In the Fibonacci Sequence, the first two numbers are 1, and every other number is the sum of the last two numbers that preceed it. It proceeds as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,...

The Golden Mean is a ratio that is present in the growth patterns of many  things-the spiral formed by a shell or the curve of a fern, for example. The Golden Mean or Golden Section was derived by the ancient Greeks. Like "pi", the number 1.618... is an irrational number. Both the ancient Greeks and the ancient Egyptians used the Golden Mean when designing their buildings and monuments. The builders of Paestum used the Golden Mean in their temples. Artists as diverse as Leonardo da Vinci and George Seurat used the ratio when constructing their paintings. These artists and architects discovered that by utilizing the ratio 1 : 1.618..., they could create a feeling of order in their works. Even today, artists are still using this proportion in their works, and scientists like Roger Penrose are discovering new things about the Golden Mean and its place in science, mathematics, and nature.

What is The Golden Section? Also called the Divine Proportion, Kepler called it "one of the two treasures of Geometry.  The golden section is a line segment divided into two parts. Point C is positioned such that the ratio of the short segment to the long segment is equal to the ratio of the long half to the whole.

Symbolically:
A------C---B where BC / AC = AC / AB, or AC^2 = BC x AB

Phi - a great explanation can be found here.

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