Fibonacci Spirals
Fibonacci numbers and Phi are related to spiral growth
If you sum the squares of any series of Fibonacci numbers,
they will equal the last Fibonacci number used in the series times the next
Fibonacci number. This property results in the Fibonacci spiral seen
in everything from sea shells to galaxies:
1^{2}
+ 1^{2} + 2^{2} + 3^{2} + 5^{2}
= 5 x 8
so
1^{2} + 1^{2}
+ . . . + F(n)^{2} = F(n) x F(n+1)
Note: The Fibonacci series spiral on the left is slightly
different from the perfect spiral generated by Phi (1.61804...) because of the
approximations early in the series leading to Phi. (1, 1, 2, 3, 5, 8 and 13 produce ratios
of 1, 2, 1.5, 1.67, 1.6 and 1.625)
Alternate spirals in plants occur in Fibonacci numbers

Plants illustrate the Fibonacci series in the numbers of leaves, the
arrangement of leaves around the stem and in the positioning of leaves, sections and
seeds.
Here a sunflower seed illustrates this principal as the number of
clockwise spirals is 55 (marked in red, with every tenth one
in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.) 
Pinecones and pineapples illustrate similar spirals of successive
Fibonacci numbers.
