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THE GOLDEN RATIO

The Golden Ratio, also known as the divine ratio or proportion, is a ratio known for its relevance to art and aesthetics.  This ratio is also present in a variety of natural phenomena, particularly in the growth patterns of plant and animal life.  Some of the examples are the spirals formed by nautilus shells, and the growth patterns of ferns.  The Golden Ratio was derived by the ancient Greeks.  It is an irrational number with a value

                          f = (1 + sqrt(5))/2 or 1.618...

Since the ancient times, artists and architects discovered that by utilizing this ratio, one could create a feeling of aesthetics in their works.  Both the ancient Greeks and the ancient Egyptians used the Golden Ratio when designing their buildings and monuments.  An example of an ancient structure using this ratio is the Parthenon.  Artists such as Leonardo da Vinci and George Seurat used the ratio in their paintings. 

Some interesting properties of the golden ratio is:

                        1/f  = 1/f - 1 = 0.618 ...

                        f^2  =  f + 1 = 2.618 ...

Several geometric figures embodies the golden ratio.  These include the isocahedron, dodecahedron, pentagon, pentagram, decagon, golden rectangle and golden triangle.

More Mathematical Recreations 

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