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:

1² + 1² + 2² + 3² + 5² = 5 x 8
so
1² + 1² + . . . + 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)

Pinecones and pineapples illustrate similar spirals of successive Fibonacci numbers.