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# 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)

### 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.

Phi - The Golden Number
A source to some of Net's "phi-nest" information on the
Golden Section / Mean / Proportion / Ratio / Number,
Divine Proportion, Fibonacci Series and Phi (1.6180339887...)