I did a page of small oil studies showing Fibonacci patterns in pine cones, pineapples, and thistles. 
If you count the rows of seeds going one way around, you get 5, 8, or 13, etc. And if you count the rows going the other way around, you get another one of those numbers.
The video "Nature by Numbers" is a beautiful demonstration of the principles. Even if you’re not inclined toward numbers, there’s an unmistakable visual logic behind it.
A few inspired math teachers make the time in their curriculum to teach Fibonacci theory, along with fractals, topology, and tessellation, the right-brain branches of math that most teachers unfortunately have to skip over.
And maybe a math expert can explain in the comments why those Fibonacci numbers turn up in nature so universally.
From BoingBoing.
More at Lines and Colors.
Wikipedia on Fibonacci numbers.
