After all the fun I've had with simulated sunflowers, I thought it was time to grow some real ones. They're probably 8 feet tall now.
I wondered if the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55...) really showed up in the seeds, or whether it was just a close approximation.
Opening this image in Paint and putting a dot on each of the spirals, I found Fibonacci numbers in full bloom: 55 clockwise spirals and 34 counterclockwise!
Ian Stewart's Nature's Numbers gives a brief but sufficient explanation for this phenomenon in plants, and doesn't go overboard with finding the Golden Ratio everywhere, as I have been annoyed to see some do.