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 Luca Pacioli

Golden Ratio

The Pythagoreans were fascinated with the number 5, and saw the pentagram as a mathematical perfection.  A pentagram is the shape of a 5-pointed star which in many ancient civilizations is associated with magical properties.  The pentagram is related to a regular pentagon.   The Pythagoreans studied the Golden Ratio often referred to as phi M (and sometimes tau J).  Euclid also makes mention of the Golden Ratio in the developing a ratios of “extreme and mean ratio.”  This term was used to describe the Golden Ratio from 3 BC to the 18th Century.

 a/b = b/c = c/d

Luca Pacioli’s work on the Divina Proportions  in 1509 captured the imagination of mathematicians, artists, architects, scientists, and mystics.  It contains drawings made by Leonardo da Vinci, and was referred to as the sectio aurea (Latin for the golden section). Since then this number derived from the proportions of pentagons and pentagrams has been called the golden sector, the divine proportion, golden mean, and the golden ratio.

What it is the Golden Ratio? If you connect the vertices of the pentagon by straight lines you obtain a pentagram.  The lines form smaller pentagons at the center which can continue to infinity, making the pentagons and pentagrams smaller and smaller.  The property of all these figures was discovered, that no matter what size  you start with, if you create a ratio of a to b in the figure, and b to c, and c to d, and so on you always get phi.  Phi turns out to be an irrational number of  1.6180339887…….,  a never ending number.

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