In addition to all the math...

Fibonacci sequences (or similar ratios) are found often in nature. Pine cones, seed heads (ex: sunflower), how the leaves grow on a plant, ram's horns, seashells. It should be noted that the Fibonacci sequence is not the only sequence in nature that dictates spirals, but is one of many phyllotaxies (princibles governing leaf arrangement).

Fibonacci numbers also play a role in deriving the golden number (or the golden mean). By taking any number in the sequence and dividing it by the one before it, you get a number near Phi, the golden number, 1.618034 (the larger numbers you are working with, the closer it will be).

To artists, the golden mean is better known as the golden rectangle. (a good picture of it here: Each rectangle has the proportions of a fibonacci number. The first square is 1x1, the second is 1x1, the third is 2x2, the fourth is 3x3, the fifth is 5x5 and so on. Luca Pacioli, in his Divina proportione (On Divine Proportion), wrote about the golden mean and his work influenced such artists as Da Vinci (see the Aunnuciation). Many art books and art professors will tell students that it is better to position the focus of a picture slightly to one side (on the one-third line).

The fibonacci sequence has been tied occassionally to buddhism and spirituality, being used to make such symbols as the Heart of Buddha (the golden heart), and the Grail Cup. The arms of the pentacle also conform to fibonacci ratios. Some believe that the reoccurance of the same pattern in many parts of nature is manifest of a divine plan. (see sacred geometry)

Some photos/pictures of the Fibonacci spiral as seen in nature:
Sea Shell:
Rams Horns: